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Volume 9
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Phys. Sci. Forum, 2023, MaxEnt 2023

The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering

Garching, Germany | 3–7 July 2023

Volume Editors:
Roland Preuss, Max-Planck-Institut for Plasmaphysics, Germany
Udo von Toussaint, Max-Planck-Institut for Plasmaphysics, Germany

Number of Papers: 26
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Cover Story (view full-size image): The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering continued a long series of MaxEnt-Workshops that started in the late 1970s of the previous [...] Read more.
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3 pages, 1339 KiB  
Editorial
Preface of the 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering—MaxEnt 2023
by Udo von Toussaint and Roland Preuss
Phys. Sci. Forum 2023, 9(1), 1; https://doi.org/10.3390/psf2023009001 - 23 Nov 2023
Viewed by 821
Abstract
The forty-second International Conference on Bayesian Inference and Maximum Entropy Methods in Science and Engineering (42nd MaxEnt’23) was held at the Max Planck Institute for Plasmaphysics (IPP) in Garching, Germany, from 3rd to 7th of July 2023 (https://www [...] Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Conference photo.</p> Full article ">

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8 pages, 465 KiB  
Proceeding Paper
An Iterative Bayesian Algorithm for 3D Image Reconstruction Using Multi-View Compton Data
by Nhan Le, Hichem Snoussi and Alain Iltis
Phys. Sci. Forum 2023, 9(1), 2; https://doi.org/10.3390/psf2023009002 - 24 Nov 2023
Viewed by 848
Abstract
Conventional maximum likelihood-based algorithms for 3D Compton image reconstruction are often stuck with slow convergence and large data volume, which could be unsuitable for some practical applications, such as nuclear engineering. Taking advantage of the Bayesian framework, we propose a fast-converging iterative maximum [...] Read more.
Conventional maximum likelihood-based algorithms for 3D Compton image reconstruction are often stuck with slow convergence and large data volume, which could be unsuitable for some practical applications, such as nuclear engineering. Taking advantage of the Bayesian framework, we propose a fast-converging iterative maximum a posteriori reconstruction algorithm under the assumption of the Poisson data model and Markov random field-based convex prior in this paper. The main originality resides in developing a new iterative maximization scheme with simultaneous updates following the line search strategy to bypass the spatial dependencies among neighboring voxels. Numerical experiments on real datasets conducted with hand-held Temporal Compton cameras developed by Damavan Imaging company and punctual 0.2 MBq 22Na sources with zero-mean Gaussian Markov random field confirm the outperformance of the proposed maximum a posteriori algorithm over various existing expectation–maximization type solutions. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Experiment setting for Compton data acquisition.</p> Full article ">Figure 2
<p>Evolution of point clouds returned by the considered algorithms (20 events/view): (<b>a</b>) eLMMLEM algorithm, (<b>b</b>) LMMaPEM algorithm, (<b>c</b>) LMMRFMaP algorithm.</p> Full article ">Figure 3
<p>Evolution of the mean and the standard deviation of SWD: (<b>a</b>) evolution with respect to the number of iterations, (<b>b</b>) evolution with respect to the number of events/view.</p> Full article ">
9 pages, 1744 KiB  
Proceeding Paper
Behavioral Influence of Social Self Perception in a Sociophysical Simulation
by Fabian Sigler, Viktoria Kainz, Torsten Enßlin, Céline Boehm and Sonja Utz
Phys. Sci. Forum 2023, 9(1), 3; https://doi.org/10.3390/psf2023009003 - 24 Nov 2023
Viewed by 854
Abstract
Humans make decisions about their actions based on a combination of their objectives and their knowledge about the state of the world surrounding them. In social interactions, one prevalent goal is the ambition to be perceived to be an honest, trustworthy person in [...] Read more.
Humans make decisions about their actions based on a combination of their objectives and their knowledge about the state of the world surrounding them. In social interactions, one prevalent goal is the ambition to be perceived to be an honest, trustworthy person in terms of having a reputation of frequently making true statements. Aiming for this goal requires the decision whether to communicate information truthfully or if deceptive lies might improve the reputation even more. The basis of this decision involves not only an individual’s belief about others, but also their understanding of others’ beliefs, described by the concept of Theory of Mind, and the mental processes from which these beliefs emerge. In the present work, we used the Reputation Game Simulation as an approach for modeling the evolution of reputation in agent-based social communication networks, in which agents treat information approximately according to Bayesian logic. We implemented a second-order Theory of Mind based message decision strategy that allows the agents to mentally simulate the impact of different communication options on the knowledge of their counterparts’ minds in order to identify the message that is expected to maximize their reputation. Analysis of the communication patterns obtained showed that deception was chosen more frequently than the truthful communication option. However, the efficacy of such deceptive behavior turned out to have a strong correlation with the accuracy of the agents’ Theory of Mind representation. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Red agent simulating black’s update of possible messages in a conversation about red, who imagines three conversations in which he tells the four options of <math display="inline"><semantics> <mrow> <mo>+</mo> <mo>,</mo> <mo>−</mo> <mo>,</mo> <mi mathvariant="normal">w</mi> </mrow> </semantics></math>, and t, as well as the blush option to <span class="html-italic">b</span> and simulates the resulting reputation in the corresponding dummies.</p> Full article ">Figure 2
<p>Red agent simulating black’s update of possible messages in a conversations about cyan, followed by simulations from cyan to black about red for each of the updated dummies. Each of the five dummies for the options of <span class="html-italic">a</span> is copied three times to simulate the effect of the subsequent message from cyan to black on red for each of the dummies from the first conversation.</p> Full article ">Figure 3
<p>Comparison of the average achieved reputation and standard deviation for the ToM-agent with the ordinary agent in 100 simulations over the time of 300 rounds. The unicolored lines represent the agents’ self-reputations, whereas the red line with black dots denotes the reputation of red in the eyes of black. The dashed lines stand for the honesty frequencies of the agents.</p> Full article ">Figure 4
<p>Correlation between the sum of the absolute deviations of first- and second-order Theory of Mind information from the real values and the success of the simulating ToM strategy in terms of the achieved mean reputation in the game.</p> Full article ">
10 pages, 7748 KiB  
Proceeding Paper
Uncertainty Quantification with Deep Ensemble Methods for Super-Resolution of Sentinel 2 Satellite Images
by David Iagaru and Nina Maria Gottschling
Phys. Sci. Forum 2023, 9(1), 4; https://doi.org/10.3390/psf2023009004 - 27 Nov 2023
Viewed by 1039
Abstract
The recently deployed Sentinel 2 satellite constellation produces images in 13 wavelength bands with a Ground Sampling Distance (GSD) of 10 m, 20 m, and 60 m. Super-resolution aims to generate all 13 bands with a spatial resolution of 10 m. This paper [...] Read more.
The recently deployed Sentinel 2 satellite constellation produces images in 13 wavelength bands with a Ground Sampling Distance (GSD) of 10 m, 20 m, and 60 m. Super-resolution aims to generate all 13 bands with a spatial resolution of 10 m. This paper investigates the performance of DSen2, a proposed convolutional neural network (CNN)-based method, for tackling super-resolution in terms of accuracy and uncertainty. As the optimization problem for obtaining the weights of a CNN is highly non-convex, there are multiple different local minima for the loss function. This results in several possible CNN models with different weights and thus implies epistemic uncertainty. In this work, methods to quantify epistemic uncertainty, termed weighted deep ensembles (WDESen2) and its variants), are proposed. These allow the quantification of predictive uncertainty estimates and, moreover, the improvement of the accuracy of the prediction by selective prediction. They involve a consideration of deep ensembles, and each model’s importance can be weighted depending on the model’s validation loss. We show that weighted deep ensembles improve the accuracy of prediction compared to state-of-the-art methods and deep ensembles. Moreover, the uncertainties can be linked to the underlying inverse problem and physical patterns on the ground. This allows us to improve the trustworthiness of CNN predictions and the predictive accuracy with selective prediction. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Architecture and pipeline of the KWDESen2 and VWDESen2 methods with <span class="html-italic">N</span> ensemble members.</p> Full article ">Figure 2
<p>The <math display="inline"><semantics> <mrow> <mi>D</mi> <mi>S</mi> <mo>:</mo> <msup> <mi mathvariant="double-struck">R</mi> <mrow> <mi>W</mi> <mo>×</mo> <mi>H</mi> </mrow> </msup> <mo>→</mo> <msup> <mi mathvariant="double-struck">R</mi> <mrow> <mi>W</mi> <mo>/</mo> <mn>2</mn> <mo>×</mo> <mi>H</mi> <mo>/</mo> <mn>2</mn> </mrow> </msup> </mrow> </semantics></math> operator. Here, <span class="html-italic">W</span> and <span class="html-italic">H</span> are the width and the height of a patch.</p> Full article ">Figure 3
<p>The composition of the <math display="inline"><semantics> <mrow> <mi>D</mi> <mi>S</mi> </mrow> </semantics></math> and <span class="html-italic">M</span> operators, <math display="inline"><semantics> <mrow> <mi>M</mi> <mi>D</mi> <mi>S</mi> <mo>:</mo> <msup> <mi mathvariant="double-struck">R</mi> <mrow> <mi>W</mi> <mo>×</mo> <mi>H</mi> </mrow> </msup> <mo>→</mo> <msup> <mi mathvariant="double-struck">R</mi> <mrow> <mi>W</mi> <mo>×</mo> <mi>H</mi> </mrow> </msup> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Kernel projection on the mean prediction (<b>left</b>) vs. variance (<b>right</b>) of WDESen2. The variance map visualizes the pixel-wise epistemic uncertainty of WDESen2 with 30 ensemble members. The patch was from the tropical data.</p> Full article ">Figure 5
<p>A and B ground-truth bands (<b>top</b>), variance-based withdrawal (<b>bottom left</b>) and variance map (<b>bottom right</b>) for VWDESen2.</p> Full article ">Figure 6
<p>The A and B bands of the patch used in <a href="#psf-09-00004-f004" class="html-fig">Figure 4</a>.</p> Full article ">Figure 7
<p>Band-wise RMSE of different deep ensemble methods (DESen2, KDESen2, VDESen2, WDESen2, KWDESen2, and VWDESen2). Each ensemble was composed of 30 members, and the baseline of DSen2 corresponds to 1 ensemble member. The RMSE was computed over 300 patches of tropical data.</p> Full article ">Figure 8
<p>Band B11’s RMSE for different numbers of ensemble members on tropical satellite data. For a given number of ensemble members <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>∈</mo> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mn>30</mn> <mo>}</mo> </mrow> </semantics></math>, a deep ensemble with <span class="html-italic">n</span> networks was randomly chosen from the 30 trained models, and this was averaged over 10 times in order to avoid the bias from the network sampling. The results were averaged over the patches.</p> Full article ">Figure 9
<p>Band B11’s RMSE for different numbers of ensemble members on mountain satellite data. For a given number of ensemble members <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>∈</mo> <mo>{</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mn>14</mn> <mo>}</mo> </mrow> </semantics></math>, a deep ensemble with <span class="html-italic">n</span> networks was randomly chosen from the 14 trained models, and this was averaged over 10 times in order to avoid the bias from the network sampling. The results were averaged over the patches.</p> Full article ">
10 pages, 1396 KiB  
Proceeding Paper
Geodesic Least Squares: Robust Regression Using Information Geometry
by Geert Verdoolaege
Phys. Sci. Forum 2023, 9(1), 5; https://doi.org/10.3390/psf2023009005 - 27 Nov 2023
Viewed by 857
Abstract
Geodesic least squares (GLS) is a regression technique that operates in spaces of probability distributions. Based on the minimization of the Rao geodesic distance between two probability models of the response variable, GLS is robust against outliers and model misspecification. The method is [...] Read more.
Geodesic least squares (GLS) is a regression technique that operates in spaces of probability distributions. Based on the minimization of the Rao geodesic distance between two probability models of the response variable, GLS is robust against outliers and model misspecification. The method is very simple, without any tuning parameters, owing to its solid foundations rooted in information geometry. Here, we illustrate the robustness properties of GLS using applications in the fields of magnetic confinement fusion and astrophysics. Additional interpretation is gained from visualizations using several models for the manifold of Gaussian probability distributions. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Illustration of the Poincaré half-plane with several half-circle geodesics in the background. The confinement time data (value and error bar) are plotted in the half-plane, as well as the modeled and predicted values obtained by GLS.</p> Full article ">Figure 2
<p>(<b>a</b>) Visualization of the confinement time data on a pseudosphere, as well as the modeled and observed distributions obtained by GLS. (<b>b</b>) Projection of the confinement data using MDS, roughly indicating the directions of increasing <math display="inline"><semantics> <mi>μ</mi> </semantics></math> and <math display="inline"><semantics> <mi>σ</mi> </semantics></math>.</p> Full article ">Figure 3
<p>Fit of the BTFR by GLS (<b>a</b>) on a logarithmic scale and (<b>b</b>) on the original scale of the data.</p> Full article ">Figure 4
<p>The BTFR data in the Poincaré half-plane, with geodesics considered by GLS shown for the case of the outlier.</p> Full article ">
11 pages, 2989 KiB  
Proceeding Paper
Magnetohydrodynamic Equilibrium Reconstruction with Consistent Uncertainties
by Robert Köberl, Robert Babin and Christopher G. Albert
Phys. Sci. Forum 2023, 9(1), 6; https://doi.org/10.3390/psf2023009006 - 27 Nov 2023
Viewed by 855
Abstract
We report on progress towards a probabilistic framework for consistent uncertainty quantification and propagation in the analysis and numerical modeling of physics in magnetically confined plasmas in the stellarator configuration. A frequent starting point in this process is the calculation of a magnetohydrodynamic [...] Read more.
We report on progress towards a probabilistic framework for consistent uncertainty quantification and propagation in the analysis and numerical modeling of physics in magnetically confined plasmas in the stellarator configuration. A frequent starting point in this process is the calculation of a magnetohydrodynamic equilibrium from plasma profiles. Profiles, and thus the equilibrium, are typically reconstructed from experimental data. What sets equilibrium reconstruction apart from usual inverse problems is that profiles are given as functions over a magnetic flux derived from the magnetic field, rather than spatial coordinates. This makes it a fixed-point problem that is traditionally left inconsistent or solved iteratively in a least-squares sense. The aim here is progressing towards a straightforward and transparent process to quantify and propagate uncertainties and their correlations for function-valued fields and profiles in this setting. We propose a framework that utilizes a low-dimensional prior distribution of equilibria, constructed with principal component analysis. A surrogate of the forward model is trained to enable faster sampling. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Samples drawn from the prior distribution of (<b>left</b>) the pressure profile shape and (<b>right</b>) the current profile shape.</p> Full article ">Figure 2
<p>Relative <math display="inline"><semantics> <msup> <mi>L</mi> <mn>2</mn> </msup> </semantics></math> error of the low-dimensional equilibrium representation, evaluated for the samples from the prior.</p> Full article ">Figure 3
<p>Input pressure profile <span class="html-italic">p</span> compared with the pressure derived from <math display="inline"><semantics> <mrow> <mo>∇</mo> <mi>p</mi> <mo>=</mo> <mi mathvariant="bold-italic">J</mi> <mo>×</mo> <mi mathvariant="bold-italic">B</mi> </mrow> </semantics></math> for a validation sample drawn from the prior and its low-dimensional representation.</p> Full article ">Figure 4
<p>Relative leave-one-out error <math display="inline"><semantics> <msubsup> <mi>ε</mi> <mrow> <mi>l</mi> <mi>o</mi> <mi>o</mi> </mrow> <mi>R</mi> </msubsup> </semantics></math> of the polynomial chaos expansion surrogate models, mapping from the low-dimensional equilibrium parameters to the synthetic diagnostics.</p> Full article ">Figure 5
<p>One-dimensional projections of the low-dimensional equilibrium parameter prior distribution, kernel density estimation (KDE) and Laplace’s approximation of the posterior distribution.</p> Full article ">Figure 6
<p>Posterior samples of the flux surface geometry in comparison to the ground truth. The solid lines are linearly spaced contours of constant <math display="inline"><semantics> <mi>ρ</mi> </semantics></math>, and the dashed lines are constant-<math display="inline"><semantics> <msup> <mi>θ</mi> <mo>★</mo> </msup> </semantics></math> contours.</p> Full article ">Figure 7
<p>Posterior kernel density estimation (KDE) of the total toroidal flux (<math display="inline"><semantics> <msub> <mi>ψ</mi> <mn>0</mn> </msub> </semantics></math>).</p> Full article ">Figure 8
<p>Rotational transform profiles <math display="inline"><semantics> <mrow> <mi>ι</mi> <mfenced open="(" close=")"> <mi>ρ</mi> </mfenced> </mrow> </semantics></math> drawn from the posterior distribution in comparison to the ground truth.</p> Full article ">Figure 9
<p>Pressure profiles calculated from the posterior using <math display="inline"><semantics> <mrow> <mo>∇</mo> <mi>p</mi> <mo>=</mo> <mi mathvariant="bold-italic">J</mi> <mo>×</mo> <mi mathvariant="bold-italic">B</mi> </mrow> </semantics></math> together with the ground-truth pressure profile.</p> Full article ">Scheme 1
<p>Overview of the proposed framework.</p> Full article ">
8 pages, 626 KiB  
Proceeding Paper
Improving Inferences about Exoplanet Habitability
by Risinie D. Perera and Kevin H. Knuth
Phys. Sci. Forum 2023, 9(1), 7; https://doi.org/10.3390/psf2023009007 - 27 Nov 2023
Cited by 1 | Viewed by 890
Abstract
Assessing the habitability of exoplanets (planets orbiting other stars) is of great importance in deciding which planets warrant further careful study. Planets in the habitable zones of stars like our Sun are sufficiently far away from the star so that the light rays [...] Read more.
Assessing the habitability of exoplanets (planets orbiting other stars) is of great importance in deciding which planets warrant further careful study. Planets in the habitable zones of stars like our Sun are sufficiently far away from the star so that the light rays from the star can be assumed to be parallel, leading to straightforward analytic models for stellar illumination of the planet’s surface. However, for planets in the close-in habitable zones of dim red dwarf stars, such as the potentially habitable planet orbiting our nearest stellar neighbor, Proxima Centauri, the analytic illumination models based on the parallel ray approximation do not hold, resulting in severe biases in the estimates of the planetary characteristics, thus impacting efforts to understand the planet’s atmosphere and climate. In this paper, we present our efforts to improve the instellation (stellar illumination) models for close-in orbiting planets and the significance of the implementation of these improved models into EXONEST, which is a Bayesian machine learning application for exoplanet characterization. The ultimate goal is to use these improved models and parameter estimates to model the climates of close-in orbiting exoplanets using planetary General Circulation Models (GCM). More specifically, we are working to apply our instellation corrections to the NASA ROCKE-3D GCM to study the climates of the potentially habitable planets in the Trappist-1 system. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Extremely close-in exoplanets exhibit four zones of illumination: a fully illuminated zone <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>F</mi> <mi>Z</mi> <mo>)</mo> </mrow> </semantics></math> defined by the inner tangents, two penumbral zones <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>P</mi> <msub> <mi>Z</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>P</mi> <msub> <mi>Z</mi> <mn>2</mn> </msub> <mo>)</mo> </mrow> </semantics></math> defined by whether more than half or less than half of the star is visible, and a dark un-illuminated zone <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>U</mi> <mi>Z</mi> <mo>)</mo> </mrow> </semantics></math> defined by the outer tangents.</p> Full article ">Figure 2
<p>Star–planet geometry involving light emission at point A on the star and incident on the planet at point B: The star with radius <math display="inline"><semantics> <msub> <mi>R</mi> <mi>S</mi> </msub> </semantics></math> is centered at <math display="inline"><semantics> <msub> <mi>O</mi> <mi>S</mi> </msub> </semantics></math> and the exoplanet with radius <math display="inline"><semantics> <msub> <mi>R</mi> <mi>P</mi> </msub> </semantics></math> is centered at <math display="inline"><semantics> <msub> <mi>O</mi> <mi>O</mi> </msub> </semantics></math>. The star–planet separation is denoted by the vector <math display="inline"><semantics> <mover accent="true"> <mi>r</mi> <mo>→</mo> </mover> </semantics></math>.</p> Full article ">Figure 3
<p>Orientations of the stellar and overall coordinate systems with origins located at <math display="inline"><semantics> <msub> <mi>O</mi> <mi>S</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>O</mi> <mi>O</mi> </msub> </semantics></math>, respectively. The stellar coordinate system, which employs spherical coordinates <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>θ</mi> <mo>,</mo> <mi>ω</mi> <mo>)</mo> </mrow> </semantics></math>, is used to define points on the star, while the overall coordinate system employs spherical coordinates <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>η</mi> <mo>,</mo> <mi>ϕ</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">
9 pages, 582 KiB  
Proceeding Paper
Bayesian Model Selection and Parameter Estimation for Complex Impedance Spectroscopy Data of Endothelial Cell Monolayers
by Franziska Zimmermann, Frauke Viola Härtel, Anupam Das, Thomas Noll and Peter Dieterich
Phys. Sci. Forum 2023, 9(1), 8; https://doi.org/10.3390/psf2023009008 - 28 Nov 2023
Viewed by 665
Abstract
Endothelial barrier function can be quantified by the determination of the transendothelial resistance (TER) via impedance spectroscopy. However, TER can only be obtained indirectly based on a mathematical model. Models usually comprise a sequence of a resistance in parallel with a capacitor (RC-circuit), [...] Read more.
Endothelial barrier function can be quantified by the determination of the transendothelial resistance (TER) via impedance spectroscopy. However, TER can only be obtained indirectly based on a mathematical model. Models usually comprise a sequence of a resistance in parallel with a capacitor (RC-circuit), each for the cell layer (including TER) and the filter substrate, one resistance (R) for the medium, and a constant phase element (CPE) for the electrode–electrolyte interface. We applied Bayesian data analysis on a variety of model variants. Phase and absolute values of impedance data were acquired over time by a commercial device for measurements of pure medium, medium and raw filter, and medium with cell-covered filter stimulated with different agents. Medium and raw filter were best described by a series of four and three RC-circuits, respectively. Parameter estimation of the TER showed a concentration-dependent decrease in response to thrombin. Model comparison indicated that even high concentrations of thrombin did not fully disrupt the endothelial barrier. Insights in the biophysical meaning of model parameters were gained through complemental cell-free measurements with sodium chloride. In summary, Bayesian analysis allows for valid parameter estimation and the selection of models with different complexity under various experimental conditions to characterize endothelial barrier function. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Scheme of the measurement setup: Isolated cells are cultivated on a filter and inserted in a so-called well, which is embedded in tissue medium. <math display="inline"><semantics> <mrow> <mo>|</mo> <mi mathvariant="bold">Z</mi> <mo>|</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mi mathvariant="bold">Ph</mi> </semantics></math> are measured via a two-electrode setup. The transendothelial resistance, <math display="inline"><semantics> <mrow> <mi>T</mi> <mi>E</mi> <mi>R</mi> </mrow> </semantics></math>, as a quantity of endothelial barrier function, is determined by an equivalent circuit model. One typically used model consists of a constant phase element (CPE), describing the non-ideal capacity behavior of the electrode–electrolyte interface, an ohmic resistance for the medium, and an RC-circuit (in this case with R = TER and C = <math display="inline"><semantics> <msub> <mi mathvariant="normal">C</mi> <mi>Cell</mi> </msub> </semantics></math>) for the cell monolayer cultivated on the filter support.</p> Full article ">Figure 2
<p>Comparison of the mean of resulting capacities of the RC-circuits of different measurement conditions. Values of the bars have to be multiplied with the respective order of the magnitude indicated by the y-axis in (<b>a</b>) or the label of the y-axis in (<b>b</b>), respectively. (<b>a</b>) Resulting capacities of the RC-circuits for medium only and medium with filter. Medium only resulted in <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>RC</mi> </msub> <mspace width="4pt"/> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (blue), whereas most of the data of medium and raw filter resulted in <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>RC</mi> </msub> <mspace width="4pt"/> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> (magenta) and only for some time points <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>RC</mi> </msub> <mspace width="4pt"/> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> (red). There was good accordance within parameter estimates of <math display="inline"><semantics> <msub> <mi mathvariant="normal">C</mi> <mrow> <mi mathvariant="normal">e</mi> <mn>1</mn> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi mathvariant="normal">C</mi> <mrow> <mi mathvariant="normal">e</mi> <mn>2</mn> </mrow> </msub> </semantics></math>, respectively, between the two measuring conditions. For <math display="inline"><semantics> <msub> <mi mathvariant="normal">C</mi> <mrow> <mi mathvariant="normal">e</mi> <mn>3</mn> </mrow> </msub> </semantics></math>, a difference of about one magnitude is noticeable, which is stable for <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>RC</mi> </msub> <mspace width="4pt"/> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>RC</mi> </msub> <mspace width="4pt"/> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>. Hence, the divergent <math display="inline"><semantics> <msub> <mi mathvariant="normal">C</mi> <mrow> <mi mathvariant="normal">e</mi> <mn>3</mn> </mrow> </msub> </semantics></math> is characteristic for the filter. (<b>b</b>) Comparison of the filter specific capacitance from measurements with raw filter and those with cell-covered filter stimulated with different concentrations of thrombin. This resulted in a stable difference, indicating that the cells have an influence on filter properties.</p> Full article ">Figure 3
<p>Time course of TER and <math display="inline"><semantics> <msub> <mi mathvariant="normal">C</mi> <mi>Cell</mi> </msub> </semantics></math> after stimulation with different concentrations of thrombin. The model analysis (CPE and <math display="inline"><semantics> <msub> <mi>n</mi> <mi>RC</mi> </msub> </semantics></math> = 4 RC-circuits) showed a concentration-dependent decrease in the TER. Magnitude and time needed for regeneration were also dependent of thrombin concentration. For all measured concentrations, the initial levels are reached within the observation time of <math display="inline"><semantics> <mrow> <mrow> <mn>1.5</mn> </mrow> <mspace width="3.33333pt"/> <mi mathvariant="normal">h</mi> </mrow> </semantics></math>. <math display="inline"><semantics> <msub> <mi mathvariant="normal">C</mi> <mi>Cell</mi> </msub> </semantics></math> shows an inversely concentration-dependent increase after stimulation with thrombin.</p> Full article ">Figure 4
<p>Resulting evidence and model probability for a model with a cell-specific RC-circuit (grey) and a filter model with <math display="inline"><semantics> <msub> <mi>n</mi> <mi>RC</mi> </msub> </semantics></math> = 3 to 5 RC-circuits (brown, green, and light blue) but without a cell-specific one. The dashed line indicates the addition of the stimulation agents (high concentration of thrombin or triton X-100) with disrupting effect on endothelial barrier function. (<b>a</b>) After adding high concentration of thrombin, there is a high rise of the evidence of the filter models (without the cell-specific RC-circuit), but the cell model (with the cell-specific RC-circuit) still remains as the one with the highest evidence. (<b>b</b>) After adding triton X-100, there is a shift in model probability to a cell-free model, indicating that the monolayer is not intact any more.</p> Full article ">Figure 5
<p>Comparison of estimated lowest capacitance <math display="inline"><semantics> <msub> <mi mathvariant="normal">C</mi> <mrow> <mi mathvariant="normal">e</mi> <mn>1</mn> </mrow> </msub> </semantics></math> as a function of sodium chloride concentration of the medium (as a ratio of the 1 M stock solution) of a well with pure medium (red) and a well with medium and raw filter (black). The capacitance, which already attracted attention in the results of measurements with cell medium because of the really small magnitude, is independent of concentration of the medium and of the presence of a filter. It was compared to calculated values (blue) according to a model for a similar setup published by [<a href="#B15-psf-09-00008" class="html-bibr">15</a>]. There is really high accordance in the magnitude and the concentration-independent course. As it can be seen by the equations in the cited publication, this capacitance mainly depends on setup criteria like electrode size and distance.</p> Full article ">Figure 6
<p>Course of the estimated resistance corresponding to the lowest capacitance as a function of sodium chloride concentration of the medium (as a ratio of the 1 M stock solution) of a well with pure medium (red) and a well with medium and raw filter (black). There is a clear dependence on concentration. Results are compared to calculated values according to a model for a similar setup published by [<a href="#B15-psf-09-00008" class="html-bibr">15</a>], considering an extra factor for real concentration of sodium chloride (green, purple, and blue), which is presumptively underestimated due to basic experimental protocol. Especially with this factor there is really good congruence to the calculated values. The underlying equations show that this resistance is mainly determined by characteristics of the medium.</p> Full article ">
9 pages, 12026 KiB  
Proceeding Paper
A Bayesian Data Analysis Method for an Experiment to Measure the Gravitational Acceleration of Antihydrogen
by Danielle Hodgkinson, Joel Fajans and Jonathan S. Wurtele
Phys. Sci. Forum 2023, 9(1), 9; https://doi.org/10.3390/psf2023009009 - 28 Nov 2023
Viewed by 787
Abstract
The ALPHA-g experiment at CERN intends to observe the effect of gravity on antihydrogen. In ALPHA-g, antihydrogen is confined to a magnetic trap with an axis aligned parallel to the Earth’s gravitational field. An imposed difference in the magnetic field of the confining [...] Read more.
The ALPHA-g experiment at CERN intends to observe the effect of gravity on antihydrogen. In ALPHA-g, antihydrogen is confined to a magnetic trap with an axis aligned parallel to the Earth’s gravitational field. An imposed difference in the magnetic field of the confining coils above and below the trapping region, known as a bias, can be delicately adjusted to compensate for the gravitational potential experienced by the trapped anti-atoms. With the bias maintained, the magnetic fields of the coils can be ramped down slowly compared to the anti-atom motion; this releases the antihydrogen and leads to annihilations on the walls of the apparatus, which are detected by a position-sensitive detector. If the bias cancels out the gravitational potential, antihydrogen will escape the trap upwards or downwards with equal probability. Determining the downward (or upward) escape probability, p, from observed annihilations is non-trivial because the annihilation detection efficiency may be up–down asymmetric; some small fraction of antihydrogen escaping downwards may be detected in the upper region (and vice versa) meaning that the precise number of trapped antihydrogen atoms is unknown. In addition, cosmic rays passing through the apparatus lead to a background annihilation rate, which may also be up–down asymmetric. We present a Bayesian method to determine p by assuming annihilations detected in the upper and lower regions are independently Poisson distributed, with the Poisson mean expressed in terms of experimental quantities. We solve for the posterior p using the Markov chain Monte Carlo integration package, Stan. Further, we present a method to determine the gravitational acceleration of antihydrogen, ag, by modifying the analysis described above to include simulation results. In the modified analysis, p is replaced by the simulated probability of downward escape, which is a function of ag. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>The ALPHA apparatus. Figure adapted from [<a href="#B5-psf-09-00009" class="html-bibr">5</a>].</p> Full article ">Figure 2
<p>ALPHA-g superconducting magnets. The diagram is rotated such that the gravitational force points to the left. For the experiment described in the text, antihydrogen is confined in the ‘up–down measurement’ region with magnetic potential produced initially by energising the long octupole (yellow), short octupole (blue) and the lower and upper coils (red). Figure courtesy of Chukman So.</p> Full article ">Figure 3
<p>An exaggerated diagram of the on-axis magnetic potential during an up–down measurement. If the magnetic field produced by the lower and upper coils is equal (black solid curve), the gravitational potential leads to the (exaggerated) up/down asymmetric potential shown. By varying the relative magnetic field produced by the coils, the gravitational potential can be compensated. The red, blue and green dashed lines show varying degrees of compensation. The regions in which detected antihydrogen is considered to have escaped downwards or upwards are highlighted in blue. The centers of the lower and upper coils are marked as <math display="inline"><semantics> <msub> <mi>z</mi> <mi>l</mi> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>z</mi> <mi>u</mi> </msub> </semantics></math>, respectively.</p> Full article ">Figure 4
<p>Estimation of simulated downward escape probability, <span class="html-italic">p</span>, as a function of bias, <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math>, for simulations assuming normal gravity (<math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>g</mi> </msub> <mo>=</mo> <mi>g</mi> </mrow> </semantics></math>, green) and repulsive gravity (<math display="inline"><semantics> <mrow> <msub> <mi>a</mi> <mi>g</mi> </msub> <mo>=</mo> <mo>−</mo> <mi>g</mi> </mrow> </semantics></math>, blue). Artificial data was generated using binomial sampling from the estimation of the normal gravity simulation (green line). The posterior <span class="html-italic">p</span> for the artificial data was determined using the Bayesian method described in the text. Red dots are the maximum a posteriori probability (MAP) estimate of the posterior <span class="html-italic">p</span> and the error bars are 68% credible intervals.</p> Full article ">Figure 5
<p>Posterior probability density of the gravitational acceleration of antihydrogen, <math display="inline"><semantics> <msub> <mi>a</mi> <mi>g</mi> </msub> </semantics></math>, for an artificial dataset. Blue and red/green vertical lines mark the maximum a posteriori probability (MAP) estimate and the 68% credible intervals, respectively.</p> Full article ">
10 pages, 963 KiB  
Proceeding Paper
Learned Harmonic Mean Estimation of the Marginal Likelihood with Normalizing Flows
by Alicja Polanska, Matthew A. Price, Alessio Spurio Mancini and Jason D. McEwen
Phys. Sci. Forum 2023, 9(1), 10; https://doi.org/10.3390/psf2023009010 - 29 Nov 2023
Cited by 1 | Viewed by 908
Abstract
Computing the marginal likelihood (also called the Bayesian model evidence) is an important task in Bayesian model selection, providing a principled quantitative way to compare models. The learned harmonic mean estimator solves the exploding variance problem of the original harmonic mean estimation of [...] Read more.
Computing the marginal likelihood (also called the Bayesian model evidence) is an important task in Bayesian model selection, providing a principled quantitative way to compare models. The learned harmonic mean estimator solves the exploding variance problem of the original harmonic mean estimation of the marginal likelihood. The learned harmonic mean estimator learns an importance sampling target distribution that approximates the optimal distribution. While the approximation need not be highly accurate, it is critical that the probability mass of the learned distribution is contained within the posterior in order to avoid the exploding variance problem. In previous work, a bespoke optimization problem is introduced when training models in order to ensure this property is satisfied. In the current article, we introduce the use of normalizing flows to represent the importance sampling target distribution. A flow-based model is trained on samples from the posterior by maximum likelihood estimation. Then, the probability density of the flow is concentrated by lowering the variance of the base distribution, i.e., by lowering its “temperature”, ensuring that its probability mass is contained within the posterior. This approach avoids the need for a bespoke optimization problem and careful fine tuning of parameters, resulting in a more robust method. Moreover, the use of normalizing flows has the potential to scale to high dimensional settings. We present preliminary experiments demonstrating the effectiveness of the use of flows for the learned harmonic mean estimator. The harmonic code implementing the learned harmonic mean, which is publicly available, has been updated to now support normalizing flows. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Diagram illustrating the concentration of the probability density of a normalizing flow. The flow is trained on samples from the posterior, giving us a normalized approximation of the posterior distribution. The temperature of the base distribution <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>∈</mo> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics></math> is reduced, which concentrates the probability density of the transformed distribution, ensuring that it is contained within the posterior. The concentrated flow can then be used as the target distribution for the learned harmonic mean estimator, avoiding the exploding variance issue of the original harmonic mean estimator.</p> Full article ">Figure 2
<p>Corner plot of samples from the posterior (red) and real NVP flow with temperature <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (blue) for the Rosenbrock benchmark problem. The target distribution given by the concentrated flow is contained within the posterior and has thinner tails, as required for the learned harmonic mean estimator.</p> Full article ">Figure 3
<p>Marginal likelihood computed by the learned harmonic mean estimator with a concentrated flow for the Rosenbrock benchmark problem. One hundred experiments are repeated to recover empirical estimates of the statistics of the estimator. In panel (<b>a</b>), the distribution of marginal likelihood values are shown (measured) along with the estimate of the standard deviation computed by the error estimator (estimated). The ground truth is indicated by the red dashed line. In panel (<b>b</b>), the distribution of the variance estimator is shown (estimated) along with the standard deviation computed by the variance-of-variance estimator (estimated). The learned harmonic mean estimator and its error estimators are highly accurate.</p> Full article ">Figure 4
<p>Corner plot of samples from the posterior (red) and real NVP flow trained on the posterior samples with temperature <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (blue) for the Normal-Gamma example with <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0.001</mn> </mrow> </semantics></math>. The target distribution given by the concentrated flow is contained within the posterior and has thinner tails, as required for the learned harmonic mean estimator.</p> Full article ">Figure 5
<p>Ratio of marginal likelihood values computed by the learned harmonic mean estimator with a concentrated flow to those computed analytically for the Normal-Gamma problem. Error bars corresponding to the estimated standard deviation of the learned harmonic estimator are also shown. Notice that the marginal likelihood values computed by the learned harmonic mean estimator are highly accurate and are indeed sensitive to changes in the prior. Predictions made with flow at temperature <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (blue) and <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0.95</mn> </mrow> </semantics></math> (green) are shown, which are slightly offset for ease of visualization, demonstrating that accuracy is not highly sensitive to the choice of <span class="html-italic">T</span>.</p> Full article ">Figure 6
<p>Corner plot of the samples from the posterior (red) and real NVP flow trained on the posterior samples with temperature <math display="inline"><semantics> <mrow> <mi>T</mi> <mo>=</mo> <mn>0.9</mn> </mrow> </semantics></math> (blue) for the Pima Indian benchmark problem for <math display="inline"><semantics> <mrow> <mi>τ</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>. The target distribution given by the concentrated flow is contained within the posterior and has thinner tails, as required for the learned harmonic mean estimator.</p> Full article ">
9 pages, 3921 KiB  
Proceeding Paper
Quantification of Endothelial Cell Migration Dynamics Using Bayesian Data Analysis
by Anselm Hohlstamm, Andreas Deussen, Stephan Speier and Peter Dieterich
Phys. Sci. Forum 2023, 9(1), 11; https://doi.org/10.3390/psf2023009011 - 30 Nov 2023
Viewed by 859
Abstract
Endothelial cells keep a tight and adaptive inner cell layer in blood vessels. Thereby, the cells develop complex dynamics through integrating active individual and collective cell migration, cell-cell interactions as well as interactions with external stimuli. It is the aim of this study [...] Read more.
Endothelial cells keep a tight and adaptive inner cell layer in blood vessels. Thereby, the cells develop complex dynamics through integrating active individual and collective cell migration, cell-cell interactions as well as interactions with external stimuli. It is the aim of this study to quantify and model these underlying dynamics. Therefore, we seeded and stained human umbilical vein endothelial cells (HUVECs) and recorded their positions every 10 min for 48 h via live-cell imaging. After image segmentation and tracking of several 10.000 cells, we applied Bayesian data analysis to models assessing the experimentally obtained cell trajectories. By analyzing the mean squared velocities, we found a dependence on the local cell density. Based on this connection, we developed a model, which approximates the time-dependent frequency of cell divisions. Furthermore, we determined two different phases of velocity deceleration, which are influenced by the emergence of correlated cell movements and time-dependent aging in this non-stationary system. By integrating the findings of correlation functions, we will be able to develop a comprehensive model to improve the understanding of endothelial cell migration in the future. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>In a section (1 mm<sup>2</sup>) of the cell layer, all cell trajectories are shown for the experiments 4 (<b>a</b>) and 6 (<b>b</b>). The trajectories are labeled sequentially with one out of seven colors in the order of their first appearance and the attributed cell number. The overall cell density in (<b>a</b>) is much lower than in (<b>b</b>). This leads to different dynamics and pattern formation.</p> Full article ">Figure 2
<p>The mean squared velocities and cell densities (inset) are shown for each time step in all ten experiments (black dots, <b>a</b>–<b>j</b>). Additionally, the mean squared velocities are fitted with two exponential functions (Equation (1), solid colored line). In certain time intervals of the experiments (especially in <b>c</b>,<b>d</b>,<b>i</b>), the fit transiently deviates from the course of the data points. The quite small uncertainties (two standard deviations) of the model fit are marked as a gray area (barely visible). The calculation of the velocities is based on all cells that could be tracked for the entire duration of the experiment (48 h). The values for the cell densities consider all cells with the exception of the boundary area. The cell numbers in the first and last nine time steps are artificially lower due to the defined minimum trajectory length during cell tracking. Experiments with cells from the same umbilical cord are shown in the same row.</p> Full article ">Figure 3
<p>The changes in cell numbers <span class="html-italic">α(t)</span> (Equation (2)) are calculated for each time step. A uniform filter with a size of 300 min was applied to smoothen the data. Due to the defined minimum length of cell trajectories during image analysis, the first and last nine time steps were discarded.</p> Full article ">Figure 4
<p>The mean and standard deviation for the estimated parameters <span class="html-italic">τ</span><sub>1</sub> (<b>a</b>), <span class="html-italic">τ</span><sub>2</sub> (<b>b</b>) and <span class="html-italic">c</span> (<b>c</b>) are shown for each experiment. Here, <span class="html-italic">τ</span><sub>1</sub> and <span class="html-italic">τ</span><sub>2</sub> describe the two time scales with which the exponential functions are decreasing. Parameter <span class="html-italic">c</span> depicts the coupling strength between the rate of cell division and the mean squared velocity.</p> Full article ">Figure 5
<p>The fits to the mean squared velocity (<b>a</b>–<b>j</b>), as shown in <a href="#psf-09-00011-f002" class="html-fig">Figure 2</a>, were improved using Equation (3). The shown data of the cell density (inset) was fitted simultaneously for each experiment according to Equation (4). The quite small uncertainties (two standard deviations) of the model fit are represented as a gray area (almost only visible in the inset of (<b>b</b>)). Note that the ranges of the y-axes are optimally adjusted to each data set in contrast to <a href="#psf-09-00011-f002" class="html-fig">Figure 2</a>.</p> Full article ">Figure 6
<p>Three models (<a href="#psf-09-00011-t001" class="html-table">Table 1</a>) were used for the simultaneous fit of the mean squared velocity and density. The evidence, obtained by Bayesian analysis, is used to compare the accuracy of the different models. The model probability is shown color-coded and in numbers for each experiment.</p> Full article ">
10 pages, 626 KiB  
Proceeding Paper
Variational Bayesian Approximation (VBA) with Exponential Families and Covariance Estimation
by Seyedeh Azadeh Fallah Mortezanejad and Ali Mohammad-Djafari
Phys. Sci. Forum 2023, 9(1), 12; https://doi.org/10.3390/psf2023009012 - 30 Nov 2023
Viewed by 864
Abstract
Variational Bayesian Approximation (VBA) is a fast technique for approximating Bayesian computation. The main idea is to assess the joint posterior distribution of all the unknown variables with a simple expression. Mean–Field Variational Bayesian Approximation (MFVBA) is a particular case developed for large–scale [...] Read more.
Variational Bayesian Approximation (VBA) is a fast technique for approximating Bayesian computation. The main idea is to assess the joint posterior distribution of all the unknown variables with a simple expression. Mean–Field Variational Bayesian Approximation (MFVBA) is a particular case developed for large–scale problems where the approximated probability law is separable in all variables. A well–known drawback of MFVBA is that it tends to underestimate the variances in the variables, even though it estimates the means well. It can lead to poor inference results. We can obtain a fixed point algorithm to evaluate the means in exponential families for the approximating distribution. However, this does not solve the problem of underestimating the variances. In this paper, we propose a modified method of VBA with exponential families to first estimate the posterior mean and then improve the estimation of the posterior covariance. We demonstrate the performance of the procedure with an example. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>The Normal–Poisson hierarchical model pattern when <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math> with sample size 100.</p> Full article ">Figure 2
<p>Sparsity patterns for the matrices using model (<a href="#FD16-psf-09-00012" class="html-disp-formula">16</a>). (<b>a</b>) VBA covariance matrix <math display="inline"><semantics> <mi mathvariant="bold-italic">V</mi> </semantics></math>; (<b>b</b>) Hessian matrix <math display="inline"><semantics> <mi mathvariant="bold-italic">H</mi> </semantics></math>; (<b>c</b>) <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic">I</mi> <mo>−</mo> <mi mathvariant="bold-italic">V</mi> <mi mathvariant="bold-italic">H</mi> </mrow> </semantics></math> matrix.</p> Full article ">Figure 3
<p>The final estimations of margin densities for each variable via MFVBA method.</p> Full article ">Figure 4
<p>The final margins via MCMC method.</p> Full article ">
10 pages, 574 KiB  
Proceeding Paper
Proximal Nested Sampling with Data-Driven Priors for Physical Scientists
by Jason D. McEwen, Tobías I. Liaudat, Matthew A. Price, Xiaohao Cai and Marcelo Pereyra
Phys. Sci. Forum 2023, 9(1), 13; https://doi.org/10.3390/psf2023009013 - 1 Dec 2023
Cited by 2 | Viewed by 1045
Abstract
Proximal nested sampling was introduced recently to open up Bayesian model selection for high-dimensional problems such as computational imaging. The framework is suitable for models with a log-convex likelihood, which are ubiquitous in the imaging sciences. The purpose of this article is two-fold. [...] Read more.
Proximal nested sampling was introduced recently to open up Bayesian model selection for high-dimensional problems such as computational imaging. The framework is suitable for models with a log-convex likelihood, which are ubiquitous in the imaging sciences. The purpose of this article is two-fold. First, we review proximal nested sampling in a pedagogical manner in an attempt to elucidate the framework for physical scientists. Second, we show how proximal nested sampling can be extended in an empirical Bayes setting to support data-driven priors, such as deep neural networks learned from training data. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Proximal nested sampling considers likelihoods that are log-convex and lower semicontinuous. A lower semicontinuous convex function has a convex and closed epigraph.</p> Full article ">Figure 2
<p>Illustration of the proximity operator (reproduced from [<a href="#B14-psf-09-00013" class="html-bibr">14</a>]). The proximal operator maps the blue points to red points (i.e., from base to head of arrows). The thick black line defines the domain boundary, while the thin black lines define level sets (iso-contours) of <span class="html-italic">f</span>. The proximity operator maps points towards the minimum of <span class="html-italic">f</span> while remaining in the proximity of the original point.</p> Full article ">Figure 3
<p>Diagram illustrating proximal nested sampling. If a sample <math display="inline"><semantics> <msup> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> </semantics></math> outside of the likelihood constraint is considered, then proximal nested sampling introduces a term in the direction of the projection of <math display="inline"><semantics> <msup> <mi>x</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> </semantics></math> onto the convex set defining the likelihood constraint, thereby acting to push the Markov chain back into the constraint set <math display="inline"><semantics> <msub> <mi mathvariant="script">B</mi> <mi>τ</mi> </msub> </semantics></math> if it wanders outside of it. A subsequent Metropolis–Hastings step can be introduced to enforce strict adherence to the convex likelihood constraint.</p> Full article ">Figure 4
<p>Results of radio interferometric imaging reconstruction problem. (<b>a</b>) Ground truth galaxy image. (<b>b</b>) Dirty reconstruction based on pseudo-inverting the measurement operator <math display="inline"><semantics> <mo>Φ</mo> </semantics></math>. (<b>c</b>) Posterior mean reconstruction computed from proximal nested samples for the hand-crafted wavelet-sparsity prior. (<b>d</b>) Posterior mean reconstruction for the data-driven prior based on a deep neural network (DnCNN) trained on example images. Reconstruction SNR is shown on each image. The computed SNR levels demonstrate that the data-driven prior results in a superior reconstruction quality, although this may not be obvious from a visual assessment of the reconstructed images. Computing the reconstructed SNR requires knowledge of the ground truth, which is not available in realistic settings. The Bayesian model evidence proves a way to compare the hand-crafted and data-driven models without requiring knowledge of the ground truth. For this example, the Bayesian evidence correctly selects the data-driven prior as the best model.</p> Full article ">
13 pages, 1585 KiB  
Proceeding Paper
Bayesian Inference and Deep Learning for Inverse Problems
by Ali Mohammad-Djafari, Ning Chu, Li Wang and Liang Yu
Phys. Sci. Forum 2023, 9(1), 14; https://doi.org/10.3390/psf2023009014 - 1 Dec 2023
Viewed by 1347
Abstract
Inverse problems arise anywhere we have an indirect measurement. In general, they are ill-posed to obtain satisfactory solutions, which needs prior knowledge. Classically, different regularization methods and Bayesian inference-based methods have been proposed. As these methods need a great number of forward and [...] Read more.
Inverse problems arise anywhere we have an indirect measurement. In general, they are ill-posed to obtain satisfactory solutions, which needs prior knowledge. Classically, different regularization methods and Bayesian inference-based methods have been proposed. As these methods need a great number of forward and backward computations, they become costly in computation, particularly when the forward or generative models are complex, and the evaluation of the likelihood becomes very costly. Using deep neural network surrogate models and approximate computation can become very helpful. However, in accounting for the uncertainties, we need first to understand Bayesian deep learning, and then we can see how we can use it for inverse problems. In this work, we focus on NN, DL, and, more specifically, the Bayesian DL particularly adapted for inverse problems. We first give details of Bayesian DL approximate computations with exponential families; then, we see how we can use them for inverse problems. We consider two cases: First, we consider the case where the forward operator is known and used as a physics constraint, and the second examines more general data-driven DL methods. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>General exponential family, with original parameters <math display="inline"><semantics> <mi mathvariant="bold-italic" mathcolor="red">θ</mi> </semantics></math>, natural parameters <math display="inline"><semantics> <mi mathvariant="bold-italic" mathcolor="blue">λ</mi> </semantics></math>, expectations parameters <math display="inline"><semantics> <mrow> <mi mathvariant="bold-italic" mathcolor="green">μ</mi> <mo>:</mo> <mo>=</mo> <msub> <mi mathvariant="normal">E</mi> <mi>q</mi> </msub> <mfenced separators="" open="{" close="}"> <mi mathvariant="bold">t</mi> <mo>(</mo> <mi mathvariant="bold-italic" mathcolor="red">θ</mi> <mo>)</mo> </mfenced> </mrow> </semantics></math>, and their relations via the Legendre Transform.</p> Full article ">Figure 2
<p>Three linear NN structures, which are derived directly from quadratic regularization inversion method. The right part of this figure is adapted from [<a href="#B16-psf-09-00014" class="html-bibr">16</a>].</p> Full article ">Figure 3
<p>A <span class="html-italic">K</span> layers DL NN equivalent to <span class="html-italic">K</span> iterations of the basic optimization algorithm.</p> Full article ">Figure 4
<p>A <span class="html-italic">K</span> layers DL NN equivalent to <span class="html-italic">K</span> iterations of a basic gradient-based optimization algorithm. A quadratic regularization results in a linear NN, while a <math display="inline"><semantics> <msub> <mo mathvariant="italic">ℓ</mo> <mn>1</mn> </msub> </semantics></math> regularization results in a classical NN with a nonlinear activation function. Left: supervised case. Right: unsupervised case. In both cases, all the <span class="html-italic">K</span> layers have the same structure.</p> Full article ">Figure 5
<p>All the <span class="html-italic">K</span> layers of DL NN equivalent to <span class="html-italic">K</span> iterations of an iterative gradient-based optimization algorithm. The simplest solution is to choose <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="bold" mathcolor="red">W</mi> <mn>0</mn> </msub> <mo>=</mo> <mi>α</mi> <mi mathvariant="bold">H</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="bold" mathcolor="red">W</mi> </mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <mo>=</mo> <mi mathvariant="bold" mathcolor="red">W</mi> <mo>=</mo> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>−</mo> <mi>α</mi> <msup> <mrow> <mi mathvariant="bold">H</mi> </mrow> <mi>t</mi> </msup> <mi mathvariant="bold">H</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>K</mi> </mrow> </semantics></math>. A more robust, but more costly approach, is to learn all the layers for <math display="inline"><semantics> <mrow> <msup> <mrow> <mi mathvariant="bold" mathcolor="red">W</mi> </mrow> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <mo>=</mo> <mrow> <mo>(</mo> <mi mathvariant="bold">I</mi> <mo>−</mo> <msup> <mi>α</mi> <mrow> <mo>(</mo> <mi>k</mi> <mo>)</mo> </mrow> </msup> <msup> <mrow> <mi mathvariant="bold">H</mi> </mrow> <mi>t</mi> </msup> <mi mathvariant="bold">H</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mo>⋯</mo> <mo>,</mo> <mi>K</mi> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Training (<b>top</b>) and testing (<b>bottom</b>) steps in the first use of physics-based ML approach.</p> Full article ">Figure 7
<p>The proposed four groups of layers of NN for denoising, deconvolution, and segmentation of IR images.</p> Full article ">Figure 8
<p>Example of expected results in deterministic methods. First row: a simulated IR image (<b>left</b>), its ground truth labels (<b>middle</b>), and the result of the deconvolution and segmentation (<b>right</b>). Second row: a real IR image (<b>left</b>) and the result of its deconvolution and segmentation (<b>right</b>).</p> Full article ">Figure 9
<p>Example of expected results in Bayesian methods. First row from left: (<b>a</b>) simulated IR image, (<b>b</b>) its ground truth labels, (<b>c</b>) the result of the deconvolution and segmentation, and (<b>d</b>) uncertainties. Second row: (<b>e</b>) a real IR image, (<b>f</b>) no ground truth, (<b>g</b>) the result of its deconvolution and segmentation, and (<b>h</b>) uncertainties.</p> Full article ">
11 pages, 2314 KiB  
Proceeding Paper
Physics-Consistency Condition for Infinite Neural Networks and Experimental Characterization
by Sascha Ranftl and Shaoheng Guan
Phys. Sci. Forum 2023, 9(1), 15; https://doi.org/10.3390/psf2023009015 - 4 Dec 2023
Viewed by 1028
Abstract
It has previously been shown that prior physics knowledge can be incorporated into the structure of an artificial neural network via neural activation functions based on (i) the correspondence under the infinite-width limit between neural networks and Gaussian processes if the central limit [...] Read more.
It has previously been shown that prior physics knowledge can be incorporated into the structure of an artificial neural network via neural activation functions based on (i) the correspondence under the infinite-width limit between neural networks and Gaussian processes if the central limit theorem holds and (ii) the construction of physics-consistent Gaussian process kernels, i.e., specialized covariance functions that ensure that the Gaussian process fulfills a priori some linear (differential) equation. Such regression models can be useful in many-query problems, e.g., inverse problems, uncertainty quantification or optimization, when a single forward solution or likelihood evaluation is costly. Based on a small set of training data, the learned model or “surrogate” can then be used as a fast approximator. The bottleneck is then for the surrogate to also learn efficiently and effectively from small data sets while at the same time ensuring physically consistent predictions. Based on this, we will further explore the properties of so-constructed neural networks. In particular, we will characterize (i) generalization behavior and (ii) the approximation quality or Gaussianity as a function of network width and discuss (iii) extensions from shallow to deep NNs. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Schematic illustration of convergence of infinite NNs toward GPs for building intuition. The right shows a simple single-layer NN with grey disks being the neurons, where we keep adding neurons in the hidden (latent) layer in the middle ad infinitum. The distribution curves next to the neurons should emphasize that each of the outputs of the neurons (i.e., the resulting values of non-linear activation after linear transform) are random variables with some probability distribution. The left side shows a GP, i.e., a stochastic process with a Gaussian distribution on <span class="html-italic">every</span> point on the real line of <span class="html-italic">x</span> (this is essentially the definition of a GP). The correlation between all those Gaussians is defined by the covariance function <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> <mo>)</mo> </mrow> </semantics></math>. The blue disks denote data points to which the GP has been adjusted (i.e., we are looking at a posterior GP trained on data, but we may as well illustrate a prior GP centered, e.g., around <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>). The <span class="html-italic">x</span>-axis of the GP has been rotated and mirrored for visualization purposes such that the orientation of the bell curves on the GP <span class="html-italic">x</span>-axis align with the distributions over the neurons r.h.s.</p> Full article ">Figure 2
<p>Generalization of neural network models. (<b>a</b>) Evolution of loss during training. (<b>b</b>) Predictions of different networks. Solid line: training error; dashed line: validation error; dotted line: test error. Generalization is typically measured in terms of the gap between training and test error. Black vertical lines indicate optimal model according to validation set.</p> Full article ">Figure 3
<p>Scatter plots of MC samples with different single-layer network widths (number of neurons, <span class="html-italic">N</span>) evaluated at fixed NN inputs <span class="html-italic">x</span>. Ordinate and abscissa values and then mean values of NN prediction for two (distinct) given inputs. Upper row: tanh-activation. Lower row: sin-activation.</p> Full article ">Figure 4
<p>Comparing kernels of NNs with <math display="inline"><semantics> <mi>ReLU</mi> </semantics></math> (left), tanh (middle column) and physics-consistent sinus (right) activation for varying widths (rows). (<b>a</b>) Infinite NN limit kernel as computed by [<a href="#B28-psf-09-00015" class="html-bibr">28</a>]. (<b>b</b>–<b>f</b>) Monte Carlo estimates with 5000 samples. <span class="html-italic">N</span> denotes number of neurons. Ordinate and abscissa mean (pairs of) input indices <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>,</mo> <msup> <mi>x</mi> <mo>′</mo> </msup> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Kullback–Leibler divergence between finite NN and GP (i.e., infinite NN) for three layer depths, two generic activations and the physics-consistent activation (<math display="inline"><semantics> <mrow> <mi>s</mi> <mi>i</mi> <mi>n</mi> </mrow> </semantics></math>) from before.</p> Full article ">
9 pages, 285 KiB  
Proceeding Paper
Quantum Measurement and Objective Classical Reality
by Vishal Johnson, Philipp Frank and Torsten Enßlin
Phys. Sci. Forum 2023, 9(1), 16; https://doi.org/10.3390/psf2023009016 - 6 Dec 2023
Viewed by 917
Abstract
We explore quantum measurement in the context of Everettian unitary quantum mechanics and construct an explicit unitary measurement procedure. We propose the existence of prior correlated states that enable this procedure to work and therefore argue that correlation is a resource that is [...] Read more.
We explore quantum measurement in the context of Everettian unitary quantum mechanics and construct an explicit unitary measurement procedure. We propose the existence of prior correlated states that enable this procedure to work and therefore argue that correlation is a resource that is consumed when measurements take place. It is also argued that a network of such measurements establishes a stable objective classical reality. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>A quantum system (the signal) is measured by an observer. Unitarity of quantum mechanics necessitates the involvement of another system (the environment) in order to facilitate quantum measurement.</p> Full article ">Figure 2
<p>Measurement procedure without environmental correction. The environment influences the measurement procedure as indicated by the term <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>∘</mo> <mi>i</mi> </mrow> </semantics></math>. The clouds indicate systems which are correlated with each other. Two clouds touching indicates that the quantum systems they are part of are correlated (and thereby entangled). The clouds are colored to distinguish between the different quantum systems participating in the measurement procedure.</p> Full article ">Figure 3
<p>Measurement procedure with environmental correction. The redundant information provided by the correlated environment is used for this correction. Also indicated are the measures of correlation in the involved subsystems.</p> Full article ">Figure 4
<p>A highly branched network of decohered states lends stability to the measurement of the signal, as there is a large amount of redundancy in the information. Also, for this information to be deleted, all the involved systems must conspire to come together and undo this correlation.</p> Full article ">Figure 5
<p>Measurement in a different basis is no longer objective. Different observers may disagree on what consititutes reality. Transparent clouds over the same quantum system indicate its being observed by observers in different bases.</p> Full article ">Figure 6
<p>Due to the redundant information stored in the environment, it is possible to recover information about the environment and thereby allow for the signal to once again be determined objectively.</p> Full article ">
5 pages, 255 KiB  
Proceeding Paper
Snowballing Nested Sampling
by Johannes Buchner
Phys. Sci. Forum 2023, 9(1), 17; https://doi.org/10.3390/psf2023009017 - 6 Dec 2023
Viewed by 831
Abstract
A new way to run nested sampling, combined with realistic MCMC proposals to generate new live points, is presented. Nested sampling is run with a fixed number of MCMC steps. Subsequently, snowballing nested sampling extends the run to more and more live points. [...] Read more.
A new way to run nested sampling, combined with realistic MCMC proposals to generate new live points, is presented. Nested sampling is run with a fixed number of MCMC steps. Subsequently, snowballing nested sampling extends the run to more and more live points. This stabilizes the MCMC proposal of later MCMC proposals, and leads to pleasant properties, including that the number of live points and number of MCMC steps do not have to be calibrated, that the evidence and posterior approximation improve as more compute is added and can be diagnosed with convergence diagnostics from the MCMC community. Snowballing nested sampling converges to a “perfect” nested sampling run with an infinite number of MCMC steps. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Estimate of <math display="inline"><semantics> <mrow> <mo form="prefix">ln</mo> <mo>(</mo> <mi>Z</mi> <mo>)</mo> </mrow> </semantics></math> for each algorithm iteration.</p> Full article ">
10 pages, 7936 KiB  
Proceeding Paper
A BRAIN Study to Tackle Image Analysis with Artificial Intelligence in the ALMA 2030 Era
by Fabrizia Guglielmetti, Michele Delli Veneri, Ivano Baronchelli, Carmen Blanco, Andrea Dosi, Torsten Enßlin, Vishal Johnson, Giuseppe Longo, Jakob Roth, Felix Stoehr, Łukasz Tychoniec and Eric Villard
Phys. Sci. Forum 2023, 9(1), 18; https://doi.org/10.3390/psf2023009018 - 13 Dec 2023
Viewed by 1119
Abstract
An ESO internal ALMA development study, BRAIN, is addressing the ill-posed inverse problem of synthesis image analysis, employing astrostatistics and astroinformatics. These emerging fields of research offer interdisciplinary approaches at the intersection of observational astronomy, statistics, algorithm development, and data science. In this [...] Read more.
An ESO internal ALMA development study, BRAIN, is addressing the ill-posed inverse problem of synthesis image analysis, employing astrostatistics and astroinformatics. These emerging fields of research offer interdisciplinary approaches at the intersection of observational astronomy, statistics, algorithm development, and data science. In this study, we provide evidence of the benefits of employing these approaches to ALMA imaging for operational and scientific purposes. We show the potential of two techniques, RESOLVE and DeepFocus, applied to ALMA-calibrated science data. Significant advantages are provided with the prospect to improve the quality and completeness of the data products stored in the science archive and the overall processing time for operations. Both approaches evidence the logical pathway to address the incoming revolution in data rates dictated by the planned electronic upgrades. Moreover, we bring to the community additional products through a new package, ALMASim, to promote advancements in these fields, providing a refined ALMA simulator usable by a large community for training and testing new algorithms. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>ALMA antennas on the Chajnantor plateau. Credit: ESO.</p> Full article ">Figure 2
<p>Application of RESOLVE to Elias 27 from the DSHARP ALMA project at 240 GHz (1.25 mm) continuum. (<b>A</b>) The fiducial image as given by the DSHARP team [<a href="#B14-psf-09-00018" class="html-bibr">14</a>]. (<b>B</b>) RESOLVE mean sky map of Elias 27. (<b>C</b>) RESOLVE uncertainty map representation.</p> Full article ">Figure 3
<p>Comparison of processing time and computing throughput with tCLEAN and DeepFocus on <math display="inline"><semantics> <mrow> <mn>29</mn> <mo>×</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </semantics></math> archived cube data from cycles 7, 8, and 9. This represents a rough estimate because at this stage of development, it is challenging to make a robust comparison between the two techniques.</p> Full article ">Figure 4
<p>Example of ALMA-simulated sources (dirty images) created with the ALMASim package: (<b>A</b>) point-like, (<b>B</b>) Gaussian shape, (<b>C</b>) extended, and (<b>D</b>) diffuse emissions.</p> Full article ">Figure 5
<p>Simplified visual explanation of the empirical approach to noise modeling. Different background and noise components measured at scales larger and shorter than the typical beam scale (central panels) are isolated from a real ALMA image (e.g., an ALMA calibrator (left panel)) and then added to a simulated image (right panel). In this example, we considered local fluctuations (center, bottom panel), a large-scale background (central panel), and high spatial frequency patterns (center, top panel). Instead of using a theoretical model to simulate noise and instrumental response, the same effects were directly measured from real observations obtained in comparable situations (telescope configuration and atmospheric conditions).</p> Full article ">
9 pages, 982 KiB  
Proceeding Paper
Inferring Evidence from Nested Sampling Data via Information Field Theory
by Margret Westerkamp, Jakob Roth, Philipp Frank, Will Handley and Torsten Enßlin
Phys. Sci. Forum 2023, 9(1), 19; https://doi.org/10.3390/psf2023009019 - 13 Dec 2023
Cited by 1 | Viewed by 911
Abstract
Nested sampling provides an estimate of the evidence of a Bayesian inference problem via probing the likelihood as a function of the enclosed prior volume. However, the lack of precise values of the enclosed prior mass of the samples introduces probing noise, which [...] Read more.
Nested sampling provides an estimate of the evidence of a Bayesian inference problem via probing the likelihood as a function of the enclosed prior volume. However, the lack of precise values of the enclosed prior mass of the samples introduces probing noise, which can hamper high-accuracy determinations of the evidence values as estimated from the likelihood-prior-volume function. We introduce an approach based on information field theory, a framework for non-parametric function reconstruction from data, that infers the likelihood-prior-volume function by exploiting its smoothness and thereby aims to improve the evidence calculation. Our method provides posterior samples of the likelihood-prior-volume function that translate into a quantification of the remaining sampling noise for the evidence estimate, or for any other quantity derived from the likelihood-prior-volume function. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>(<b>Left</b>): Visualisation of the nested sampling dead point logarithmic likelihoods, <math display="inline"><semantics> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">→</mo> </mover> <mi>L</mi> </msub> </semantics></math>, as a function of logarithmic prior mass data, <math display="inline"><semantics> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">→</mo> </mover> <mi>X</mi> </msub> </semantics></math>, for the normalized simple Gaussian in Equation (<a href="#FD14-psf-09-00019" class="html-disp-formula">14</a>) (<math display="inline"><semantics> <mrow> <msub> <mi>σ</mi> <mi>X</mi> </msub> <mo>=</mo> <mn>0.01</mn> <mo>,</mo> <mspace width="3.33333pt"/> <mi>D</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>). The corresponding data was generated by the software package anesthetic [<a href="#B9-psf-09-00019" class="html-bibr">9</a>]. (<b>Right</b>): Visualisation of the reparametrised nested sampling dead point logarithmic likelihoods according to Equation (<a href="#FD13-psf-09-00019" class="html-disp-formula">13</a>) as a function of logarithmic prior mass for the same case as shown left.</p> Full article ">Figure 2
<p>Reconstruction prior samples of the likelihood-prior-volume function plotted together with the ground truth. (<b>Left</b>): Log-log-scale. (<b>Right</b>): Parametrisation according to Equation (<a href="#FD13-psf-09-00019" class="html-disp-formula">13</a>).</p> Full article ">Figure 3
<p>Reconstruction results for the likelihood-prior-volume function for the simple Gaussian example in Equation (<a href="#FD14-psf-09-00019" class="html-disp-formula">14</a>). The plots show the data, the ground truth and the reconstruction as well as its uncertainty. (<b>Left</b>): Reconstruction results on log-log-scale. (<b>Right</b>): Reconstruction results in reparametrised coordinates according to Equation (<a href="#FD13-psf-09-00019" class="html-disp-formula">13</a>).</p> Full article ">Figure 4
<p>Reconstruction results for the prior volumes given the likelihood data <math display="inline"><semantics> <msub> <mover accent="true"> <mi>d</mi> <mo stretchy="false">→</mo> </mover> <mi>L</mi> </msub> </semantics></math> for the simple Gaussian example in Equation (<a href="#FD14-psf-09-00019" class="html-disp-formula">14</a>). The plots show the data, the ground truth and the reconstruction as well as its uncertainty. (<b>Left</b>): Reconstruction results on log-log-scale. (<b>Right</b>): Reconstruction results in reparametrised coordinates according to Equation (<a href="#FD13-psf-09-00019" class="html-disp-formula">13</a>).</p> Full article ">Figure 5
<p>Comparison of histograms for logarithmic evidences for <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>samp</mi> </msub> <mo>=</mo> <mn>200</mn> </mrow> </semantics></math> samples for the classical nested sampling (NSL) approach and the reconstructed prior volumes.</p> Full article ">
8 pages, 2635 KiB  
Proceeding Paper
Knowledge-Based Image Analysis: Bayesian Evidences Enable the Comparison of Different Image Segmentation Pipelines
by Mats Leif Moskopp, Andreas Deussen and Peter Dieterich
Phys. Sci. Forum 2023, 9(1), 20; https://doi.org/10.3390/psf2023009020 - 4 Jan 2024
Viewed by 939
Abstract
The analysis and evaluation of microscopic image data is essential in life sciences. Increasing temporal and spatial digital image resolution and the size of data sets promotes the necessity of automated image analysis. Previously, our group proposed a Bayesian formalism that allows for [...] Read more.
The analysis and evaluation of microscopic image data is essential in life sciences. Increasing temporal and spatial digital image resolution and the size of data sets promotes the necessity of automated image analysis. Previously, our group proposed a Bayesian formalism that allows for converting the experimenter’s knowledge, in the form of a manually segmented image, into machine-readable probability distributions of the parameters of an image segmentation pipeline. This approach preserved the level of detail provided by expert knowledge and interobserver variability and has proven robust to a variety of recording qualities and imaging artifacts. In the present work, Bayesian evidences were used to compare different image processing pipelines. As an illustrative example, a microscopic phase contrast image of a wound healing assay and its manual segmentation by the experimenter (ground truth) are used. Six different variations of image segmentation pipelines are introduced. The aim was to find the image segmentation pipeline that is best to automatically segment the input image given the expert knowledge with respect to the principle of Occam’s razor to avoid unnecessary complexity and computation. While none of the introduced image segmentation pipelines fail completely, it is illustrated that assessing the quality of the image segmentation with the naked eye is not feasible. Bayesian evidence (and the intrinsically estimated uncertainty σ of the image segmentation) is used to choose the best image processing pipeline for the given image. This work illustrates a proof of principle and is extendable to a diverse range of image segmentation problems. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p><b>Image data and manual segmentation.</b> (<b>A</b>) A typical region of interest on an image of a wound healing assay can be seen. (<b>B</b>) The same region of interest as in panel (<b>A</b>) is shown with enhanced contrast for illustrative purposes. (<b>C</b>) A manual image segmentation for a cell-free (black) and cell-covered (white) area is shown. (<b>D</b>) The boundary between the black and the white pixels of the manual segmentation is indicated by a green line. This boundary is essential for distances between manually segmented images and pipeline-segmented images (see below). (<b>E</b>) An overlay of the contrast-enhanced original image and the boundary is given for visual clarification.</p> Full article ">Figure 2
<p><b>Image segmentation pipelines.</b> The image segmentation pipelines (Models 1–6) consist of a sequence of image filters and algorithms that depend on one (1P) or two (2P) parameters. Further, some algorithms are applied with a fixed set of parameters (0P), so that no free parameters were used during parameter estimation. Differences of the applied filters with respect to Model 1 are highlighted in yellow. The original image is displayed with enhanced contrast for illustrative purposes only—calculations and shown results are based on the native original image (see <a href="#psf-09-00020-f001" class="html-fig">Figure 1</a>A).</p> Full article ">Figure 3
<p><b>Evaluation of the necessary number of live points.</b> This figure shows the results for estimated logarithmic evidence <math display="inline"><semantics> <mrow> <mi>l</mi> <mi>n</mi> <mo>(</mo> <mi>Z</mi> <mo>)</mo> </mrow> </semantics></math>, estimated uncertainty <math display="inline"><semantics> <mi>σ</mi> </semantics></math>, and the total number of likelihood evaluations (N) for three independent applications of the previously introduced formalism using either 20, 50, 100, 200, 400, or 800 live points. With 100 live points or more, the estimated evidences <math display="inline"><semantics> <mrow> <mi>l</mi> <mi>n</mi> <mo>(</mo> <mi>Z</mi> <mo>)</mo> </mrow> </semantics></math> and estimated uncertainties <math display="inline"><semantics> <mi>σ</mi> </semantics></math> remain stable. Of the tested values with stable results, 100 live points require the least likelihood evaluations and are therefore computationally the most effective. (Data are shown as mean and Bayesian uncertainty (error bars) of the posterior distribution. Each estimation was independently run three times to evaluate reproducibility. In <span class="html-italic">RUN 3</span> with 20 live points, results are <math display="inline"><semantics> <mrow> <mo>−</mo> <mn>3095.7</mn> <mspace width="0.277778em"/> <mo>±</mo> <mspace width="0.277778em"/> <mn>1.2</mn> </mrow> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>l</mi> <mi>n</mi> <mo>(</mo> <mi>Z</mi> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mn>40.5</mn> <mspace width="0.277778em"/> <mo>±</mo> <mspace width="0.277778em"/> <mn>0.1</mn> <mspace width="0.277778em"/> <mi>p</mi> <mi>i</mi> <mi>x</mi> <mi>e</mi> <mi>l</mi> </mrow> </semantics></math> for <math display="inline"><semantics> <mi>σ</mi> </semantics></math>; both are off the charts. These extreme results indicate a failure due to very few live points. For illustrative purposes, they were not taken into consideration for the limits of the y-axes.)</p> Full article ">Figure 4
<p><b>Image segmentation for Models 1–6.</b> After applying the above-introduced formalism, estimated posterior parameters were used to obtain one pipeline-segmented image for Models 1–6. These images are overlays of the original input image (see <a href="#psf-09-00020-f001" class="html-fig">Figure 1</a>A). The pipeline-generated image is superimposed with dark red indicating a cell-free area and blue indicating a cell-covered area. The green line represents the boundary between a cell-free and a cell-covered area in the manually segmented image (see <a href="#psf-09-00020-f001" class="html-fig">Figure 1</a>D). The white box shows a region of interest, which is magnified in the lower part of the figure to magnify the details.</p> Full article ">
9 pages, 1337 KiB  
Proceeding Paper
Flow Annealed Kalman Inversion for Gradient-Free Inference in Bayesian Inverse Problems
by Richard D. P. Grumitt, Minas Karamanis and Uroš Seljak
Phys. Sci. Forum 2023, 9(1), 21; https://doi.org/10.3390/psf2023009021 - 4 Jan 2024
Viewed by 789
Abstract
For many scientific inverse problems, we are required to evaluate an expensive forward model. Moreover, the model is often given in such a form that it is unrealistic to access its gradients. In such a scenario, standard Markov Chain Monte Carlo algorithms quickly [...] Read more.
For many scientific inverse problems, we are required to evaluate an expensive forward model. Moreover, the model is often given in such a form that it is unrealistic to access its gradients. In such a scenario, standard Markov Chain Monte Carlo algorithms quickly become impractical, requiring a large number of serial model evaluations to converge on the target distribution. In this paper, we introduce Flow Annealed Kalman Inversion (FAKI). This is a generalization of Ensemble Kalman Inversion (EKI) where we embed the Kalman filter updates in a temperature annealing scheme and use normalizing flows (NFs) to map the intermediate measures corresponding to each temperature level to the standard Gaussian. Thus, we relax the Gaussian ansatz for the intermediate measures used in standard EKI, allowing us to achieve higher-fidelity approximations to non-Gaussian targets. We demonstrate the performance of FAKI on two numerical benchmarks, showing dramatic improvements over standard EKI in terms of accuracy whilst accelerating its already rapid convergence properties (typically in O(10) steps). Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Pair plots for the Rosenbrock target. Panel (<b>a</b>): pair-plot comparison of samples from EKI and a long HMC run. Panel (<b>b</b>): pair-plot comparison of samples from FAKI and a long HMC run. Samples from FAKI were able to correctly capture the highly non-linear target geometry. Standard EKI struggled to fill the tails of the target and required ∼100 iterations to converge, compared to ∼34 iterations for FAKI.</p> Full article ">Figure 2
<p>Comparison of first and second moment estimates along each dimension for the stochastic Lorenz system. Panel (<b>a</b>): comparison between the mean estimates from EKI and a long HMC run. Panel (<b>b</b>): comparison between the mean estimates from FAKI and a long HMC run. Panel (<b>c</b>): comparison between the standard deviation estimates from EKI and a long HMC run. Panel (<b>d</b>): comparison between the standard deviation estimates from FAKI and a long HMC run. Blue bars indicate the moment estimates obtained via HMC along each dimension, with the adjacent orange bars showing the estimates obtained through EKI/FAKI. EKI was unable to obtain accurate mean estimates for much of the <math display="inline"><semantics> <msub> <mi>Z</mi> <mi>t</mi> </msub> </semantics></math> trajectory, whilst FAKI was able to obtain accurate mean estimates for each dimension. FAKI outperformed EKI in its estimates of the marginal standard deviations, with EKI drastically overestimating the standard deviations along many dimensions.</p> Full article ">
8 pages, 2089 KiB  
Proceeding Paper
Nested Sampling—The Idea
by John Skilling
Phys. Sci. Forum 2023, 9(1), 22; https://doi.org/10.3390/psf2023009022 - 8 Jan 2024
Viewed by 996
Abstract
We seek to add up Q=∫fdX over unit volume in arbitrary dimension. Nested sampling locates the bulk of Q by geometrical compression, using a Monte Carlo ensemble constrained within a progressively more restrictive lower limit [...] Read more.
We seek to add up Q=∫fdX over unit volume in arbitrary dimension. Nested sampling locates the bulk of Q by geometrical compression, using a Monte Carlo ensemble constrained within a progressively more restrictive lower limit f≤f*. This domain is divided into a core f>f* and a shell f=f*, with the core kept adequately populated. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Sample ranked <span class="html-italic">r</span> out of <span class="html-italic">n</span> encloses about <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>/</mo> <mi>n</mi> </mrow> </semantics></math> of the volume.</p> Full article ">Figure 2
<p>(<b>left</b>) unit volume <span class="html-italic">V</span> modulated by <span class="html-italic">F</span>; (<b>right</b>) volume <math display="inline"><semantics> <mrow> <mi>X</mi> <mo stretchy="false">(</mo> <mi>f</mi> <mo stretchy="false">)</mo> </mrow> </semantics></math> covering <math display="inline"><semantics> <mrow> <mi>F</mi> <mo>⩾</mo> <mi>f</mi> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Nested sampling iterate with ensemble size 4.</p> Full article ">Figure 4
<p>Nested sampling trajectory.</p> Full article ">Figure 5
<p>Riemann and Lebesgue.</p> Full article ">Figure 6
<p>(<b>left</b>) set prior object by MC; (<b>right</b>) generate new object by MCMC.</p> Full article ">
10 pages, 743 KiB  
Proceeding Paper
Preconditioned Monte Carlo for Gradient-Free Bayesian Inference in the Physical Sciences
by Minas Karamanis and Uroš Seljak
Phys. Sci. Forum 2023, 9(1), 23; https://doi.org/10.3390/psf2023009023 - 9 Jan 2024
Cited by 1 | Viewed by 902
Abstract
We present preconditioned Monte Carlo (PMC), a novel Monte Carlo method for Bayesian inference in complex probability distributions. PMC incorporates a normalizing flow (NF) and an adaptive Sequential Monte Carlo (SMC) scheme, along with a novel past resampling scheme to boost the number [...] Read more.
We present preconditioned Monte Carlo (PMC), a novel Monte Carlo method for Bayesian inference in complex probability distributions. PMC incorporates a normalizing flow (NF) and an adaptive Sequential Monte Carlo (SMC) scheme, along with a novel past resampling scheme to boost the number of propagated particles without extra computational costs. Additionally, we utilize preconditioned Crank–Nicolson updates, enabling PMC to scale to higher dimensions without the gradient of target distribution. The efficacy of PMC in producing samples, estimating model evidence, and executing robust inference is showcased through two challenging case studies, highlighting its superior performance compared to conventional methods. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Two-dimensional marginal posteriors of 10-D Rosenbrock (<b>left</b>) and 61-D logistic regression with sonar data (<b>right</b>) as obtained using PMC-PR-p<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>CN (blue) and SMC-RWM (orange).</p> Full article ">
7 pages, 269 KiB  
Proceeding Paper
Analysis of Ecological Networks: Linear Inverse Modeling and Information Theory Tools
by Valérie Girardin, Théo Grente, Nathalie Niquil and Philippe Regnault
Phys. Sci. Forum 2023, 9(1), 24; https://doi.org/10.3390/psf2023009024 - 20 Feb 2024
Viewed by 955
Abstract
In marine ecology, the most studied interactions are trophic and are in networks called food webs. Trophic modeling is mainly based on weighted networks, where each weighted edge corresponds to a flow of organic matter between two trophic compartments, containing individuals of similar [...] Read more.
In marine ecology, the most studied interactions are trophic and are in networks called food webs. Trophic modeling is mainly based on weighted networks, where each weighted edge corresponds to a flow of organic matter between two trophic compartments, containing individuals of similar feeding behaviors and metabolisms and with the same predators. To take into account the unknown flow values within food webs, a class of methods called Linear Inverse Modeling was developed. The total linear constraints, equations and inequations defines a multidimensional convex-bounded polyhedron, called a polytope, within which lie all realistic solutions to the problem. To describe this polytope, a possible method is to calculate a representative sample of solutions by using the Monte Carlo Markov Chain approach. In order to extract a unique solution from the simulated sample, several goal (cost) functions—also called Ecological Network Analysis indices—have been introduced in the literature as criteria of fitness to the ecosystems. These tools are all related to information theory. Here we introduce new functions that potentially provide a better fit of the estimated model to the ecosystem. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
6 pages, 669 KiB  
Proceeding Paper
Manifold-Based Geometric Exploration of Optimization Solutions
by Guillaume Lebonvallet, Faicel Hnaien and Hichem Snoussi
Phys. Sci. Forum 2023, 9(1), 25; https://doi.org/10.3390/psf2023009025 - 16 May 2024
Viewed by 777
Abstract
This work introduces a new method for the exploration of solutions space in complex problems. This method consists of the build of a latent space which gives a new encoding of the solution space. We map the objective function on the latent space [...] Read more.
This work introduces a new method for the exploration of solutions space in complex problems. This method consists of the build of a latent space which gives a new encoding of the solution space. We map the objective function on the latent space using a manifold, i.e., a mathematical object defined by an equations system. The latent space is built with some knowledge of the objective function to make the mapping of the manifold easier. In this work, we introduce a new encoding for the Travelling Salesman Problem (TSP) and we give a new method for finding the optimal round. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Encoding tree: each intermediate node (round shape) represents a splitting of the solution space based on the use of a link (the left branch keeps the solutions using the link, the right branch the solutions without the link), the number in square brackets indicates the number of solutions; the leaf nodes (rectangle shape) represents the solutions.</p> Full article ">Figure 2
<p>Graph of the new encoding: the horizontal axis represents the rank of a solution (0 is the optimum) and the vertical axis the encoding of that solution.</p> Full article ">Figure 3
<p>Simulations of the new encoding on several TSP problems with 7 nodes.</p> Full article ">Figure 3 Cont.
<p>Simulations of the new encoding on several TSP problems with 7 nodes.</p> Full article ">
7 pages, 967 KiB  
Proceeding Paper
Nested Sampling for Detection and Localization of Sound Sources Using a Spherical Microphone Array
by Ning Xiang and Tomislav Jasa
Phys. Sci. Forum 2023, 9(1), 26; https://doi.org/10.3390/psf2023009026 - 20 May 2024
Viewed by 890
Abstract
Since its inception in 2004, nested sampling has been used in acoustics applications. This work applies nested sampling within a Bayesian framework to the detection and localization of sound sources using a spherical microphone array. Beyond an existing work, this source localization task [...] Read more.
Since its inception in 2004, nested sampling has been used in acoustics applications. This work applies nested sampling within a Bayesian framework to the detection and localization of sound sources using a spherical microphone array. Beyond an existing work, this source localization task relies on spherical harmonics to establish parametric models that distinguish the background sound environment from the presence of sound sources. Upon a positive detection, the parametric models are also involved to estimate an unknown number of potentially multiple sound sources. For the purpose of source detection, a no-source scenario needs to be considered in addition to the presence of at least one sound source. Specifically, the spherical microphone array senses the sound environment. The acoustic data are analyzed via spherical Fourier transforms using a Bayesian model comparison of two different models accounting for the absence and presence of sound sources for the source detection. Upon a positive detection, potentially multiple source models are involved to analyze direction of arrivals (DoAs) using Bayesian model selection and parameter estimation for the sound source enumeration and localization. These are two levels (enumeration and localization) of inferential estimations necessary to correctly localize potentially multiple sound sources. This paper discusses an efficient implementation of the nested sampling algorithm applied to the sound source detection and localization within the Bayesian framework. Full article
(This article belongs to the Proceedings of The 42nd International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Engineering)
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<p>Spherical microphone array of radius <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>3.5</mn> </mrow> </semantics></math> cm. Altogether, 32 microphones are nearly uniformly flush-mounted over the rigid spherical surface.</p> Full article ">Figure 2
<p>Beamforming superposition of two sound sources using a spherical order <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Comparison between the experimental data (<b>a</b>) processed according to Equation (<a href="#FD1-psf-09-00026" class="html-disp-formula">1</a>) with the prediction model (<b>b</b>) in Equation (<a href="#FD4-psf-09-00026" class="html-disp-formula">4</a>) of two sound sources using a spherical order <math display="inline"><semantics> <mrow> <mi>N</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Sound source detection based on Bayesian model comparison. Bayesian evidence is estimated using both ’no-source’ model <math display="inline"><semantics> <msub> <mi>M</mi> <mn>0</mn> </msub> </semantics></math> and one-source model <math display="inline"><semantics> <msub> <mi>M</mi> <mn>1</mn> </msub> </semantics></math>. The evidence is expressed in unit [decibans] in honor of Thomas Bayes [<a href="#B8-psf-09-00026" class="html-bibr">8</a>].</p> Full article ">Figure 5
<p>The sound source enumeration based on Bayes factor estimation. The Bayes factors are expressed in unit [decibans] in honor of Thomas Bayes [<a href="#B8-psf-09-00026" class="html-bibr">8</a>]. A two-source model is preferred by the Bayesian model selection. The evidence estimated using nested sampling also provides the posterior as a byproduct.</p> Full article ">
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