Abstract
In this paper, we consider two-layer Bayesian networks. The first layer consists of hidden (unobservable) variables and the second layer consists of observed variables. All variables are assumed to be binary. The variables in the second layer depend on the variables in the first layer. The dependence is characterised by conditional probability tables representing Noisy-AND or simple Noisy-AND. We will refer to this class of models as BN2A models. We found that the models known in the Bayesian network community as Noisy-AND and simple Noisy-AND are also used in the cognitive diagnostic modelling known in the psychometric community under the names of RRUM and DINA, respectively. In this domain, the hidden variables of BN2A models correspond to skills and the observed variables to students’ responses to test questions. In this paper we analyse the identifiability of these models. Identifiability is an important concept because without it we cannot hope to learn correct models. We present necessary conditions for the identifiability of BN2As with Noisy-AND models. We also propose and test a numerical approach for testing identifiability.
This work was supported by grants 22-11101S and 21-03658S of the Czech Science Foundation.
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Notes
- 1.
Symbol \(\textbf{x}_{pa(\ell )}\) denotes the subvector of \(\textbf{x}\) whose values corresponds to variables \(X_i, i \in pa(Y_{\ell })\).
- 2.
The set of exceptions has Lebesgue measure zero.
- 3.
We do not claim that this list is exclusive.
- 4.
We emphasize that there is no hope of getting correct results with finite-precision real arithmetic since, e.g., in one run, the absolute values of the computed determinants were in the interval \([10^{-37},10^{-72}]\) for this model.
References
D’Ambrosio, B.: Symbolic probabilistic inference in large BN2O networks. In: de Mantaras, R.L., Poole, D. (eds.) Uncertainty in Artificial Intelligence (UAI’94) Proceedings, pp. 128–135. Morgan Kaufmann, San Francisco (CA) (1994). https://doi.org/10.1016/B978-1-55860-332-5.50022-5
de la Torre, J.: The generalized DINA model framework. Psychometrika 76, 179–199 (2011)
Díez, F.J., Druzdzel, M.J.: Canonical probabilistic models for knowledge engineering. Technical report CISIAD-06-01, UNED, Madrid, Spain (2006)
Geiger, D., Heckerman, D., Meek, C.: Asymptotic model selection for directed networks with hidden variables. In: Proceedings of the Twelfth Conference on Uncertainty in Artificial Intelligence (UAI-96). pp. 283–290 (1996)
Gu, Y., Xu, G.: Sufficient and necessary conditions for the identifiability of the Q-matrix. Stat. Sin. 31, 449–472 (2021). https://doi.org/10.5705/ss.202018.0410
Halpern, Y., Sontag, D.: Unsupervised learning of noisy-or Bayesian networks. In: Proceedings of the Twenty-Ninth Conference on Uncertainty in Artificial Intelligence, pp. 272–281. UAI 2013, AUAI Press, Arlington, Virginia, USA (2013)
Hartz, S.M.: A Bayesian framework for the unified model for assessing cognitive abilities: Blending theory with practicality. Ph.D. thesis, University of Illinois at Urbana-Champaign (1996)
Heller, J.: Identifiability in probabilistic knowledge structures. J. Math. Psychol. 77, 46–57 (2017). https://doi.org/10.1016/j.jmp.2016.07.008
Henrion, M.: Some practical issues in constructing belief networks. In: Proceedings of the Third Conference on Uncertainty in Artificial Intelligence (UAI-87), pp. 161–173. Elsevier Science Publishers B.V. (North Holland) (1987)
Jensen, F.V., Nielsen, T.D.: Bayesian Networks and Decision Graphs. Information Science and Statistics, 2nd edn. Springer, New York (2007). https://doi.org/10.1007/978-0-387-68282-2
Junker, B.W., Sijtsma, K.: Cognitive assessment models with few assumptions, and connections with nonparametric item response theory. Appl. Psychol. Meas. 25, 258–272 (2001)
Koller, D., Friedman, N.: Probabilistic Graphical Models: Principles and Techniques. The MIT Press, Cambridge (2009)
Pearl, J.: Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann Publishers Inc., San Francisco (1988)
Sullivant, S.: Algebraic Statistics. American Mathematical Society, Providence (2018)
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Pérez, I., Vomlel, J. (2024). On Identifiability of BN2A Networks. In: Bouraoui, Z., Vesic, S. (eds) Symbolic and Quantitative Approaches to Reasoning with Uncertainty. ECSQARU 2023. Lecture Notes in Computer Science(), vol 14294. Springer, Cham. https://doi.org/10.1007/978-3-031-45608-4_11
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