A Bayesian nonparametric multi-sample test in any dimension

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Abstract

This paper considers a general Bayesian test for the multi-sample problem. Specifically, for M independent samples, the interest is to determine whether the M samples are generated from the same multivariate population. First, M Dirichlet processes are considered as priors for the true distributions generated the data. Then, the concentration of the distribution of the total distance between the M posterior processes is compared to the concentration of the distribution of the total distance between the M prior processes through the relative belief ratio. The total distance between processes is established based on the energy distance. Various interesting theoretical results of the approach are derived. Several examples covering the high dimensional case are considered to illustrate the approach.

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References

Abdelrazeq, I., Al-Labadi, L., Alzaatreh, A.: On one-sample Bayesian tests for the Mean. Statistics 54(2), 424–440 (2020)
Article MathSciNet Google Scholar
Al-Labadi, L.: The two-sample problem via relative belief ratio. Comput. Stat. (2020). https://doi.org/10.1007/s00180-020-00988-y
Article MathSciNet MATH Google Scholar
Al-Labadi, L., Baskurt, Z., Evans, M.: Goodness of fit for the logistic regression model using relative belief. J. Stat. Distrib. Appl. 4(1), 1 (2017)
Article Google Scholar
Al-Labadi, L., Baskurt, Z., Evans, M.: Statistical reasoning: choosing and checking the ingredients, inferences based on a measure of statistical evidence with some applications. Entropy 20(4), 289 (2018)
Article Google Scholar
Al-Labadi, L., Evans, M.: Prior-based model checking. Can. J. Stat. 46(3), 380–398 (2018)
Article MathSciNet Google Scholar
Al-Labadi, L., Fazeli Asl, F., Saberi, Z.: A Bayesian semiparametric Gaussian copula approach to a multivariate normality test. J. Stat. Comput. Simul. (2020). https://doi.org/10.1080/00949655.2020.1820504
Article MATH Google Scholar
Al-Labadi, L., Zarepour, M.: Two-sample Kolmogorov-Smirnov test using a Bayesian nonparametric approach. Math. Methods Stat. 26, 212–225 (2017)
Article MathSciNet Google Scholar
Baringhaus, L., Franz, C.: On a new multivariate two-sample test. J. Multivar. Anal. 88, 190–206 (2004)
Article MathSciNet Google Scholar
Bickel, P.J., Breiman, L.: Sums of functions of nearest neighbor distances, moment bounds, limit theorems and a goodness of fit test. Ann. Probab 11, 185–214 (1983)
Article MathSciNet Google Scholar
Biswas, M., Ghosh, A.K.: A nonparametric two-sample test applicable to high dimensional data. J. Multivar. Anal. 123, 160–171 (2014)
Article MathSciNet Google Scholar
Chen, Y., Hanson, T.: Bayesian nonparametric k-sample tests for censored and uncensored data. Comput. Stat. Data Anal. 71, 335–346 (2014)
Article MathSciNet Google Scholar
Evans, M. (2015). Measuring Statistical Evidence Using Relative Belief. volume 144 of Monographs on Statistics and Applied Probability. CRC Press, Boca Raton, FL
Fehrman, E., Muhammad, A.K., Mirkes, E.M., Egan, V., Gorban, A.N.: The five factor model of personality and evaluation of drug consumption risk. Data Sci. (2017). https://doi.org/10.1007/978-3-319-55723-6_18
Article Google Scholar
Ferguson, T.S.: A Bayesian analysis of some nonparametric problems. Ann. Stat. 1, 209–230 (1973)
Article MathSciNet Google Scholar
Friedman, J.H., Rafsky, L.C.: Multivariate generalizations of the Wald-Wolfowitz and Smirnov two sample tests. Ann. Stat. 7, 697–717 (1979)
Article MathSciNet Google Scholar
Heller, R., Jensen, S.T., Rosenbaum, P.R., Small, D.S.: Sensitivity analysis for the cross-match test, with applications in genomics. J. Am. Stat. Assoc. 105, 1005–1013 (2010)
Article MathSciNet Google Scholar
Henze, N.: A multivariate two-sample test based on the number of nearest neighbor type coincidences. Ann. Stat. 16, 772–783 (1988)
Article MathSciNet Google Scholar
Holmes, C.C., Caron, F., Griffin, J.E., Stephens, D.A.: Two-sample Bayesian nonparametric hypothesis testing. Bayesian Anal. 2, 297–320 (2015)
MathSciNet MATH Google Scholar
Ishwaran, H., Zarepour, M.: Exact and approximate sum representations for the Dirichlet process. Can. J. Stat. 30, 269–283 (2003)
Article MathSciNet Google Scholar
Kuipers, J.B.: Quaternions and rotation sequences. Princeton University Press, Princeton (1999)
Book Google Scholar
Mukherjee, S., Agarwal, D., Zhang, N.R., Bhattacharya, B.B.: Distribution-free multisample tests based on optimal matchings with applications to single cell genomics. J. Am. Stat. Assoc. 18, 1–12 (2020)
Article Google Scholar
Mukhopadhyay, S., Wang, K.: A nonparametric approach to high-dimensional $k$-sample comparison problems. Biometrica (2020). https://doi.org/10.1093/biomet/asaa015
Article MATH Google Scholar
Oja, H.: Multivariate nonparametric methods with R: an approach based on spatial signs and ranks. Springer, New York (2010)
Book Google Scholar
Oja, H., Randles, R.H.: Multivariate nonparametric tests. Stat. Sci. 19, 598–605 (2004)
Article MathSciNet Google Scholar
Petrie, A.: Graph-theoretic multisample tests of equality in distribution for high dimensional data. Comput. Stat. Data Anal. 96, 145–158 (2016)
Article MathSciNet Google Scholar
Rosenbaum, P.R.: An exact distribution-free test comparing two multivariate distributions based on adjacency. J. R. Stat. Soc. Ser. B Stat. Methodol. 67, 515–530 (2005)
Article MathSciNet Google Scholar
Schilling, M.F.: Multivariate two-sample tests based on nearest neighbors. J. Am. Stat. Assoc. 81, 799–806 (1986)
Article MathSciNet Google Scholar
Székely, G.: E-statistics: Energy of statistical samples, pp. 03–05. Bowling Green State University, Department of Mathematics and Statistics Technical Report No (2003)
Székely, G. and Rizzo, M. (2004). Testing for equal distributions in high dimension. Interstat
Székely, G., Rizzo, M.: The energy of data. Ann. Rev. Stat. Appl. 4(1), 447–479 (2017)
Article Google Scholar
Tsukada, S.: High dimensional two-sample test based on the inter-point distance. Comput. Stat. 34, 599–615 (2019)
Article MathSciNet Google Scholar
Zhang, Q., Filippi, S., Flaxman, S., Sejdinovic, D.: Bayesian kernel two-sample testing. Technical Report (2020). arXiv:2002.05550
Weinstein, J.N., Collisson, E.A., Mills, G.B., Shaw, K.R., Ozenberger, B.A., et al.: The cancer genome atlas pan-cancer analysis project. Nat. Gen. 45(10), 1113–1120 (2013)
Article Google Scholar

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Authors and Affiliations

Department of Mathematical and Computational Sciences, University of Toronto Mississauga, Mississauga, Ontario, L5L 1C6, Canada
Luai Al-Labadi
Department of Mathematical Sciences, Isfahan University of Technology, Isfahan, 84156-83111, Iran
Forough Fazeli Asl & Zahra Saberi

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Luai Al-Labadi
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Zahra Saberi
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Appendices

Appendix A proof of lemma 1

To prove (i), from properties of Dirichlet distribution, $E_{P^{(1)}_N}(J^{(1)}_{i,N})=E_{P^{(2)}_N}(J^{(2)}_{i,N})=1/N$ and $E_{P^{(1)}_{N}}(J^{(1)}_{i,N}J^{(1)}_{j,N})=E_{P^{(2)}_{N}}(J^{(2)}_{i,N}J^{(2)}_{j,N})=\frac{a}{(a+1)N^2}$. Since $J^{(1)}_{i,N}$ and $J^{(2)}_{i,N}$ are independent, we have

$$\begin{aligned} E_{P^{(1)}_{N},P^{(2)}_{N}}\left( d_{{\mathcal {E}},a,N}(F^{pri}_{1},F^{pri}_{2})\right)&=\dfrac{2}{N^{2}}\sum _{i,j=1}^{N}||\mathbf {X}^{(1)}_i-\mathbf {X}^{(2)}_j||-\dfrac{a}{(a+1)N^2}\sum _{i,j=1}^{N}||\mathbf {X}^{(1)}_i-\mathbf {X}^{(1)}_j||\nonumber \\&\quad -\dfrac{a}{(a+1)N^2}\sum _{i,j=1}^{N}||\mathbf {X}^{(2)}_i-\mathbf {X}^{(2)}_j||. \end{aligned}$$

(17)

The proof is immediately followed by letting $a\rightarrow \infty $ in (17). To prove (ii), For any $a>0$, we have

$$\begin{aligned} E\left( d_{{\mathcal {E}},a,N}(F^{pri}_{1},F^{pri}_{2})\right)&=E_{G_{1},G_{2}}\big \lbrace E_{P^{(1)}_{N},P^{(2)}_{N}}\big (d_{{\mathcal {E}},a,N}(F^{pri}_{1},F^{pri}_{2})\big )\big \rbrace \nonumber \\&=E_{G_{1},G_{2}}\big \lbrace \dfrac{2}{N^{2}}\sum _{i,j=1}^{N}||\mathbf {X}^{(1)}_i-\mathbf {X}^{(2)}_j||-\dfrac{a}{(a+1)N^2}\sum _{i,j=1}^{N}||\mathbf {X}^{(1)}_i-\mathbf {X}^{(1)}_j||\nonumber \\& \quad -\dfrac{a}{(a+1)N^2}\sum _{i,j=1}^{N}||\mathbf {X}^{(2)}_i-\mathbf {X}^{(2)}_j||.\big \rbrace \end{aligned}$$

(18)

Now, to compute the limit of (18) when $a\rightarrow \infty $, we use the monotone convergence theorem as below. For this, let

$$\begin{aligned} h(a)&=\dfrac{2}{N^{2}}\sum _{i,j=1}^{N}||\mathbf {X}^{(1)}_i-\mathbf {X}^{(2)}_j||-\dfrac{a}{(a+1)N^2}\sum _{i,j=1}^{N}||\mathbf {X}^{(1)}_i-\mathbf {X}^{(1)}_j||\nonumber \\&\quad -\dfrac{a}{(a+1)N^2}\sum _{i,j=1}^{N}||\mathbf {X}^{(2)}_i-\mathbf {X}^{(2)}_j||. \end{aligned}$$

(19)

Note that, since $\dfrac{a}{(a+1)N^2}<\dfrac{1}{N^2}$, we have

$$\begin{aligned} h(a)&>\dfrac{2}{N^{2}}\sum _{i,j=1}^{N}||\mathbf {X}^{(1)}_i-\mathbf {X}^{(2)}_j||-\dfrac{1}{N^2}\sum _{i,j=1}^{N}||\mathbf {X}^{(1)}_i-\mathbf {X}^{(1)}_j|| -\dfrac{1}{N^2}\sum _{i,j=1}^{N}||\mathbf {X}^{(2)}_i-\mathbf {X}^{(2)}_j||. \end{aligned}$$

(20)

Recalling equation (8), the right-hand side of (20) is $d_{{\mathcal {E}},N}(G_{1},G_{2})$. Since $d_{{\mathcal {E}},N}(G_{1},G_{2})\ge 0$, we have $h(a)>0$ (h(a) is non-negative). On the other hand, since $\frac{a}{a+1}<\frac{a+1}{a+2}$, we get $h(a+1)<h(a)$ (h(a) is monotone). Also, since $\dfrac{a}{(a+1)N^2}\rightarrow \dfrac{1}{N^2}$ as $a\rightarrow \infty $, by letting $a\rightarrow \infty $ in (19), we have $h(a)\rightarrow d_{{\mathcal {E}},N}(G_{1},G_{2})$ as $a\rightarrow \infty $. Hence, the monotone convergence theorem implies

$$\begin{aligned} E_{G_{1},G_{2}}(h(a))\rightarrow E_{G_{1},G_{2}}(d_{{\mathcal {E}},N}(G_{1},G_{2})), \end{aligned}$$

as $a\rightarrow \infty $. From Székely and Rizzo Székely and Rizzo (2004), we have

$$\begin{aligned} E_{G_{1},G_{2}}(d_{{\mathcal {E}},N}(G_{1},G_{2}))&=2E||\mathbf {X}^{(1)}-\mathbf {X}^{(2)}||-E||\mathbf {X}^{(1)}-\mathbf {X}^{\prime ^{(1)}}||-E||\mathbf {X}^{(2)}-\mathbf {X}^{\prime ^{(2)}}||\nonumber \\&\quad +\dfrac{E||\mathbf {X}^{(1)}-\mathbf {X}^{\prime ^{(1)}}||+E||\mathbf {X}^{(2)}-\mathbf {X}^{\prime ^{(2)}}||}{N}, \end{aligned}$$

(21)

where $\mathbf {X}^{(1)},\mathbf {X}^{\prime ^{(1)}}\overset{i.i.d.}{\sim } G_{1}$ and $\mathbf {X}^{(2)},\mathbf {X}^{\prime ^{(2)}}\overset{i.i.d.}{\sim } G_{2}$. By letting $N\rightarrow \infty $ in (21), we have

$$\begin{aligned} E_{G_{1},G_{2}}(d_{{\mathcal {E}},N}(G_{1},G_{2}))\rightarrow 2E||\mathbf {X}^{(1)}-\mathbf {X}^{(2)}||-E||\mathbf {X}^{(1)}-\mathbf {X}^{\prime ^{(1)}}||-E||\mathbf {X}^{(2)}-\mathbf {X}^{\prime ^{(2)}}||. \end{aligned}$$

(22)

Recalling Eq. (7), the right-hand side of (22) is $d_{{\mathcal {E}}}(G_{1},G_{2})$. This completes the proof of (ii).

Appendix B proof of theorem 1

By considering the conjugacy property of Dirichlet process in the proof of part (ii) of Lemma 1, we have $E\left( d_{{\mathcal {E}},a,N}(F^{pos}_{1},F^{pos}_{2})\right) \rightarrow E_{G^*_1,G^*_2}\left( d_{{\mathcal {E}},N}(G^*_1,G^*_2)\right) $ as $n_1,n_2\rightarrow \infty $. On the other hand, by Glivenko-Cantelli theorem, when $n_1,n_2\rightarrow \infty $, $G^{*}_1$ and $G^{*}_2$ converge almost surely to the true distribution of data $F_1$ and $F_2$. Thus, $E\left( d_{{\mathcal {E}},a,N}(F^{pos}_{1},F^{pos}_{2})\right) \xrightarrow {a.s.} E_{F_{1},F_{2}}(d_{{\mathcal {E}},N}(F_1,F_2))$ as $n_1,n_2\rightarrow \infty $. Similar to the proof of part (ii) of Lemma 1, as $N\rightarrow \infty $, $E_{F_{1},F_{2}}(d_{{\mathcal {E}},N}(F_{1},F_{2}))\rightarrow d_{{\mathcal {E}}}(F_{1},F_{2})$. To prove (ii), we have $E\left( d_{{\mathcal {E}},a,N}(F^{pos}_{1},F^{pos}_{2})\right) $ $\rightarrow E_{G^*_1,G^*_2}\left( d_{{\mathcal {E}},N}(G^*_1,G^*_2)\right) $ as $a\rightarrow \infty $. Also, from (2), by letting $a\rightarrow \infty $, $G^{*}_1$ and $G^{*}_2$ converge to $G_1$ and $G_2$, respectively. Hence, $E\left( d_{{\mathcal {E}},a,N}(F^{pos}_{1},F^{pos}_{2})\right) \rightarrow E_{G_{1},G_{2}}(d_{{\mathcal {E}},N}(G_1,G_2))$ as $a\rightarrow \infty $. Now, similar to the proof of (i), the result follows as $N\rightarrow \infty $.

Appendix C proof of proposition 1

From the linear property of the expectation, we have

$$\begin{aligned} E\left( d_{T,N}(F^{pri}_1,\ldots , F^{pri}_M)\right) =\sum _{1\le i<j \le M}E\left( d_{{\mathcal {E}},a,N}(F^{pri}_{i},F^{pri}_{j})\right) , \end{aligned}$$

(23)

$$\begin{aligned} E\Big (d_{T,N}(F^{pos}_1,\ldots , F^{pos}_M)\Big )=\sum _{1\le i<j \le M}E\left( d_{{\mathcal {E}},a,N}(F^{pos}_{i},F^{pos}_{j})\right) . \end{aligned}$$

(24)

By using Lemma 1, part (ii), and Theorem 1 in (23) and (24) the proof is concluded.

Appendix D proof of corollary 1

To prove (i), from part (i) of Proposition 1, we have $E(d_{T,N}(F^{pri}_1,\ldots , F^{pri}_M))\rightarrow d_{T}(G_1,\ldots , G_M)$, as $a\rightarrow \infty $ and $N\rightarrow \infty $. Now, from Székely and Rizzo Székely and Rizzo (2004), $d_{T}(G_1,\ldots , G_M)=0$ if and only if $G_1=\cdots =G_M$ and the proof is completed. The proof of (ii) and (iii) are similar to the proof of (i) and are omitted.

Appendix E pseudocode algorithm of the multi-sample test

Appendix F relevant notations

Table 9 Description of notations

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Al-Labadi, L., Asl, F.F. & Saberi, Z. A Bayesian nonparametric multi-sample test in any dimension. AStA Adv Stat Anal 106, 217–242 (2022). https://doi.org/10.1007/s10182-021-00419-3

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Received: 18 December 2020
Accepted: 10 September 2021
Published: 28 September 2021
Issue Date: June 2022
DOI: https://doi.org/10.1007/s10182-021-00419-3