Bayesian inference for latent chain graphs

Deng Lu; Maria De Iorio; Ajay Jasra; Gary L. Rosner

doi:10.3934/fods.2020003

Article Contents

2020, Volume 2, Issue 1: 35-54. Doi: 10.3934/fods.2020003

This issue Previous Article Semi-supervised classification on graphs using explicit diffusion dynamics Next Article Bayesian inference of chaotic dynamics by merging data assimilation, machine learning and expectation-maximization

Bayesian inference for latent chain graphs

1.
Department of Statistics & Applied Probability, National University of Singapore, Singapore, 117546, SG
2.
Yale-NUS, Singapore, 138527, SG. & , Department of Statistical Science, University College London, UK
3.
Computer, Electrical and Mathematical Sciences and Engineering Division, King Abdullah University of Science and Technology, Thuwal, 23955, KSA
4.
Oncology Biostatistics and Bioinformatics, Sidney Kimmel Comprehensive Cancer Center, Johns Hopkins University, Baltimore, MD 21205, USA

Published: March 2020

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Abstract

In this article we consider Bayesian inference for partially observed Andersson-Madigan-Perlman (AMP) Gaussian chain graph (CG) models. Such models are of particular interest in applications such as biological networks and financial time series. The model itself features a variety of constraints which make both prior modeling and computational inference challenging. We develop a framework for the aforementioned challenges, using a sequential Monte Carlo (SMC) method for statistical inference. Our approach is illustrated on both simulated data as well as real case studies from university graduation rates and a pharmacokinetics study.

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Mathematics Subject Classification: 62F15.

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Figure 1. Simulation results for the independent case: (a) ESS in each SMC step; (b) plot of $ \Omega[1,1] $ across on particles; (c) acceptance rates in each SMC step; (d) distribution of the log(target) (i.e., log of $ \pi(B,\Omega,(a_{ij})_{i<j}\mid y_{1:m},\alpha) $) at the end of the algorithm

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Figure 2. Chain Graph Estimate presented in [12]

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Figure 3. Empirical Graph

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Figure 4. posterior estimated chain graph using a Dirichlet prior with $ \alpha = (0.39 , 0.25 , 0.36 , 0.05) $

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Figure 5. posterior estimated chain graph using a Dirichlet prior with $ \alpha = (1 , 1 , 1 , 1) $

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Figure 6. posterior estimated chain graph using a Dirichlet prior with $ \alpha = (1 , 3 , 3 , 3) $

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Figure 7. Chain graph with highest posterior probability

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Table 1. Posterior probability $ \mathbb{P} (a_{ij} = 0 | y_{1:m},\alpha), 1\leq i < j \leq p $

Nodes	$ 2 $	$ 3 $	$ 4 $	$ 5 $	$ 6 $	$ 7 $	$ 8 $	$ 9 $	$ 10 $
$ \; \; 1 $	0.882	0.892	0.920	0.932	0.920	0.870	0.874	0.908	0.934
$ \; \; 2 $		0.902	0.906	0.828	0.898	0.934	0.784	0.804	0.906
$ \; \; 3 $			0.914	0.890	0.880	0.890	0.908	0.900	0.890
$ \; \; 4 $				0.900	0.866	0.882	0.770	0.918	0.900
$ \; \; 5 $					0.788	0.952	0.912	0.830	0.918
$ \; \; 6 $						0.806	0.932	0.906	0.908
$ \; \; 7 $							0.904	0.738	0.916
$ \; \; 8 $								0.918	0.740
$ \; \; 9 $									0.952

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Table 2. The adjacency matrix corresponding to chain graph in Figure 2

	strat	spend	salar	top10	tstsc	rejr	pacc	apgra
strat	0	1	1	2	0	0	0	0
spend	1	0	1	2	2	2	0	0
salar	1	1	0	0	2	2	2	2
top10	3	3	0	0	1	0	1	0
tstsc	0	3	3	1	0	1	0	2
rejr	0	3	3	0	1	0	1	0
pacc	0	0	3	1	0	1	0	2
apgra	0	0	3	0	3	0	3	0

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Table 3. The adjacency matrix corresponding to the chain graph in Figure 3

	strat	spend	salar	top10	tstsc	rejr	pacc	apgra
strat	0	1	2	2	2	2	2	2
spend	1	0	1	2	2	2	2	2
salar	3	1	0	1	2	2	3	2
top10	3	3	1	0	2	1	0	2
tstsc	3	3	3	3	0	1	0	2
rejr	3	3	3	1	1	0	3	0
pacc	3	3	2	0	0	2	0	2
apgra	3	3	3	3	3	0	3	0

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Table 4. Summaries of different chain graphs using package SEM

Base chain graph				Chain graph selected by SIN
Edge	p-value	Edge	p-value	Edge	p-value
strat — spend	1.630e-14	pacc $ \rightarrow $ salar	1.137e-06	strat — spend	1.630e-14
strat $ \rightarrow $ salar	1.382e-06	pacc $ \rightarrow $ rejr	1.470e-03	strat — salar	1.082e-05
spend — salar	2.629e-11	strat $ \rightarrow $ apgra	8.237e-02	strat $ \rightarrow $ top10	1.935e-09
strat $ \rightarrow $ top10	4.743e-07	spend $ \rightarrow $ apgra	6.067e-02	spend — salar	7.156e-13
spend $ \rightarrow $ top10	2.822e-28	salar $ \rightarrow $ apgra	4.794e-03	spend $ \rightarrow $ top10	5.979e-34
top10 — salar	8.931e-03	top10 $ \rightarrow $ apgra	4.253e-01	spend $ \rightarrow $ tstsc	3.995e-12
strat $ \rightarrow $ tstsc	3.140e-03	tstsc $ \rightarrow $ apgra	1.096e-10	spend $ \rightarrow $ rejr	2.909e-03
spend $ \rightarrow $ tstsc	6.634e-01	pacc $ \rightarrow $ apgra	2.711e-03	salar $ \rightarrow $ tstsc	2.350e-05
salar $ \rightarrow $ tstsc	2.008e-04			salar $ \rightarrow $ rejr	1.323e-03
top10 $ \rightarrow $ tstsc	6.831e-19			salar $ \rightarrow $ pacc	1.827e-14
strat $ \rightarrow $ rejr	1.954e-01			salar $ \rightarrow $ apgra	1.570e-02
spend $ \rightarrow $ rejr	3.621e-03			top10 — tstsc	1.256e-09
salar $ \rightarrow $ rejr	2.575e-04			top10 — pacc	5.020e-01
top10 — rejr	1.816e-04			tstsc — rejr	8.297e-03
tstsc — rejr	1.003e-02			tstsc $ \rightarrow $ apgra	8.352e-19
strat $ \rightarrow $ pacc	2.585e-02			rejr — pacc	5.617e-03
spend $ \rightarrow $ pacc	4.109e-07			pacc $ \rightarrow $ apgra	5.481e-03
AIC		BIC		AIC	BIC
67.887		-13.319		80.838	-24.919
Chain graph selected by algorithm		Chain graph selected by algorithm		Chain graph selected by algorithm
($ \alpha=(0.39 , 0.25 , 0.36 , 0.05) $)		($ \alpha=(1,1,1,1) $)		($ \alpha=(1,3,3,3) $)
Edge	p-value	Edge	p-value	Edge	p-value
strat — spend	1.630e-14	spend $ \rightarrow $ strat	2.150e-52	spend $ \rightarrow $ strat	1.536e-42
strat $ \rightarrow $ salar	1.727e-06	strat $ \rightarrow $ salar	9.127e-07	salar $ \rightarrow $ strat	4.952e-06
strat $ \rightarrow $ top10	3.597e-09	spend $ \rightarrow $ salar	2.484e-24	strat $ \rightarrow $ top10	1.636e-07
spend $ \rightarrow $ salar	4.956e-33	top10 $ \rightarrow $ spend	2.068e-25	tstsc $ \rightarrow $ strat	8.237e-01
spend $ \rightarrow $ top10	1.086e-33	salar $ \rightarrow $ pacc	7.304e-10	salar $ \rightarrow $ spend	9.659e-11
spend $ \rightarrow $ tstsc	4.059e-12	apgra $ \rightarrow $ salar	3.751e-11	spend $ \rightarrow $ top10	3.881e-10
salar $ \rightarrow $ tstsc	5.524e-05	top10 — tstsc	2.163e-14	tstsc $ \rightarrow $ spend	2.886e-08
salar $ \rightarrow $ pacc	1.012e-18	top10 $ \rightarrow $ rejr	2.504e-03	tstsc $ \rightarrow $ salar	3.654e-27
salar $ \rightarrow $ apgra	3.201e-05	tstsc $ \rightarrow $ rejr	2.847e-04	salar $ \rightarrow $ rejr	8.844e-02
top10 — tstsc	2.295e-10	tstsc $ \rightarrow $ apgra	1.428e-43	salar $ \rightarrow $ pacc	1.062e-08
top10 $ \rightarrow $ rejr	1.951e-03	rejr $ \rightarrow $ pacc	1.480e-04	salar $ \rightarrow $ apgra	1.036e-04
tstsc $ \rightarrow $ rejr	2.348e-04	apgra $ \rightarrow $ pacc	3.587e-03	tstsc $ \rightarrow $ top10	1.033e-20
tstsc $ \rightarrow $ apgra	1.367e-18			top10 $ \rightarrow $ rejr	9.160e-03
rejr $ \rightarrow $ pacc	1.032e-03			tstsc $ \rightarrow $ rejr	5.443e-03
pacc — apgra	5.315e-03			tstsc $ \rightarrow $ apgra	2.644e-17
				rejr $ \rightarrow $ pacc	3.048e-04
				apgra $ \rightarrow $ pacc	5.759e-03
AIC	BIC	AIC	BIC	AIC	BIC
61.372	-50.522	102.93	-18.169	58.401	-47.356

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Table 5. Tissues and cell types examined in the PK studies

Compound	Compartment	Notation
TFV	Blood plasma	TFV$ _{plasma} $
TFV	Rectal biopsy tissue	TFV$ _{tissue} $
TFV	Rectal fluid	TFV$ _{rectal} $
TFVdp	Rectal biopsy tissue	TFVdp$ _{tissue} $
TFVdp	Total mononuclear cells in rectal tissue	Total$ _{\text{MMC}} $
TFVdp	CD4$ ^+ $ lymphocytes from MMC	CD4$ ^+_{\text{MMC}} $
TFVdp	CD4$ ^- $ lymphocytes from MMC	CD4$ ^-_{\text{MMC}} $

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Table 6. Summaries of two different chain graphs using package SEM

Chain graph selected by algorithm		Chain graph selected by algorithm
($ \alpha=(1,1,1,1) $)		($ \alpha=(1,3,3,3) $)
Edge	p-value	Edge	p-value
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ Total$ _{\text{MMC}} $	9.881e-19	CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ TFVdp$ _{tissue} $	5.687e-01
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ CD4$ ^+_{\text{MMC}} $	4.091e-81	CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ CD4$ ^-_{\text{MMC}} $	2.941e-46
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ TFV$ _{rectal} $	1.496e-01	CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ TFV$ _{plasma} $	1.210e-02
Total$ _{\text{MMC}} $ — TFVdp$ _{tissue} $	1.589e-02	CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ Total$ _{\text{MMC}} $	7.028e-10
TFVdp$ _{tissue} $ — CD4$ ^+_{\text{MMC}} $	1.812e-02	CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ TFV$ _{rectal} $	1.874e-03
CD4$ ^+_{\text{MMC}} $ — TFV$ _{rectal} $	8.583e-03	TFVdp$ _{tissue} $ — CD4$ ^-_{\text{MMC}} $	8.477e-02
Total$ _{\text{MMC}} $ $ \rightarrow $ TFV$ _{plasma} $	1.815e-03	TFVdp$ _{tissue} $ $ \rightarrow $ TFV$ _{rectal} $	2.991e-01
CD4$ ^+_{\text{MMC}} $ $ \rightarrow $ TFV$ _{tissue} $	7.043e-01	TFVdp$ _{tissue} $ $ \rightarrow $ TFV$ _{plasma} $	2.584e-01
		TFVdp$ _{tissue} $ $ \rightarrow $ Total$ _{\text{MMC}} $	1.162e-13
		CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ Total$ _{\text{MMC}} $	2.352e-22
		TFV$ _{plasma} $ $ \rightarrow $ TFV$ _{tissue} $	4.259e-01
		TFV$ _{plasma} $ — Total$ _{\text{MMC}} $	5.719e-02
		TFV$ _{tissue} $ $ \rightarrow $ TFV$ _{rectal} $	7.550e-01
		Total$ _{\text{MMC}} $ $ \rightarrow $ TFV$ _{rectal} $	2.337e-02
AIC	BIC	AIC	BIC
Inf	Inf	46.875	-11.909

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Table 7. Summaries of modification indices for the model corresponding to the chain graph obtained under prior with $ \alpha = (1,3,3,3) $

5 largest modification indices, A matrix		5 largest modification indices, P matrix
(regression coefficients)		(variances/covariances)
TFV$ _{rectal} $ $ \rightarrow $ Total$ _{\text{MMC}} $	3.281	TFV$ _{rectal} $ — Total$ _{\text{MMC}} $	3.394
TFV$ _{rectal} $ $ \rightarrow $ TFV$ _{plasma} $	2.219	TFV$ _{rectal} $ — TFV$ _{plasma} $	2.881
CD4$ ^-_{\text{MMC}} $ $ \rightarrow $ TFV$ _{rectal} $	0.709	TFV$ _{rectal} $ — CD4$ ^-_{\text{MMC}} $	0.709
TFV$ _{rectal} $ $ \rightarrow $ TFVdp$ _{tissue} $	0.654	TFV$ _{rectal} $ — TFVdp$ _{tissue} $	0.709
TFV$ _{rectal} $ $ \rightarrow $ CD4$ ^-_{\text{MMC}} $	0.555	TFV$ _{tissue} $ — TFVdp$ _{tissue} $	0.389

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