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Bayesian inference for latent chain graphs

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  • 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.

    Mathematics Subject Classification: 62F15.

    Citation:

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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

    Figure 2.  Chain Graph Estimate presented in [12]

    Figure 3.  Empirical Graph

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

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

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

    Figure 7.  Chain graph with highest posterior probability

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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}} $
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV

    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
     | Show Table
    DownLoad: CSV
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