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A Bayesian nonparametric test for conditional independence

Supported by EPSRC grant EP/R013519/1
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  • This article introduces a Bayesian nonparametric method for quantifying the relative evidence in a dataset in favour of the dependence or independence of two variables conditional on a third. The approach uses Pólya tree priors on spaces of conditional probability densities, accounting for uncertainty in the form of the underlying distributions in a nonparametric way. The Bayesian perspective provides an inherently symmetric probability measure of conditional dependence or independence, a feature particularly advantageous in causal discovery and not employed in existing procedures of this type.

    Mathematics Subject Classification: Primary: 62-08; Secondary: 62G10.


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  • Figure 1.  Construction of a Pólya tree distribution on $ \Omega = [0,1] $. From each set $ C_\ast $, a particle of probability mass passes to the left with (random) probability $ \theta_{\ast0} $ and to the right with probability $ \theta_{\ast1} = 1-\theta_{\ast0} $, with all $ \theta_\ast $ being independently Beta-distributed as described in the main text

    Figure 2.  Pseudocode for the proposed Bayesian nonparametric test for conditional independence

    Figure 3.  Application of the proposed Bayesian testing procedure to four synthetic datasets supported on $ [0,1]^3 $, chosen such that all combinations of unconditional and conditional dependence/independence are represented. The final column gives the ensemble of probabilities of conditional dependence $ p(H_1|W) $ output by the test over 100 repetitions at varying values of data size $ N $, with the blue line representing the median, and the dark and light shaded regions representing the (25, 75)-percentile and (5, 95)-percentile ranges respectively

    Figure 4.  Marginal scatter plots from the CalCOFI Bottle dataset showing the pairwise relationships between $\texttt{Salnty}$, $\texttt{Oxy_µmol.Kg}$ and $\texttt{T_degC}$. The nonlinear nature of the dependences is immediately apparent

    Figure 5.  Example pairwise dependence graphs output by the Bayesian conditional independence test for five variables from the CalCOFI dataset, conditional on $\texttt{T_degC}$, for four different sizes of subsample drawn from the complete dataset. The numbers associated with each edge are the posterior probabilities of conditional dependence $ p(H_1|W^{(N)}) $ and are given to two decimal places; where no edge is shown, this indicates $ p(H_1|W^{(N)})<0.005 $

    Figure 6.  Box-plots giving the output posterior probability of conditional dependence $ p(H_1|W^{(N)}) $ for 100 repetitions of the Bayesian conditional independence test applied to randomly-drawn subsamples of various sizes $ N $ from the CalCOFI dataset. The left-hand plot gives a representative example of a pair of variables conditionally dependent given $\texttt{T_degC}$, while the right-hand plot gives a representative conditionally independent pair

    Figure 7.  Top left: Heat map of conditional marginal likelihood values for the three constituent models over $ \Omega_X $, $ \Omega_Y $ and $ \Omega_XY $ for the second and third models of Figure 3. Top right: 'Slices' from this heatmap with $ \rho = 0.5 $. Bottom: Test outputs for 100 repetitions of the second and third models of Figure 3. Red plots fix $ c = 1 $ (output identical to Figure 3), while the blue plots use the optimising values $ \hat{c} $ from the plot above

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