Paper: Nov 20,2019
stat.CO
ID:1911.08725
Assessment and adjustment of approximate inference algorithms using the law of total variance
A common method for assessing validity of Bayesian sampling or approximate
inference methods makes use of simulated data replicates for parameters drawn
from the prior. Under continuity assumptions, quantiles of functions of the
simulated parameter values within corresponding posterior distributions are
uniformly distributed. Checking for uniformity when a posterior density is
approximated numerically provides a diagnostic for algorithm validity.
Furthermore, adjustments to achieve uniformity can improve the quality of
approximate inference methods. A weakness of this general approach is that it
seems difficult to extend beyond scalar functions of interest. The present
article develops an alternative to quantile-based checking and adjustment
methods which is inherently multivariate. The new approach is based on use of
the tower property of conditional expectation and the law of total variance for
relating prior and posterior expectations and covariances. For adjustment,
approximate inferences are modified so that the correct prior to posterior
relationships hold. We illustrate the method in three examples. The first uses
an auxiliary model in a likelihood-free inference problem. The second considers
corrections for variational Bayes approximations in a deep neural network
generalized linear mixed model. Our final application considers a deep neural
network surrogate for approximating Gaussian process regression predictive
inference.
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Paper Author: Xuejun Yu,David J. Nott,Minh-Ngoc Tran,Nadja Klein
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