MCMC-based local parametric sensitivity estimations

Robust Inference with Variational Bayes

“…[15], following many previous theoretical works [4], exchanges the integral in a posterior expectation with the derivative with respect to prior perturbations, giving a robustness estimate that can be evaluated from MCMC samples. [16,17] extends this idea. These approaches exploit importance sampling and / or closed forms for derivatives or posterior densities, and care must be taken to control the variance of the MCMC estimates.…”

Measuring prior sensitivity and prior informativeness in large Bayesian models

“…The posterior density p is derived from p and the likelihood function l via pðyÞ ¼ pðyÞlðyÞ= Now embed the baseline prior density in a family p a with p 0 ¼ p, score function s a ðyÞ ¼ d log p a ðyÞ=da and dð R yp a ðyÞ dyÞ=da ¼ 1, so that at least for values of a close to zero, the prior mean is approximately R yp a ðyÞ dy % m p þa. The posterior mean as a function of a then equals m p ðaÞ ¼ R yp a ðyÞlðyÞ dy= R p a ðyÞlðyÞ dy, and under weak regularity conditions that justify differentiation under the integral (see, for instance, Perez et al, 2006 for details),…”

“…Berger (1994), Gustafson (2000) and Sivaganesan (2000) provide overviews and references. More specifically, Basu et al (1996), Geweke (1999) and Perez et al (2006) study the local sensitivity of the posterior mean in a parametric class of priors, which amounts to the computation of the posterior mean derivative with respect to the prior hyperparameter. Millar (2004) observes that if the scalar marginal prior distribution is in the exponential family, then the derivative with respect to the prior mean is simply given by the ratio of the posterior to prior variance.…”

Posterior Covariance Information Criterion for Weighted Inference

“…The derivation of waic (Watanabe, 2018b) implies that the information required for evaluating the predictive ability of a model is contained in posterior distributions and can be extracted in the form of posterior covariance, at least in the lowest order of an asymptotic expansion with respect to the sample size. This principle is also seen in sensitivity estimation from mcmc outputs (e.g., Pérez et al, 2006;Giordano et al, 2018). This study aims to further pursue this idea and generalise waic for a wide range of predictive settings, including covariate-shift adaptation (e.g., Shimodaira, 2000;Sugiyama et al, 2007), counterfactual prediction (e.g., Platt et al, 2013;Baba et al, 2017), and quasi-Bayesian prediction (e.g., Konishi & Kitagawa, 1996;Yin, 2009).…”