Estimating Posterior Sensitivities with Application to Structural Analysis of Bayesian Vector Autoregressions

Abstract

An inherent feature of Bayesian inference is posterior/prior (parameter) dependence. Locally, it can
be explained by the changes in posterior inference resulting from local changes in prior parameters.
Yet, computing such sensitivities remains challenging given the analytical complexities and computational intensity of the widely used Markov Chain Monte Carlo. This paper extends Infinitesimal Perturbation Analysis, widely used in the classical simulation context to assess input sensitivities of stochastic dynamic systems, to compute prior parameter sensitivities of posterior statistics from Markov Chain Monte Carlo inference via Gibbs sampling. We show that, unlike the traditional finite differencing method for computing derivatives, our proposed method provides asymptotic unbiased and consistent derivative estimators of posterior statistics with respect to prior input parameters. Efficient computation of exact sensitivities is possible alongside the estimation algorithm via Automatic Differentiation tools, which avoid rerunning the chain. We illustrate the methods within Bayesian Vector Autoregression (BVAR) models, a benchmark tool in macroeconomic policy analysis and forecasting due to their ability to utilise informative shrinkage priors. We assess the prior robustness of both posterior parameter and impulse response inference in a standard smaller scale policy BVAR
for US macroeconomic time series data under two versions of a common Minnesota shrinkage prior. The first formal sensitivity analysis shows that the exact prior specification can have substantial and complex influences on inference regarding the GDP responses under a government spending shock. Our results suggest that in the particular setting, the posterior impulse response function is less sensitive to the changes in the prior hyper-parameters if the Minnesota shrinkage pushes the model towards a random walk instead of a random walk with drifts..

Keywords: MCMC, Prior Robustness, Convergence, Automatic Differentiation

JEL Classification: C01, C11