Informative Bayesian priors are often difficult to elicit, and when this is the case, modelers usually turn to noninformative or objective priors. However, objective priors such as the Jeffreys and reference priors are not tractable to derive for many models of interest. We address this issue by proposing techniques for learning reference prior approximations: we select a parametric family and optimize a black-box lower bound on the reference prior objective to find the member of the family that serves as a good approximation. We experimentally demonstrate the method's effectiveness by recovering Jeffreys priors and learning the Variational Autoencoder's reference prior.
Lafferty & Wasserman (2001) proposed a Markov Chain Monte Carlo (MCMC) method for sampling from a RP. Their approach involves running the Metropolis-Hastings algorithm on the following ratio @cite_14 : where @math is the iteration index, @math is the entropy of the likelihood function, and @math where @math are the parameter samples collected during the previous iteration.
@cite_14
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We present an iterative Markov chain Monte Carlo algorithm for computing reference priors and minimax risk for general parametric families. Our approach uses MCMC techniques based on the Blahut-Arimoto algorithm for computing channel capacity in information theory. We give a statistical analysis of the algorithm, bounding the number of samples required for the stochastic algorithm to closely approximate the deterministic algorithm in each iteration. Simulations are presented for several examples from exponential families. Although we focus on applications to reference priors and minimax risk, the methods and analysis we develop are applicable to a much broader class of optimization problems and iterative algorithms.