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.
The third approach, and the only other that we are aware of for finding approximate RPs, is the (RDM) proposed by (2015). This method focuses on finding a joint RP by minimizing the divergence between a parametric family and the marginal RPs @cite_6 . Since we are concerned with models for which even the marginal RPs are intractable, the RDM is not a relevant point for comparison.
@cite_6
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In multi-parameter models, reference priors typically depend on the parameter or quantity of interest, and it is well known that this is necessary to produce objective posterior distributions with optimal properties. There are, however, many situations where one is simultaneously interested in all the parameters of the model or, more realistically, in functions of them that include aspects such as prediction, and it would then be useful to have a single objective prior that could safely be used to produce reasonable posterior inferences for all the quantities of interest. In this paper, we consider three methods for selecting a single objective prior and study, in a variety of problems including the multinomial problem, whether or not the resulting prior is a reasonable overall prior.