Robust Bayesian Inference Using the Mixture Priors

In Bayesian estimation the posterior distribution is proportional to likelihood function and the prior density of the parameter. Thus Bayesian inference, most often, is affected by the prior density. In this paper we look at how we can make Bayesian inference more robust against a poorly specified prior. We find that using a mixture of conjugate priors enables us to do this. We allow a small prior probability that our prior is misspecified. If the likelihood is very different than what would be expected under the prior, the posterior probability of misspecification is large and our posterior distribution will depend mostly on the likelihood.