Duality between Approximate Bayesian Methods and Prior Robustness

In this paper we show that there is a link between approximate Bayesian methods and prior robustness. We show that what is typically recognized as an approximation to the likelihood, either due to the simulated data as in the Approximate Bayesian Computation (ABC) methods or due to the functional approximation to the likelihood, can instead also be viewed upon as an implicit exercise in prior robustness. We first define two new classes of priors for the cases where the sufficient statistics is available, establish their mathematical properties and show, for a simple illustrative example, that these classes of priors can also be used to obtain the posterior distribution that would be obtained by implementing ABC. We then generalize and define two further classes of priors that are applicable in very general scenarios; one where the sufficient statistics is not available and another where the likelihood is approximated using a functional approximation. We then discuss the interpretation and elicitation aspects of the classes proposed here as well as their potential applications and possible computational benefits. These classes establish the duality between approximate Bayesian inference and prior robustness for a wide category of Bayesian inference methods.