  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.
