A Continuation Method in Bayesian Inference

We present a continuation method that entails generating a sequence of transition probability density functions from the prior to the posterior in the context of Bayesian inference for parameter estimation problems. The characterization of transition distributions, by tempering the likelihood function, results in a homogeneous nonlinear partial integro-differential equation for which existence and uniqueness of solutions are addressed. The posterior probability distribution is obtained as the interpretation of the final state of a path of transition distributions. A computationally stable scaling domain for the likelihood is explored for approximation of the expected deviance, where we restrict the evaluations of the forward predictive model at the prior stage. To obtain a solution formulation for the expected deviance, we derive a partial differential equation governing the moment-generating function of the log-likelihood. We show also that a spectral formulation of the expected deviance can be obtained for low-dimensional problems under certain conditions. The effectiveness of the proposed method is demonstrated using four numerical examples. These focus on analyzing the computational bias generated by the method, assessing its use in Bayesian inference with non-Gaussian noise, evaluating its ability to invert a multimodal parameter of interest, and quantifying its performance in terms of computational cost.