A Diffusion Process Perspective on Posterior Contraction Rates for Parameters

We show that diffusion processes can be exploited to study the posterior contraction rates of parameters in Bayesian models. By treating the posterior distribution as a stationary distribution of a stochastic differential equation (SDE), posterior convergence rates can be established via control of the moments of the corresponding SDE. Our results depend on the structure of the population log-likelihood function, obtained in the limit of an infinite sample sample size, and stochastic perturbation bounds between the population and sample log-likelihood functions. When the population log-likelihood is strongly concave, we establish posterior convergence of a $d$-dimensional parameter at the optimal rate $(d/n)^{1/ 2}$. In the weakly concave setting, we show that the convergence rate is determined by the unique solution of a non-linear equation that arises from the interplay between the degree of weak concavity and the stochastic perturbation bounds. We illustrate this general theory by deriving posterior convergence rates for three concrete examples: Bayesian logistic regression models, Bayesian single index models, and over-specified Bayesian mixture models.