Bernstein -- von Mises theorems for statistical inverse problems II: Compound Poisson processes

TL;DR

This paper develops Bayesian methods for nonparametric inference of jump process parameters, deriving optimal contraction rates and establishing Bernstein-von Mises theorems to ensure asymptotic normality and validity of credible sets.

Contribution

It introduces a wavelet series prior for the Lévy measure and proves contraction rates and Bernstein-von Mises theorems for the posterior distribution in compound Poisson processes.

Findings

Optimal contraction rates up to logarithmic factors.

Posterior distribution approximates a Gaussian measure asymptotically.

Credible sets are asymptotically valid and optimal.

Abstract

We study nonparametric Bayesian statistical inference for the parameters governing a pure jump process of the form $$Y_t = \sum_{k=1}^{N(t)} Z_k,~~~ t \ge 0,$$ where $N(t)$ is a standard Poisson process of intensity $\lambda$, and $Z_k$ are drawn i.i.d.~from jump measure $\mu$. A high-dimensional wavelet series prior for the L\'evy measure $\nu = \lambda \mu$ is devised and the posterior distribution arises from observing discrete samples $Y_\Delta, Y_{2\Delta}, \dots, Y_{n\Delta}$ at fixed observation distance $\Delta$, giving rise to a nonlinear inverse inference problem. We derive contraction rates in uniform norm for the posterior distribution around the true L\'evy density that are optimal up to logarithmic factors over H\"older classes, as sample size $n$ increases. We prove a functional Bernstein-von Mises theorem for the distribution functions of both $\mu$ and $\nu$, as well as for the intensity $\lambda$, establishing the fact that the posterior distribution is approximated by an infinite-dimensional Gaussian measure whose covariance structure is shown to attain the information lower bound for this inverse problem. As a consequence posterior based inferences, such as nonparametric credible sets, are asymptotically valid and optimal from a frequentist point of view.