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

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 Lévy 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 Lévy density that are optimal up to logarithmic factors over Hölder 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.

51 pages, to appear in Electronic Journal of Statistics

References in corpus (8)

• Nonparametric Bayesian posterior contraction rates for discretely observed scalar diffusions
• Rates of contraction for posterior distributions in $\bolds{L^r}$-metrics, $\bolds{1\le r\le\infty}$
• Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions
• Bernstein -- von Mises theorems for statistical inverse problems II: Compound Poisson processes
• Efficient nonparametric inference for discretely observed compound Poisson processes
• Nonparametric Bayesian inference for multidimensional compound Poisson processes
• Adaptive nonparametric estimation for compound Poisson processes robust to the discrete-observation scheme
• Bernstein-von Mises theorems and uncertainty quantification for linear inverse problems

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• Nonparametric statistical inference for drift vector fields of multi-dimensional diffusions
• Bernstein -- von Mises theorems for statistical inverse problems II: Compound Poisson processes
• Consistency of Bayesian inference with Gaussian process priors for a parabolic inverse problem
• On polynomial-time computation of high-dimensional posterior measures by Langevin-type algorithms
• On the Bernstein-von Mises theorem for the Dirichlet process
• Bayesian inverse problems with unknown operators
• Sup-norm adaptive simultaneous drift estimation for ergodic diffusions
• Adaptive nonparametric estimation for compound Poisson processes robust to the discrete-observation scheme
• Decompounding discrete distributions: A non-parametric Bayesian approach
• Semiparametric inference using fractional posteriors
• Bayesian inference for nonlinear inverse problems
• Reconciling the Bayes Factor and Likelihood Ratio for Two Non-Nested Model Selection Problems
• The nonparametric LAN expansion for discretely observed diffusions
• Finite sample Bernstein-von Mises theorems for functionals and spectral projectors of the covariance matrix