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 formwhere N (t) is a standard Poisson process of intensity λ, and Z k are drawn i.i.d. from jump measure µ. A high-dimensional wavelet series prior for the Lévy measure ν = λµ is devised and the posterior distribution arises from observing discrete samples Y ∆ , Y 2∆ , . . . , Y n∆ at fixed observation distance ∆, 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 µ and ν, as well as for the intensity λ, 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.MSC 2000 subject classification: 62G20, 65N21, 60G51, 60J75

Bernstein--von Mises Theorems and Uncertainty Quantification for Linear Inverse Problems

“…The frequentist analysis of nonparametric Bayesian procedures for inverse problems has received increasing interest in the last decade, and several contributions in the linear setting have established consistency results and derived posterior contraction rates; see [1, 2, 26-30, 44, 58] among others. We also mention [40,42,43] for results for non-linear inverse problems.…”

“…Utilising a Wassersteintype metric [7,8] achieve weak convergence of the posterior distribution to a prior-independent infinitedimensional Gaussian distribution on a large enough function space. More recently similar techniques were used in the inverse setting [38], for the linear X-ray transform problem, obtaining a semiparametric BvM theorem relative to smooth functionals of the unknown, while [40] proved a nonparametric result for a non-linear problem arising in partial differential equations. See also [41,42] for further related results.…”

“…More recently similar techniques were used in the inverse setting [38], for the linear X-ray transform problem, obtaining a semiparametric BvM theorem relative to smooth functionals of the unknown, while [40] proved a nonparametric result for a non-linear problem arising in partial differential equations. See also [41,42] for further related results. Positive results have also been obtained in [29,33,53], for priors defined on the SVD basis of the forward operator.…”

“…Theorem 11). Adapting the program laid out in [42] to the problem at hand, we show that the asymptotic approximation of the marginal distributions holds uniformly across a suitable collection of test functions, leading to the formulation of a nonparametric BvM theorem. This entails the convergence of the posterior distribution to a limiting Gaussian probability measure with minimal covariance in suitable function spaces (cf.…”

Hyperparameter Estimation in Bayesian MAP Estimation: Parameterizations and Consistency

“…The subject has developing mathematical foundations and attendant stability and approximation theories [16,28,29,44,39]. Furthermore, the subject of Bayesian posterior consistency is being systematically developed [4,6,35,37,27,26,41,19,20]. Furthermore, the paper [25] was the first to establish consistency in the context of hyperparameter learning, as we do here, and in doing so demonstrates that Bayesian methods have comparable capabilities to frequentist methods, regarding adaptation to smoothness, whilst also quantifying uncertainty.…”