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

Richard Nickl, Jakob Söhl
2019 arXiv   pre-print
We study nonparametric Bayesian statistical inference for the parameters governing a pure jump process of the form Y_t = ∑_k=1^N(t) Z_k, t > 0, where 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_Δ, Y_2Δ, ..., Y_nΔ at fixed observation distance Δ, giving rise to a nonlinear inverse inference
more » ... em. 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 Cramér-Rao 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.
arXiv:1709.07752v2 fatcat:piy6sfcaxngunjpt46ip54fsbm