View Submission - EcoSta2018

A0573
**Title: **Nonparametric Bayesian contraction rates for compound Poisson processes observed discretely at arbitrary frequencies
**Authors: **Alberto J Coca - University of Cambridge (United Kingdom) **[presenting]**

**Abstract: **Compound Poisson processes (CPPs) are the textbook example of pure jump stochastic processes, and they approximate arbitrarily well much richer classes of processes such as L{\'e}vy processes. Two parameters characterise them: the drift, and the L{\'e}vy jump distribution, $N$, driving the frequency at which jumps (randomly) occur and their (random) sizes. In most applications, the CPP is not perfectly observed: only discrete observations over a finite-time interval are available. Thus, the process may jump several times between two observations and estimating $N$ is a non-linear statistical inverse problem. In the recent years, understanding the asymptotic behaviour of the Bayesian method in inverse problems and, in particular, in this problem has received considerable attention. We will present some recent results on posterior contraction rates for the density $\nu$ of $N$: we show two-sided stability estimates between $\nu$ and its image through the forward operator that allow to use existing classical theory; furthermore, these are robust to the observation interval, i.e. optimal adaptive inference can be made without specification of whether the regime is of high- or low-frequency; and, lastly, we propose an efficient $\infty$-MCMC procedure to draw from the posterior using mixture and Gaussian priors that can handle the multidimensional setting.