# Van Dantzig Seminar

#### nationwide series of lectures in statistics

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## Van Dantzig Seminar: 26 October 2016

#### Programme: (click names or scroll down for titles and abstracts)

 14:00 - 14:05 Opening 14:05 - 15:05 Jim Griffin (University of Kent) 15:05 - 15:25 Break 15:25 - 16:25 Jakob Söhl (Delft University of Technology) 16:30 - 17:30 Reception
 Location: Leiden University, Snellius Building, Room 402 (Directions)

## Titles and abstracts

• Jim Griffin

Compound random measures and their use in Bayesian nonparametrics

In Bayesian nonparametrics, a prior is placed on an infinite dimensional object such as a function or distribution. In this talk, I will consider the estimation of related distributions and describe a new class of dependent random measures which we call compound random measures. These priors are parametrized by a distribution and a Lévy process and and their dependence can be characterized using both the Lévy copula and correlation function. A normalized version of this random measure can be used as dependent priors for related distributions. I will describe an MCMC algorithm for posterior inference when the parametric distribution has a known moment generating function and a pseudo-marginal method for more general models (for example, where the parametric distribution is given by a regression model). The approach will be illustrated with data examples.

• Jakob Söhl

Bayesian nonparametric inference for diffusion models with discrete sampling

We consider nonparametric Bayesian inference in a reflected diffusion model $$dX_t=b(X_t)dt+\sigma(X_t)dW_t$$, with discretely sampled observations $$X_0,X_\Delta,\ldots, X_{n\Delta}$$. We analyse the nonlinear inverse problem corresponding to the `low-frequency sampling' regime where $$\Delta > 0$$ is fixed and $$n \to \infty$$. A general theorem is proved that gives conditions for prior distributions $$\Pi$$, on the diffusion coefficient $$\sigma$$ and the drift function $$b$$ that ensure minimax optimal contraction rates of the posterior distribution over Hölder-Sobolev smoothness classes. These conditions are verified for natural examples of nonparametric random wavelet series priors. For the proofs we derive new concentration inequalities for empirical processes arising from discretely observed diffusions that are of independent interest.