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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.
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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.
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