Posterior Contraction Rates for Bayesian Inverse Problems

Stochastic Analysis Seminar Series

We consider the inverse problem of recovering u from a noisy, indirect observation. We adopt a Bayesian approach, in which the aim is to determine the posterior distribution _y on the unknown u, given some prior information about u in the form of a prior distribution _0, together with the observation y. We are interested in the question of posterior consistency, which is the characterization of the behaviour of _y as more data become available.  We work in a separable Hilbert space X, assuming a Gaussian prior _0 = N(0; _ 2C0).  The theory is developed using two concrete problems: i) a family of linear inverse problems in which we want to _nd u from y where y = A_1u+ p1 n_ for _ Gaussian noise. Here, we study posterior consistency in the small noise limit, n ! 1, [1, 2];

ii) the problem of nonparametrically estimating the drift u of an SDE dYt = u(Yt)dt + dWt by observing the solution Y = fYtgt2[0;T ]. We study posterior consistency as T ! 1, [3].

In both cases _y is also Gaussian, with mean the minimizer of a Tikhonov- Phillips  regularized least squares functional. We work with the unbounded inverse covariance operators of the relevant Gaussian measures, which enables us to use PDE methodology to obtain rates of contraction of the posterior distribution to a Dirac measure centred on the truth underlying the data. The rates are optimized by choosing the scaling _ 2 in the prior as an appropriate function of the level of the noise 1=n and the time-length of the observationTrespectively in the two problems. Finally, we identify the common structure of the above problems and propose an abstract theory.

References

[1] S. Agapiou, S. Larsson and A. M. Stuart, Posterior Contraction Rates for the Bayesian Approach to Linear Ill-Posed Inverse Problems, http://arxiv.org/abs/1203.5753

[2] S. Agapiou, A. M. Stuart and Y. X. Zhang, Bayesian Posterior Contraction Rates for the Bayesian Approach to Linear Severely Ill-Posed Inverse Problems, http://arxiv.org/abs/1210.1563

[3] Y. Pokern, A. M. Stuart and J. H. Van Zanten, Posterior consistency via precision operators for nonparametric drift estimation in SDEs, http://arxiv.org/abs/1202.0976

Location:
Speaker(s):

Sergios Agapiou (Warwick University)

Start date:
Monday, January 21, 2013 - 14:15
End date:
Thursday, December 13, 2012 - 15:15