6 March 2014

14:00

Professor Andrew Stuart

Abstract

Many problems in the physical sciences
require the determination of an unknown
function from a finite set of indirect measurements.
Examples include oceanography, oil recovery,
water resource management and weather forecasting.
The Bayesian approach to these problems
is natural for many reasons, including the
under-determined and ill-posed nature of the inversion,
the noise in the data and the uncertainty in
the differential equation models used to describe
complex mutiscale physics. The object of interest
in the Bayesian approach is the posterior
probability distribution on the unknown field [1].
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However the Bayesian approach presents a
computationally formidable task as it
results in the need to probe a probability
measure on separable Banach space. Monte
Carlo Markov Chain methods (MCMC) may be
used to achieve this [2], but can be
prohibitively expensive. In this talk I
will discuss approximation of probability measures
by a Gaussian measure, looking for the closest
approximation with respect to the Kullback-Leibler
divergence. This methodology is widely
used in machine-learning [3]. In the context of
target measures on separable Banach space
which themselves have density with respect to
a Gaussian, I will show how to make sense of the
resulting problem in the calculus of variations [4].
Furthermore I will show how the approximate
Gaussians can be used to speed-up MCMC
sampling of the posterior distribution [5].
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[1] A.M. Stuart. "Inverse problems: a Bayesian
perspective." Acta Numerica 19(2010) and
http://arxiv.org/abs/1302.6989
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[2] S.L.Cotter, G.O.Roberts, A.M. Stuart and D. White,
"MCMC methods for functions: modifying old algorithms
to make them faster". Statistical Science 28(2013).
http://arxiv.org/abs/1202.0709
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[3] C.M. Bishop, "Pattern recognition and machine learning".
Springer, 2006.
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[4] F.J. Pinski G. Simpson A.M. Stuart H. Weber, "Kullback-Leibler
Approximations for measures on infinite dimensional spaces."
http://arxiv.org/abs/1310.7845
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[5] F.J. Pinski G. Simpson A.M. Stuart H. Weber, "Algorithms
for Kullback-Leibler approximation of probability measures in
infinite dimensions." In preparation.