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Abstract
A basic problem in statistics is to make inferences about physical parameters from experimental data.
Because of measurement and modeling errors, data do not determine parameter values exactly. Bayesian
statistics represents the remaining parameter uncertainty as a “posterior” probability distribution of the
parameters conditional on the measurements and prior information. For high dimensional problems, it
is hard to represent this posterior except as a collection of random samples from it. This talk discusses
the problem of producing such samples.
We describe MCMC ( Markov chain Monte Carlo) samplers, which are guaranteed to “work” in principle
but may be too slow in practical applications. There are several theoretical approaches to understanding
the convergence of MCMC samplers, including Sobolev and log Sovolev inequalities, the beautiful methods
of the Lovasz school (which are based on “Cheeger’s inequality”), and work of Hairer and Stuart. Each of
these has led to better samplers for specific problems.
Some of my work is motivated by an analogy between MCMC samplers and optimization algorithms.
One idea from optimization is that a good method for generic problems should be invariant under affine
transformations. This allows the method to work well for poorly conditioned problems. I present two
affine invariant sampling algorithms, one of which is the basis of a popular software package that will
be described later this afternoon by David Hogg. Line search is another method from optimization that
we have imported to MCMC samplers. I will list several open problems and opportunities. |
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