## Posterior Approximation by Interpolation for Bayesian Inference in Computationally Expensive Statistical Models

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Markov Chain Monte Carlo (MCMC) is nowadays a standard
approach to numerical computation of integrals of the
posterior density $\pi$ of the parameter vector $\eta$.
Unfortunately, Bayesian inference using MCMC is
computationally intractable when $\pi$ is expensive to
evaluate. In this work, we develop practical methods that
approximate $\pi$ with radial basis functions (RBFs) and
Gaussian processes (GPs) interpolants and use the resulting
cheap-to-evaluate surfaces in MCMC.
In Chapter 1, $\pi$
arises from a nonlinear regression model with transformation
and dependence. To build the RBF approximation, we limit
evaluation of the computationally expensive regression
function to points chosen on a high posterior density (HPD)
region found using a local quadratic approximation of
$\log(\pi)$ at its mode. We illustrate our approach on
simulated data for a pollutant diffusion problem and study
the frequentist coverage properties of credible intervals.
In Chapter 2, we relax the assumptions about $\pi$ made in
Chapter 1 and develop a derivative-free procedure GRIMA
to approximate the logarithm of $\pi$ using RBF
interpolation over a HPD region of $\pi$ estimated using
the RBF surface. We use GRIMA for Bayesian inference in a
computationally intensive nonlinear regression model for
real measured streamflow data in the Town Brook watershed.
In Chapter 3, we study statistical models where it is
possible to identify a minimal subvector $\beta$ of $\eta$
responsible for the expensive computation in the evaluation
of $\pi$. We propose two approaches to approximate $\pi$ by
interpolation that exploit this computational structure. Our
primary contribution is derivation of a GP interpolant that
provably improves over some of the existing approaches by
reducing the effective dimension of the interpolation
problem from $\dim(\eta)$ to $\dim(\beta)$. When $\dim(\eta)
$ is high but $\dim(\beta)$ is low, this allows one to
dramatically reduce the number of expensive evaluations
necessary to construct an accurate approximation of $\pi$.
Our experiments indicate that our methods produce results similar to those
when the true expensive posterior density is sampled by MCMC while reducing
computational costs by well over an order of magnitude.

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NSF DMS-04-538

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2008-07-10T15:44:07Z

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inverse problems; response surface

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bibid: 6397183

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dissertation or thesis