Nikolay Bliznyuk
Department of Statistics 
University of Florida

TITLE: On Bayesian Calibration of Computationally Expensive 
Environmental Models.

We present a Bayesian approach to model calibration when the
model is specified by a computationally expensive black-box
computer code $f$. Here, calibration is a nonlinear
regression problem: given a data vector $Y$ corresponding to
the regression model $f(\beta)$, find plausible values of
$\beta$. As an intermediate step, $Y$ and $f$ are embedded
into a statistical model allowing transformation and
dependence.

Typically, this problem is solved by MCMC sampling from the
posterior density $\pi$ of $\beta$ given $Y$. However, since
each evaluation of $\pi$ requires an expensive run of $f$,
naive sampling of $\pi$ by MCMC to obtain a nontrivial
effective sample size is computationally prohibitive. To
reduce computational burden, we limit evaluation of $f$ to a
small number of points chosen on a high-probability region
of $\pi$ reached by optimization. Then, we approximate the
logarithm of $\pi$ using radial basis functions and use the
resulting cheap-to-evaluate surface in MCMC. The main
challenge is to determine the approximation region properly.

The methodology is subsequently extended to statistical models,
in which it is possible to identify a minimal subvector $\beta$
of the whole parameter vector $\eta$ responsible for the
expensive computation in the evaluation of $\pi$.  We propose two
approaches, DOSKA and INDA, that approximate $\pi$ by
interpolation in ways that exploit this computational structure
to mitigate the curse of dimensionality.  DOSKA interpolates
$\pi$ directly while INDA interpolates $\pi$ indirectly by
interpolating functions, e.g., a regression function, upon which
$\pi$ depends. 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 $\dim(\beta)$. This
allows a dramatic reduction of the number of expensive
evaluations necessary to construct an accurate approximation of
$\pi$ when $\dim(\eta)$ is high but $\dim(\beta)$ is low.

We illustrate the proposed approaches are illustrated on models
arising in environmental engineering and environmental health.

This is joint work with David Ruppert and Christine Shoemaker.