WHIM: function approximation where it matters

Abstract

If f is a function of interest, typically either a likelihood or posterior density function on a parameter space Θ, and $\widehat{\theta }$ is the MLE (maximum likelihood estimate) or MAP (maximum a posteriori estimate), it can be of interest to find the region of Θ where $f(\theta )$ or $\text{log}\hspace{0.17em}f(\theta )$ is nearly as large as $f(\widehat{\theta })$ or $\text{log}\hspace{0.17em}f(\widehat{\theta }).$ Typical tools for working with f, such as optimizers and MCMC, can fail when f is multimodal or has a plateau. This paper describes an algorithm called WHIM – for function approximation WHere It Matters – that finds the region of Θ where f is large and that is guaranteed not to fail for f’s arising from the large class of models described here, even when those f’s are multimodal or have plateaus.

WHIM was introduced in Lavine, Bray, and Hodges, where the focus was on linear mixed models with exactly two variances. Here, the focus is on the algorithm and what can be learned by finding the region where $f(\theta )$ is large, rather than finding just $\widehat{\theta }.$

Keywords:

• Likelihood
• Optimization
• Statistical inference