Optimizing and Adapting the Metropolis Algorithm

Many modern scientific questions involve high dimensional data and complicated statistical models. For example, data on weather consist of huge numbers of measurements across spatial grids, over a period of time. Even in simpler settings, data can be complex: for example, Bartolucci et al. (2007) consider recurrence rates for melanoma (skin cancer) patients after surgery. The probability of recurrence for an individual may depend on physical or biological characteristics of their cancerous lesion, as well as other factors. A statistical model in this context may involve a large number of variables and a correspondingly large number of parameters, which are often represented by a vector θ of some dimension d. To assess the relevance of specific variables for disease recurrence, and to build models that give a risk of recurrence for any given individual, researchers often use Bayesian analysis (see e.g. Box and Tiao, 1973; Gelman et al., 2003; Carlin and Louis, 2008). In this framework, the parameter vector is assumed to follow some probability distribution (of dimension d), and the challenge is to combine a “prior” distribution for θ (typically based on background information about the scientific area) with data that are collected, so as to produce a “posterior” distribution for θ. This probability distribution (call it π(θ)) can then be used to answer important scientific questions (e.g., is the size of a cancerous lesion related to the risk of recurrence after surgery?) and to calculate specific probabilities (e.g., this person has a 20% probability of a recurrence within the next five years). One challenge for Bayesian analysis in situations where the data and parameter vectors are high dimensional is that it is difficult or impossible to compute probabilities based on the posterior distribution. If there is some outcome A of interest (e.g., the outcome that a specific individual’s cancer