Many applied data are comprised of a collection of dependent random variables. Probabilistic models for such data typically have complicated correlation structures and unknown normalizing constants. It can be difficult to both sample from and make inference on these models. Standard methods for sampling (e.g., Metropolis) are often prohibitively slow, and standard methods for inference (e.g., pseudolikelihood) prohibitively inaccurate. Domain-specific Monte Carlo methods can provide an accurate, computationally efficient alternative. We introduce Monte Carlo methods for three particular applications: functional magnetic resonance imaging brain scans, social networks and protein folding. The first two settings are inferential: we consider the problems of parameter estimation for an exponential random graph model (social networks) and of hypothesis testing for smooth Gaussian random fields (brain scans). Specifically, we propose a Monte Carlo maximum likelihood method for parameter estimation and a thresholding test for hypothesis testing. The third setting (protein folding) involves sampling an atomic conformation from the Boltzmann distribution, for which we develop a fast configurational bias sampler. In each case we consider both speed and accuracy, finding substantial gains for our methods over leading alternatives.
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