Methods of approximate inference

The large and high-dimensional data sets typically seen in contemporary science and engineering applications often preclude the direct application of optimal inference procedures. Here we describe three example scenarios in which exact inference is unachievable, and present alternative strategies that yield approximate results in a computationally efficient manner: (1) Stochastic differential equations are widely used as probability models in science, engineering, and financial mathematics. As such models are specified in continuous time, however, any inference based on discretely-observed data necessitates the integration of the model over the elapsed time between observations. We present a new multiresolution Markov chain Monte Carlo sampler which relies on the use of multiple approximations and Richardson extrapolation. We find it is significantly faster than known strategies based on Gibbs sampling. (2) Video microscopy is widely used in experimental biology, particularly to study intracellular calcium dynamics. Researchers remain uncertain regarding the role localized releases of free calcium ions play in cell regulation. In order to detect these local events and to estimate their extent and duration from typical video microscopy data sets, we present a model for their observation via a digital video camera. Though exact inference is readily achieved for small sample sizes, the sheer number of dependent observations prevalent in video data sets necessitates approximate methods. We utilize a simple two-stage approach to hypothesis testing which greatly reduces the computational burden of likelihood evaluations, yet retains much of the exact test's inherent power. (3) Networks and graph-valued data sets are increasingly prevalent across a variety of fields. A typical inferential task is to determine whether or not some form of group structure exists within a network data set. Calculating the usual test statistic of the likelihood ratio, however, is computationally infeasible, even for the simplest structure models. As an alternative, we describe how algorithms for graph cuts can be used to obtain fast approximations to the likelihood ratio test statistic for the case of popular random graph models, and how in turn the power of such approximations can be evaluated through the use of stochastic simulation tools.