A robust Bayesian two-sample test for detecting intervals of differential gene expression in microarray time series.

Understanding the regulatory mechanisms that are responsible for an organism's response to environmental change is an important issue in molecular biology. A first and important step towards this goal is to detect genes whose expression levels are affected by altered external conditions. A range of methods to test for differential gene expression, both in static as well as in time-course experiments, have been proposed. While these tests answer the question whether a gene is differentially expressed, they do not explicitly address the question when a gene is differentially expressed, although this information may provide insights into the course and causal structure of regulatory programs. In this article, we propose a two-sample test for identifying intervals of differential gene expression in microarray time series. Our approach is based on Gaussian process regression, can deal with arbitrary numbers of replicates, and is robust with respect to outliers. We apply our algorithm to study the response of Arabidopsis thaliana genes to an infection by a fungal pathogen using a microarray time series dataset covering 30,336 gene probes at 24 observed time points. In classification experiments, our test compares favorably with existing methods and provides additional insights into time-dependent differential expression.

Two problems with variational expectation maximisation for time-series models

Variational methods are a key component of the approximate inference and learning toolbox. These methods fill an important middle ground, retaining distributional information about uncertainty in latent variables, unlike maximum a posteriori methods (MAP), and yet generally requiring less computational time than Monte Carlo Markov Chain methods. In particular the variational Expectation Maximisation (vEM) and variational Bayes algorithms, both involving variational optimisation of a free-energy, are widely used in time-series modelling. Here, we investigate the success of vEM in simple probabilistic time-series models. First we consider the inference step of vEM, and show that a consequence of the well-known compactness property of variational inference is a failure to propagate uncertainty in time, thus limiting the usefulness of the retained distributional information. In particular, the uncertainty may appear to be smallest precisely when the approximation is poorest. Second, we consider parameter learning and analytically reveal systematic biases in the parameters found by vEM. Surprisingly, simpler variational approximations (such a mean-field) can lead to less bias than more complicated structured approximations.