Convex optimisation and stochastic simulation are two powerful computational methodologies for performing statistical inference in high-dimensional inverse problems. It is widely acknowledged that these methodologies can complement each other very well, yet they are generally studied and used separately. This talk presents a new Langevin Markov chain Monte Carlo method that uses elements of convex analysis and proximal optimisation to simulate efficiently from high-dimensional densities that are log-concave, a class of probability distributions that is widely used in modern high-dimensional statistics and data analysis. The method is based on a new first-order approximation for Langevin diffusions that uses Moreau-Yoshida approximations and proximity mappings to capture the log-concavity of the target density and construct Markov chains with favourable convergence properties. This approximation is closely related to Moreau-Yoshida regularisations for convex functions and uses proximity mappings instead of gradient mappings to approximate the continuous-time process. The proposed method complements existing Langevin algorithms in two ways. First, the method is shown to have very robust stability properties and to converge geometrically for many target densities for which other algorithms are not geometric, or only if the time step is sufficiently small. Second, the method can be applied to high-dimensional target densities that are not continuously differentiable, a class of distributions that is increasingly used in image processing and machine learning and that is beyond the scope of existing Langevin and Hamiltonian Monte Carlo algorithms. The proposed methodology is demonstrated on two challenging models related to image resolution enhancement and low-rank matrix estimation, which are not well addressed by existing MCMC methodology.

Magnus Rattray (Manchester)
Gaussian process modelling for omic time course data
We are developing methods based on Gaussian process inference for analysing data from high-throughput biological time course data. Applications range from classical statistical problems such as clustering and differential expression through to systems biology models of cellular processes such as transcription and it's regulation. Our focus is on developing tractable Bayesian methods which scale to genome-wide applications. I will describe our approach to a number of problems: (1) non-parametric clustering of replicated time course data; (2) inferring the full posterior of the perturbation time point from two-sample time course data; (3) inferring the pre-mRNA elongation rate from RNA polymerase ChIP-Seq time course data; (4) uncovering transcriptional delays by integrating pol-II and RNA time course data through a simple differential equation model.