Stat Colloquium: Dr. Riten Mitra

Title: Bayesian non-parametric learning of Stochastic Differential Equations using Reproducible Kernels

Abstract: Stochastic Differential Equations (SDEs)are relevant to a wide array of applications involving complex temporal dynamics, ranging from finance to epidemic modeling. A little less explored is their application to systems biology, where they arise as continuous approximations of Continuous time Markov Chains that modulate the temporal expressions of genomic entities in a pathway. The structure of the Markov processes is informed fully or partially by a known system of biochemical equations among the entities with typically unknown reaction rates. Inferring the rates allow us insights into long-term dynamics of these processes. A fully known reaction structure lends itself to a parametric SDE model with rates appearing as unknown parameters in the drift function thereof. However, this knowledge is unrealistic for most stochastic systems. To address this, we propose a non-parametric modeling of the drift function based on the Representer Theorem in Reproducing Kernel Hilbert Spaces. For rigorous uncertainty quantification, we further embed this in a Bayesian hierarchical model which we show to be amenable to a clean Gibbs sample procedure. This allows visual illustration of uncertainty on the function space through MCMC samples. Our proposed method not only mirrored the true drift function quite closely in simulations from some well-known biophysical systems (MM kinetics, Cook’s model) but also showed good predictive performance with respect to the long-term stationary distributions.  To our knowledge, this is the first instance of merging of SDEs, RKHS and Bayesian hierarchies in a single framework. I will conclude this talk with some future extensions on partially observed data and extension of the Representer Theorem to complex SDEs. This is a joint work with my collaborator Dr. Arnab Ganguly of LSU and Jinpu Zhou, a doctoral candidate at LSU.