Data-Based Inference of Generators for Markov Jump Processes Using Convex Optimization

A variational approach to the estimation of generators for Markov jump processes from discretely sampled data is discussed and generalized. In this approach, one first calculates the spectrum of the discrete maximum likelihood estimator for the transition matrix consistent with the discrete data. Then the generator that best matches the spectrum is determined by solving a convex quadratic minimization problem with linear constraints (quadratic program). Here, we discuss the method in detail and position it in the context of maximum likelihood inference of generators from discretely sampled data. Furthermore, we show how the approach can be generalized to estimation from data sampled at nonconstant time intervals. Finally, we discuss numerical aspects of the algorithm for estimation of processes with high-dimensional state spaces. Numerical examples are presented throughout the paper.