Estimation of parametric sensitivities for continuous time Markov processes on integer lattices

Muruhan Rathinam
University of Maryland Baltimore County

We consider a class of continuous time Markov processes with the $n$ dimensional integer vectors as state space. The class under consideration is motivated by the well-stirred model of stochastic chemical kinetics which is essential in the modeling of intra-cellular biochemical reactions. This class of models also appear in epidemiology, predator-prey interactions and other forms of population processes.
An important feature of chemical kinetics or populations models is the so-called density dependence of the event rate functions. This property, in the large population limit yields a deterministic, ordinary differential equation (ODE) model.

Estimation of parametric sensitivities is an essential ingredient in the process of parameter fitting based on data. For deterministic systems governed by ordinary differential equations (ODEs) the sensitivity defined by a partial derivative of a function of the state $f(X(t))$ with respect to a parameter $c$ is relatively easy to compute. In contrast, several radically different approaches are available for the computation of the parametric sensitivity defined by the partial derivative of $E(f(X(t)))$ with respect to a parameter $c$. When the dimension of the state space is large, usually the Monte Carlo methods are the most efficient and even among these there are several different approaches.

We provide an overview of some of the major Monte Carlo approaches, namely the Finite Difference (FD), Pathwise Derivative (PD) and the Girsanov Transformation (GT). The efficiency of a Monte Carlo approach depends on the estimator having a low variance. It has been numerically observed that in several examples the FD and PD methods tend to have lower variance than the GT estimator while the latter has the advantage of being unbiased.
We present a theoretical explanation for the larger variance of the GT approach when compared to the FD or PD methods. It turns out that the density dependent form of rates result in larger variance of the GT method method compared to FD or PD methods when system size is even modestly large.

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