  Parameter estimation for discretely observed Markov processes is a
challenging problem. However, simulation of Markov processes is straightforward
using the Gillespie algorithm. We exploit this ease of simulation to develop an
effective sequential Monte Carlo (SMC) algorithm for obtaining samples from the
posterior distribution of the parameters. In particular, we introduce two key
innovations, coupled simulations, which allow us to study multiple parameter
values on the basis of a single simulation, and a simple, yet effective,
importance sampling scheme for steering simulations towards the observed data.
These innovations substantially improve the efficiency of the SMC algorithm
with minimal effect on the speed of the simulation process. The SMC algorithm
is successfully applied to two examples, a Lotka-Volterra model and a
Repressilator model.
