  We consider the problem of efficiently simulating stochastic models of
chemical kinetics. The Gillespie Stochastic Simulation algorithm (SSA) is often
used to simulate these models, however, in many scenarios of interest, the
computational cost quickly becomes prohibitive. This is further exasperated in
the Bayesian inference context when estimating parameters of chemical models,
as the intractability of the likelihood requires multiple simulations of the
underlying system. To deal with issues of computational complexity in this
paper, we propose a novel hybrid $\tau$-leap algorithm for simulating
well-mixed chemical systems. In particular, the algorithm uses $\tau$-leap when
appropriate (high population densities), and SSA when necessary (low population
densities, when discrete effects become non-negligible). In the intermediate
regime, a combination of the two methods, which leverages the properties of the
underlying Poisson formulation, is employed. As illustrated through a number of
numerical experiments the hybrid $\tau$ offers significant computational
savings when compared to SSA without however sacrificing the overall accuracy.
This feature is particularly welcomed in the Bayesian inference context, as it
allows for parameter estimation of stochastic chemical kinetics at reduced
computational cost.
