  In this paper we address the problem of Monte Carlo approximation of
posterior probability distributions in stochastic kinetic models (SKMs). SKMs
are multivariate Markov jump processes that model the interactions among
species in biochemical systems according to a set of uncertain parameters.
Markov chain Monte Carlo (MCMC) methods have been typically preferred for this
Bayesian inference problem. Specifically, the particle MCMC (pMCMC) method has
been recently shown to be an effective, while computationally demanding, method
applicable to this problem. Within the pMCMC framework, importance sampling
(IS) has been used only as the basis of the sequential Monte Carlo (SMC)
approximation of the acceptance ratio in the Metropolis-Hastings kernel.
However, the recently proposed nonlinear population Monte Carlo (NPMC)
algorithm, based on an iterative IS scheme, has also been shown to be effective
as a Bayesian inference tool for low dimensional (predator-prey) SKMs. In this
paper, we provide an extensive performance comparison of pMCMC versus NPMC,
when applied to the challenging prokaryotic autoregulatory network. We show how
the NPMC method can greatly outperform the pMCMC algorithm in this scenario,
with an overall moderate computational effort. We complement the numerical
comparison of the two techniques with an asymptotic convergence analysis of the
nonlinear IS scheme at the core of the proposed method when the importance
weights can only be computed approximately.
