  For many stochastic models of interest in systems biology, such as those
describing biochemical reaction networks, exact quantification of parameter
uncertainty through statistical inference is intractable. Likelihood-free
computational inference techniques enable parameter inference when the
likelihood function for the model is intractable but the generation of many
sample paths is feasible through stochastic simulation of the forward problem.
The most common likelihood-free method in systems biology is approximate
Bayesian computation that accepts parameters that result in low discrepancy
between stochastic simulations and measured data. However, it can be difficult
to assess how the accuracy of the resulting inferences are affected by the
choice of acceptance threshold and discrepancy function. The pseudo-marginal
approach is an alternative likelihood-free inference method that utilises a
Monte Carlo estimate of the likelihood function. This approach has several
advantages, particularly in the context of noisy, partially observed,
time-course data typical in biochemical reaction network studies. Specifically,
the pseudo-marginal approach facilitates exact inference and uncertainty
quantification, and may be efficiently combined with particle filters for low
variance, high-accuracy likelihood estimation. In this review, we provide a
practical introduction to the pseudo-marginal approach using inference for
biochemical reaction networks as a series of case studies. Implementations of
key algorithms and examples are provided using the Julia programming language;
a high performance, open source programming language for scientific computing.
