  The complexity of mathematical models in biology has rendered model reduction
an essential tool in the quantitative biologist's toolkit. For stochastic
reaction networks described using the Chemical Master Equation, commonly used
methods include time-scale separation, the Linear Mapping Approximation and
state-space lumping. Despite the success of these techniques, they appear to be
rather disparate and at present no general-purpose approach to model reduction
for stochastic reaction networks is known. In this paper we show that most
common model reduction approaches for the Chemical Master Equation can be seen
as minimising a well-known information-theoretic quantity between the full
model and its reduction, the Kullback-Leibler divergence defined on the space
of trajectories. This allows us to recast the task of model reduction as a
variational problem that can be tackled using standard numerical optimisation
approaches. In addition we derive general expressions for the propensities of a
reduced system that generalise those found using classical methods. We show
that the Kullback-Leibler divergence is a useful metric to assess model
discrepancy and to compare different model reduction techniques using three
examples from the literature: an autoregulatory feedback loop, the
Michaelis-Menten enzyme system and a genetic oscillator.
