  The simplest, and most common, stochastic model for population processes,
including those from biochemistry and cell biology, are continuous time Markov
chains. Simulation of such models is often relatively straightforward as there
are easily implementable methods for the generation of exact sample paths.
However, when using ensemble averages to approximate expected values, the
computational complexity can become prohibitive as the number of computations
per path scales linearly with the number of jumps of the process. When such
methods become computationally intractable, approximate methods, which
introduce a bias, can become advantageous. In this paper, we provide a general
framework for understanding the weak error, or bias, induced by different
numerical approximation techniques in the current setting. The analysis takes
into account both the natural scalings within a given system and the step-size
of the numerical method. Examples are provided to demonstrate the main
analytical results as well as the reduction in computational complexity
achieved by the approximate methods.
