Characterizing complex biochemical networks in terms of their input-output behavior is at the core of rational circuit design. The theory of stochastic filtering offers an elegant methodology to describe such behavior by means of conditional probability distributions. The latter are relevant to various applications such as biomolecular estimation and control, robustness analysis and information transmission, to name a few. However, finding the conditional distributions as a function of time relies on the solution of a Kushner-Stratonovich equation that can be solved analytically in only a few exceptional cases. Here we develop an approximate filtering solution for the case when both input and output are described by one-dimensional discrete-valued Markov chains, whereas the output depends on the input through arbitrary rate functions. We analyze the accuracy of the approximation using a few examples and demonstrate its use for devising robust biochemical networks.
