  Many natural and engineered systems can be modeled as discrete state Markov
processes. Often, only a subset of states are directly observable. Inferring
the conditional probability that a system occupies a particular hidden state,
given the partial observation, is a problem with broad application. In this
paper, we introduce a continuous-time formulation of the sum-product algorithm,
which is a well-known discrete-time method for finding the hidden states'
conditional probabilities, given a set of finite, discrete-time observations.
From our new formulation, we can explicitly solve for the conditional
probability of occupying any state, given the transition rates and observations
within a finite time window. We apply our algorithm to a realistic model of the
cystic fibrosis transmembrane conductance regulator (CFTR) protein for exact
inference of the conditional occupancy probability, given a finite time series
of partial observations.
