An integer grid bridge sampler for the Bayesian inference of incomplete birth-death records

A one-to-one correspondence is established between the bridge path space of birth-death processes and the exclusive union of the product spaces of simplexes and integer grids. Formulae are derived for the exact counting of the integer grid bridges with fixed number of upward jumps. Then a uniform sampler over such restricted bridge path space is constructed. This leads to a Monte Carlo scheme, the integer grid bridge sampler, IGBS, to evaluate the transition probabilities of birth-death processes. Even the near zero probability of rare event could now be evaluated with controlled relative error. The IGBS based Bayesian inference for the incomplete birth-death observations is readily performed in demonstrating examples and in the analysis of a severely incomplete data set recording a real epidemic event. Comparison is performed with the basic bootstrap filter, an elementary sequential importance resampling algorithm. The haunting filtering failure has found no position in the new scheme.