Fearnhead, Paul and Meligkotsidou, Loukia
(2004)
*Exact filtering for partially-oberved continuous-time models.*
Journal of the Royal Statistical Society: Series B (Statistical Methodology), 66 (3).
pp. 771-789.
ISSN 1369-7412

## Abstract

Summary. The forward–backward algorithm is an exact filtering algorithm which can efficiently calculate likelihoods, and which can be used to simulate from posterior distributions. Using a simple result which relates gamma random variables with different rates, we show how the forward–backward algorithm can be used to calculate the distribution of a sum of gamma random variables, and to simulate from their joint distribution given their sum. One application is to calculating the density of the time of a specific event in a Markov process, as this time is the sum of exponentially distributed interevent times. This enables us to apply the forward–backward algorithm to a range of new problems. We demonstrate our method on three problems: calculating likelihoods and simulating allele frequencies under a non-neutral population genetic model, analysing a stochastic epidemic model and simulating speciation times in phylogenetics.

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