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Many modern statistical applications involve inference for complicatedstochastic models for which the likelihood function is difficult or evenimpossible to calculate, and hence conventional likelihood-based inferentialechniques cannot be used. In such settings, Bayesian inference can be performedusing Approximate Bayesian Computation (ABC). However, in spite of many recentdevelopments to ABC methodology, in many applications the computational cost ofABC necessitates the choice of summary statistics and tolerances that canpotentially severely bias the estimate of the posterior.  We propose a new "piecewise" ABC approach suitable for discretely observedMarkov models that involves writing the posterior density of the parameters asa product of factors, each a function of only a subset of the data, and thenusing ABC within each factor. The approach has the advantage of side-steppingthe need to choose a summary statistic and it enables a stringent tolerance tobe set, making the posterior "less approximate". We investigate two methods forestimating the posterior density based on ABC samples for each of the factors:the first is to use a Gaussian approximation for each factor, and the second isto use a kernel density estimate. Both methods have their merits. The Gaussianapproximation is simple, fast, and probably adequate for many applications. Onthe other hand, using instead a kernel density estimate has the benefit ofconsistently estimating the true ABC posterior as the number of ABC samplestends to infinity. We illustrate the piecewise ABC approach for three examples;in each case, the approach enables "exact matching" between simulations anddata and offers fast and accurate inference.

Fast Approximate Bayesian Computation for discretely observed Markov models using a factorised posterior distribution