  Many modern statistical applications involve inference for complicated
stochastic models for which the likelihood function is difficult or even
impossible to calculate, and hence conventional likelihood-based inferential
echniques cannot be used. In such settings, Bayesian inference can be performed
using Approximate Bayesian Computation (ABC). However, in spite of many recent
developments to ABC methodology, in many applications the computational cost of
ABC necessitates the choice of summary statistics and tolerances that can
potentially severely bias the estimate of the posterior.
  We propose a new "piecewise" ABC approach suitable for discretely observed
Markov models that involves writing the posterior density of the parameters as
a product of factors, each a function of only a subset of the data, and then
using ABC within each factor. The approach has the advantage of side-stepping
the need to choose a summary statistic and it enables a stringent tolerance to
be set, making the posterior "less approximate". We investigate two methods for
estimating 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 is
to use a kernel density estimate. Both methods have their merits. The Gaussian
approximation is simple, fast, and probably adequate for many applications. On
the other hand, using instead a kernel density estimate has the benefit of
consistently estimating the true ABC posterior as the number of ABC samples
tends to infinity. We illustrate the piecewise ABC approach for three examples;
in each case, the approach enables "exact matching" between simulations and
data and offers fast and accurate inference.
