|details:||Stuart Barber, Jochen Voss and Mark Webster: The Rate of Convergence for Approximate Bayesian Computation. Electronic Journal of Statistics, vol. 9, pp. 80–105, 2015.|
|keywords:||Approximate Bayesian Computation, likelihood-free inference, Monte Carlo methods, convergence of estimators, rate of convergence|
Approximate Bayesian Computation (ABC) is a popular computational
method for likelihood-free Bayesian inference. The term
likelihood-free refers to problems where the likelihood is
intractable to compute or estimate directly, but where it is possible to
generate simulated data X relatively easily given a candidate
set of parameters θ simulated from a prior distribution.
Parameters which generate simulated data within some
tolerance δ of the observed data xast
are regarded as plausible, and a collection of such θ is used to
estimate the posterior distribution
θ,|,X!=!xast. Suitable choice
of δ is vital for ABC methods to return good approximations
to θ in reasonable computational time. While ABC methods are
widely used in practice, particularly in population genetics, study of the
mathematical properties of ABC estimators is still in its infancy. We
prove that ABC estimates converge to the exact solution under very weak
assumptions and, under slightly stronger assumptions, quantify the rate of
this convergence. Our results can be used to guide the choice of the
tolerance parameter δ.
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