BigDataFr recommends: Behaviour of ABC for Big Data

‘Many statistical applications involve models that it is difficult to evaluate the likelihood, but relatively easy to sample from, which is called intractable likelihood. Approximate Bayesian computation (ABC) is a useful Monte Carlo method for inference of the unknown parameter in the intractable likelihood problem under Bayesian framework. Without evaluating the likelihood function, ABC approximately samples from the posterior by jointly simulating the parameter and the data and accepting/rejecting the parameter according to the distance between the simulated and observed data.

Many successful applications have been seen in population genetics, systematic biology, ecology etc. In this work, we analyse the asymptotic properties of ABC as the number of data points goes to infinity, under the assumption that the data is summarised by a fixed-dimensional statistic, and this statistic obeys a central limit theorem. We show that the ABC posterior mean for estimating a function of the parameter can be asymptotically normal, centred on the true value of the function, and with a mean square error that is equal to that of the maximum likelihood estimator based on the summary statistic.

We further analyse the efficiency of importance sampling ABC for fixed Monte Carlo sample size. For a wide-range of proposal distributions importance sampling ABC can be efficient, in the sense that the Monte Carlo error of ABC increases the mean square error of our estimate by a factor that is just 1+O(1/N), where N is the Monte Carlo sample size. ‘

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By Wentao Li, Paul Fearnhead
Source: arxiv.org