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How best to summarize large and complex datasets is a problem that arises in many areas of science.  We approach it from the point of view of seeking data summaries that minimize the average squared error of the posterior distribution for a parameter of interest under approximate Bayesian computation (ABC).  In ABC, simulation under the model replaces computation of the likelihood, which is convenient for many complex models. Simulated and observed datasets are usually compared using summary statistics, typically in practice chosen on the basis of the investigator's intuition and established practice in the field.  We propose two algorithms for automated choice of efficient data summaries.  Firstly, we motivate minimisation of the estimated entropy of the posterior approximation as a heuristic for the selection of summary statistics.  Secondly, we propose a two-stage procedure: the minimum-entropy algorithm  is used to identify simulated datasets close to that observed, and these are each successively regarded as observed datasets for which the mean root integrated squared error of the ABC posterior approximation is minimized over sets of summary statistics.  In a simulation study, we both singly and jointly inferred the scaled mutation and recombination parameters from a population sample of DNA sequences.  The computationally-fast minimum entropy algorithm showed a modest improvement over existing methods while our two-stage procedure showed substantial and highly-significant further improvement for both univariate and bivariate inferences.  We found that the optimal set of summary statistics was highly dataset specific, suggesting that more generally there may be no globally-optimal choice, which argues for a new selection for each dataset even if the model and target of inference are unchanged.

On Optimal Selection of Summary Statistics for Approximate Bayesian Computation