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Register a new userA Parzen-based distance between probability measures as an alternative of summary statistics in Approximate Bayesian Computation
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Added by
Mauricio A. \\'Alvarez
Publication date
2015
fields
Mathematical Statistics
and research's language is
English
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Approximate Bayesian Computation (ABC) are likelihood-free Monte Carlo methods. ABC methods use a comparison between simulated data, using different parameters drew from a prior distribution, and observed data. This comparison process is based on computing a distance between the summary statistics from the simulated data and the observed data. For complex models, it is usually difficult to define a methodology for choosing or constructing the summary statistics. Recently, a nonparametric ABC has been proposed, that uses a dissimilarity measure between discrete distributions based on empirical kernel embeddings as an alternative for summary statistics. The nonparametric ABC outperforms other methods including ABC, kernel ABC or synthetic likelihood ABC. However, it assumes that the probability distributions are discrete, and it is not robust when dealing with few observations. In this paper, we propose to apply kernel embeddings using an smoother density estimator or Parzen estimator for comparing the empirical data distributions, and computing the ABC posterior. Synthetic data and real data were used to test the Bayesian inference of our method. We compare our method with respect to state-of-the-art methods, and demonstrate that our method is a robust estimator of the posterior distribution in terms of the number of observations.
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We present a novel family of deep neural architectures, named partially exchangeable networks (PENs) that leverage probabilistic symmetries. By design, PENs are invariant to block-switch transformations, which characterize the partial exchangeability properties of conditionally Markovian processes. Moreover, we show that any block-switch invariant function has a PEN-like representation. The DeepSets architecture is a special case of PEN and we can therefore also target fully exchangeable data. We employ PENs to learn summary statistics in approximate Bayesian computation (ABC). When comparing PENs to previous deep learning methods for learning summary statistics, our results are highly competitive, both considering time series and static models. Indeed, PENs provide more reliable posterior samples even when using less training data.
This report is a collection of comments on the Read Paper of Fearnhead and Prangle (2011), to appear in the Journal of the Royal Statistical Society Series B, along with a reply from the authors.
Approximate Bayesian Computation (ABC) methods are used to approximate posterior distributions in models with unknown or computationally intractable likelihoods. Both the accuracy and computational efficiency of ABC depend on the choice of summary statistic, but outside of special cases where the optimal summary statistics are known, it is unclear which guiding principles can be used to construct effective summary statistics. In this paper we explore the possibility of automating the process of constructing summary statistics by training deep neural networks to predict the parameters from artificially generated data: the resulting summary statistics are approximately posterior means of the parameters. With minimal model-specific tuning, our method constructs summary statistics for the Ising model and the moving-average model, which match or exceed theoretically-motivated summary statistics in terms of the accuracies of the resulting posteriors.
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