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Inference in generative models using the Wasserstein distance Bernton, Espen


In purely generative models, one can simulate data given parameters but not necessarily evaluate the likelihood. We use Wasserstein distances between empirical distributions of observed data and empirical distributions of synthetic data drawn from such models to estimate their parameters. Previous interest in the Wasserstein distance for statistical inference has been mainly theoretical, due to computational limitations. Thanks to recent advances in numerical transport, the computation of these distances has become feasible for relatively large data sets, up to controllable approximation errors. We leverage these advances to propose point estimators and quasi-Bayesian distributions for parameter inference, first for independent data. For dependent data, we extend the approach by using delay reconstruction, residual reconstruction, and curve matching techniques. For large data sets, we also propose an alternative distance using the Hilbert space-filling curve, whose computation scales as $n\log n$ where $n$ is the size of the data. We provide a theoretical study of the proposed estimators, and adaptive Monte Carlo algorithms to approximate them. The approach is illustrated on several examples, including a toggle switch model from systems biology, a Lotka-Volterra model for plankton population sizes, and a L\'evy-driven stochastic volatility model.

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