Likelihood-free inference methods can perform Bayesian inference when evaluating the likelihood is impractical but simulating synthetic data from the model is feasible. Approximate Bayesian computation (ABC) is a well-established likelihood-free approach that constructs particle posterior approximations by evaluating the similarity between simulated and observed data using a distance function, which is used in rejection or weighting steps. Here we extend previous work on adaptive distance learning for ABC to misspecified time series, while also exploring applications in neural posterior estimation using prior-data fitted networks (NPE-PFN) with localization. The adaptation of the distance that we consider optimizes out-of-sample predictive performance using a scoring rule. We also establish a connection between linear pooling for forecast combination and our posterior estimation methods with randomized distances, showing that empirical estimation of pooling weights can be interpreted as another form of adaptive distance learning. For both ABC algorithms and NPE-PFN methods with localization, adaptive distance learning improves forecasting performance in simulated and real examples.

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