Information-based inference for singular models and finite sample sizes
A central problem in statistics is model selection, the choice between competing models of a stochastic process whose observables are corrupted by noise. In the information-based paradigm of inference, model selection is performed by estimating the predictive performance of the com- peting models. The candidate model with the best estimated predictive performance is selected. Information-based inference is dependent on the accuracy of the estimate of the predictive complexity, a measure of the flexibility of the model in fitting the data. A large-sample-size approximation for the performance is the Akaike Information Criterion (AIC). The AIC approximation fails in a wide range of important applications, either significantly under or over-estimating the complexity. We introduce an improved approximation for the complexity which we use to define a new information criterion: the frequentist information criterion (FIC). FIC extends the applicability of information-based infer- ence to the finite-sample-size regime of regular models and to singular models. We demonstrate the power of the approach in a number of example problems.