Entropy and inference, revisited

We study properties of popular near-uniform (Dirichlet) priors for learning undersampled probability distributions on discrete nonmetric spaces and show that they lead to disastrous results. However, an Occam-style phase space argument expands the priors into their infinite mixture and resolves most of the observed problems. This leads to a surprisingly good estimator of entropies of discrete distributions.Comment: LaTex2e, 9 pages, 5 figures; references added, minor revisions introduced, formatting errors correcte

A Parsimonious Tour of Bayesian Model Uncertainty

Modern statistical software and machine learning libraries are enabling semi-automated statistical inference. Within this context, it appears easier and easier to try and fit many models to the data at hand, reversing thereby the Fisherian way of conducting science by collecting data after the scientific hypothesis (and hence the model) has been determined. The renewed goal of the statistician becomes to help the practitioner choose within such large and heterogeneous families of models, a task known as model selection. The Bayesian paradigm offers a systematized way of assessing this problem. This approach, launched by Harold Jeffreys in his 1935 book Theory of Probability, has witnessed a remarkable evolution in the last decades, that has brought about several new theoretical and methodological advances. Some of these recent developments are the focus of this survey, which tries to present a unifying perspective on work carried out by different communities. In particular, we focus on non-asymptotic out-of-sample performance of Bayesian model selection and averaging techniques, and draw connections with penalized maximum likelihood. We also describe recent extensions to wider classes of probabilistic frameworks including high-dimensional, unidentifiable, or likelihood-free models

Prediction of particle type from measurements of particle location: A physicist's approach to Bayesian classification

The Bayesian approach to the prediction of particle type given measurements of particle location is explored, using a parametric model whose prior is based on the transformation group. Two types of particle are considered, and locations are expressed in terms of a single spatial coordinate. Several cases corresponding to different states of prior knowledge are evaluated, including the effect of measurement uncertainty. Comparisons are made to nearest neighbor classification and kernel density estimation. How one can evaluate the reliability of the prediction solely from the available data is discussed.Comment: 22 pages, 11 figures, 4 tables, minor revisio