Quantum Statistical Inference for Density Estimation

Book excerpt: A new penalized likelihood method for non-parametric density estimation is proposed, which is based on a mathematical analogy to quantum statistical physics. The mathematical procedure for density estimation is related to maximum entropy methods for inverse problems; the penalty function is a convex information divergence enforcing global smoothing toward default models, positivity, extensivity and normalization. The novel feature is the replacement of classical entropy by quantum entropy, so that local smoothing may be enforced by constraints on the expectation values of differential operators. Although the hyperparameters, covariance, and linear response to perturbations can be estimated by a variety of statistical methods, we develop the Bayesian interpretation. The linear response of the MAP estimate is proportional to the covariance. The hyperparameters are estimated by type-II maximum likelihood. The method is demonstrated on standard data sets.