Entropic Priors for Discrete Probabilistic Networks and for Mixtures of Gaussians Models

The ongoing unprecedented exponential explosion of available computing power, has radically transformed the methods of statistical inference. What used to be a small minority of statisticians advocating for the use of priors and a strict adherence to bayes theorem, it is now becoming the norm across disciplines. The evolutionary direction is now clear. The trend is towards more realistic, flexible and complex likelihoods characterized by an ever increasing number of parameters. This makes the old question of: What should the prior be? to acquire a new central importance in the modern bayesian theory of inference. Entropic priors provide one answer to the problem of prior selection. The general definition of an entropic prior has existed since 1988, but it was not until 1998 that it was found that they provide a new notion of complete ignorance. This paper re-introduces the family of entropic priors as minimizers of mutual information between the data and the parameters, as in [rodriguez98b], but with a small change and a correction. The general formalism is then applied to two large classes of models: Discrete probabilistic networks and univariate finite mixtures of gaussians. It is also shown how to perform inference by efficiently sampling the corresponding posterior distributions.Comment: 24 pages, 3 figures, Presented at MaxEnt2001, APL Johns Hopkins University, August 4-9 2001. See also http://omega.albany.edu:8008

Wrong Priors

All priors are not created equal. There are right and there are wrong priors. That is the main conclusion of this contribution. I use, a cooked-up example designed to create drama, and a typical textbook example to show the pervasiveness of wrong priors in standard statistical practice.Comment: 9 pages, 8 figures. MaxEnt2007.org pape

Optimal Recovery of Local Truth

Probability mass curves the data space with horizons. Let f be a multivariate probability density function with continuous second order partial derivatives. Consider the problem of estimating the true value of f(z) > 0 at a single point z, from n independent observations. It is shown that, the fastest possible estimators (like the k-nearest neighbor and kernel) have minimum asymptotic mean square errors when the space of observations is thought as conformally curved. The optimal metric is shown to be generated by the Hessian of f in the regions where the Hessian is definite. Thus, the peaks and valleys of f are surrounded by singular horizons when the Hessian changes signature from Riemannian to pseudo-Riemannian. Adaptive estimators based on the optimal variable metric show considerable theoretical and practical improvements over traditional methods. The formulas simplify dramatically when the dimension of the data space is 4. The similarities with General Relativity are striking but possibly illusory at this point. However, these results suggest that nonparametric density estimation may have something new to say about current physical theory.Comment: To appear in Proceedings of Maximum Entropy and Bayesian Methods 1999. Check also: http://omega.albany.edu:8008

A triple comparison between anticipating stochastic integrals in financial modeling

We consider a simplified version of the problem of insider trading in a financial market. We approach it by means of anticipating stochastic calculus and compare the use of the Hitsuda-Skorokhod, the Ayed-Kuo, and the Russo-Vallois forward integrals within this context. Our results give some indication that, while the forward integral yields results with a suitable financial meaning, the Hitsuda-Skorokhod and the Ayed-Kuo integrals do not provide an appropriate formulation of this problem. Further results regarding the use of the Ayed-Kuo integral in this context are also provided, including the proof of the fact that the expectation of a Russo-Vallois solution is strictly greater than that of an Ayed-Kuo solution. Finally, we conjecture the explicit solution of an Ayed-Kuo stochastic differential equation that possesses discontinuous sample paths with finite probability

Tilings of quadriculated annuli

Tilings of a quadriculated annulus A are counted according to volume (in the formal variable q) and flux (in p). We consider algebraic properties of the resulting generating function Phi_A(p,q). For q = -1, the non-zero roots in p must be roots of unity and for q > 0, real negative.Comment: 33 pages, 12 figures; Minor changes were made to make some passages cleare