In this chapter I review Bayesian statistics as used for induction and relate it to logic based abduction Much reasoning under un certainty including induction is based on Bayes rule Bayes rule is interesting precisely because it provides a mechanism for abduc tion I review work of Buntine that argues that much of the work on Bayesian learning can be best viewed in terms of graphical models suc...

The chain-rule processor is a method of constructing an optimal Bayes classifier from a bank of processors. Each processor is a feature extractor designed to separate the given class from a class-dependent reference hypothesis, thereby avoiding the curse of dimensionality. This work builds upon prior work in optimal classifier design using class-specific features. The chain-rule processor is an...

This lecture covers the basics of core concepts in probability and statistics to be used in the course. These include random variables, continuous and discrete distributions, joint and conditional distributions, the chain rule, marginalization, Bayes Rule, independence and conditional independence, and expectation. Probability models are discussed along with the concepts of independently and id...

A complete algorithm is presented for the sharpening of imprecise information, based on the methodology of kernel estimators and the Bayes decision rule, including conditioning factors. The use of the Bayes rule with a nonsymmetrical loss function enables the inclusion of different results of an underand overestimation of a sharp value (real number), as well as minimizing potential losses. A co...

In recent years, a considerable amount of work has been devoted to generalizing linear discriminant analysis to overcome its incompetence for high-dimensional classification (Witten and Tibshirani, 2011, Cai and Liu, 2011, Mai et al., 2012 and Fan et al., 2012). In this paper, we develop high-dimensional sparse semiparametric discriminant analysis (SSDA) that generalizes the normal-theory discr...

The year 2013 marks the 250th anniversary of Bayes rule, one of the two fundamental inferential principles of mathematical statistics. The rule has been influential over the entire period—and controversial over most of it. Its reliance on prior beliefs has been challenged by frequentism, which focuses instead on the behavior of specific estimates and tests under repeated use. Twentieth-century ...

Recently, Brafman and Engel (2009) proposed new concepts of marginal and conditional utility that obey additive analogues of the chain rule and Bayes rule, which they employed to obtain a directed graphical model of utility functions that resembles Bayes nets. In this paper we carry this analogy a step farther by showing that the notion of utility independence, built on conditional utility, sat...

In this chapter I review Bayesian statistics as used for induction and relate it to logic-based abduction. Much reasoning under uncertainty , including induction, is based on Bayes' rule. Bayes' rule is interesting precisely because it provides a mechanism for abduction. I review work of Buntine that argues that much of the work on Bayesian learning can be best viewed in terms of graphical mode...

In this note we consider the problem of, given a sample, selecting the number of bins in a histogram. A loss function is introduced which reflects the idea that smooth distributions should have fewer bins than rough distributions. A stepwise Bayes rule, based on the Bayesian bootstrap, is found and is shown to be admissible. Some simulation results are presented to show how the rule works in pr...

Bayes’ inference cannot reliably account for uncertainty in mental processes. The reason is that based on the assumption order which relevant features are evaluated indifferent, not case most of Instead rule, a more general, probabilistic rule capable accounting these effects established. This new can be used to improve current Bayesian models cognition. Moreover, it should play an essential ro...