The classical Bayes rule computes the posterior model probability from the prior probability and the data likelihood. We generalize this rule to the case when the prior is a density matrix (symmetric positive definite and trace one) and the data likelihood a covariance matrix. The classical Bayes rule is retained as the special case when the matrices are diagonal. In the classical setting, the ...

We present the system nFOIL. It tightly integrates the naı̈ve Bayes learning scheme with the inductive logic programming rule-learner FOIL. In contrast to previous combinations, which have employed naı̈ve Bayes only for post-processing the rule sets, nFOIL employs the naı̈ve Bayes criterion to directly guide its search. Experimental evidence shows that nFOIL performs better than both its base line...

This paper provides an axiomatic characterization of Bayes’ Rule that is widely used for updating beliefs. Bayes’ Rule is viewed as a revision rule. Consider an agent whose belief about a set of states is characterized by a point in a unit simplex of appropriate dimension. Now new information emerges that rules out the possible occurrence of some of the states. The revision rule then assigns ne...

This paper axiomatizes updating rules for preferences that are not necessarily in the expected utility class. Two sets of results are presented. The first is the axiomatization of conditional preferences. The second consists of the axiomatization of three updating rules: the traditional Bayes rule, the Dempster-Shafer rule, and the generalized Bayes rule. The last rule can be regarded as the up...

Within the Kolmogorov theory of probability, Bayes’ rule allows one to perform statistical inference by relating conditional probabilities to unconditional probabilities. As we show here, however, there is a continuous set of alternative inference rules that yield the same results, and that may have computational or practical advantages for certain problems. We formulate generalized axioms for ...

We present the system nFOIL. It tightly integrates the naı̈ve Bayes learning scheme with the inductive logic programming rule-learner FOIL. In contrast to previous combinations, which have employed naı̈ve Bayes only for post-processing the rule sets, nFOIL employs the naı̈ve Bayes criterion to directly guide its search. Experimental evidence shows that nFOIL performs better than both its base line...

There is a current emphasis on making the introductory statistics class more dataoriented. Data distributions are used to motivate probability distributions, such as the normal, and sampling distributions. However, di culties remain in communicating the basic tenets of traditional statistical procedures such as con dence intervals and hypothesis tests. It is shown that skills in understanding r...

In this paper, we re-visit the foundations of the statistical approach to machine translation and study two forms of the Bayes decision rule: the common rule for minimizing the number of string errors and a novel rule for minimizing the number of symbol errors. The Bayes decision rule for minimizing the number of string errors is widely used, but its justification is rarely questioned. We study...

Bayes’ rule is a fundamental and general mechanism that composes hypotheses and data, which is arguably the purpose of many computer programs, and yet few programming languages or libraries incorporate Bayes’ rule as an abstraction. Bayes’ rule has several benefits. (1) It provides a formalism for programs to express composition of data from multiple sources to improve the program’s accuracy, e...

We present a novel approach for reconciling hierarchical forecasts, based on Bayes’ rule. define prior distribution the bottom time series of hierarchy, base forecasts. Then we update their via rule, forecasts upper series. Under Gaussian assumption, derive updating in closed-form. two algorithms, which differ as assumed independencies. discuss relation with MinT reconciliation algorithm and Ka...