See also:Book ReviewSome Tests of Significance, Treated by the Theory of Probability. Mathematical Proceedings of the Cambridge Philosophical Society, olume 31, Issue 02, April 1935, pp 203-222.Universidad de la República, Montevideo, Uruguay Facultad de Agronomia.LXXV. On some aspects of the theory of probability. Philosophical Magazine Series 6, Volume 38, Issue 228, 1919.Probability, Statistics, and the Theory of Errors. Proceedings of the Royal Society of London, Vol. 140, No. 842, Jun. 1, 1933.Clarendon Press

Leppert, Robert, Zimmermann, Karl-Heinz

Machine learning provides algorithms that can learn from data and make inferences or predictions on data. Bayesian networks are a class of graphical models that allow to represent a collection of random variables and their condititional dependencies by directed acyclic graphs. In this paper, an inference algorithm for the hidden random variables of a Bayesian network is given by using the tropicalization of the marginal distribution of the observed variables. By restricting the topological structure to graded networks, an inference algorithm for graded Bayesian networks will be established that evaluates the hidden random variables rank by rank and in this way yields the most probable states of the hidden variables. This algorithm can be viewed as a generalized version of the Viterbi algorithm for graded Bayesian networks.

A new method is developed to represent probabilistic relations on multiple random events. Where previously knowledge bases containing probabilistic rules were used for this purpose, here a probability distribution over the relations is directly represented by a Bayesian network. By using a powerful way of specifying conditional probability distributions in these networks, the resulting formalism is more expressive than the previous ones. Particularly, it provides for constraints on equalities of events, and it allows to define complex, nested combination functions.

Welcome to the next episode in my series of answering questions and provoking thought in Data Analytics and Machine learning. An intern at Optisol who is undergoing the Data Analytics boot camp that we are running mentioned that Bayesian statistics topic was very dry and boring at the Amity university online degree on Big Data that she is pursuing. No offense to the Amity people, but my response is if Bayesian seems boring, it has to be the instructor or the curriculum that should take the blame.

This paper is concerned with two theories of probability judgment: the Bayesian theory and the theory of belief functions. It illustrates these theories with some simple examples and discusses some of the issues that arise when we try to implement them in expert systems. The Bayesian theory is well known; its main ideas go back to the work of Thomas Bayes (1702-1761). The theory of belief functions, often called the Dempster-Shafer theory in the artificial intelligence community, is less well known, but it has even older antecedents; belief-function arguments appear in the work of George Hooper (16401723) and James Bernoulli (1654-1705). For elementary expositions of the theory of belief functions, see Shafer (1976, 1985).