Bayesian Test of Significance for Conditional Independence
Pablo de Morais Andrade, Julio Michael Stern, Carlos Alberto de, Braganca Pereira · 2014
Pablo de Morais Andrade *, Julio Michael Stern and Carlos Alberto de Braganc¸a PereiraInstituto de Matematica e Estat´ ´istica, Universidade de S ao Paulo (IME-USP) Rua do Mat˜ ˜ao, 1010,Cidade Universitaria, S´ ˜ao Paulo, SP/Brasil, CEP: 05508-090; E-Mails: [email protected] (J.M.S.);[email protected] (C.A.B.P.)* Author to whom correspondence should be addressed; E-Mail: [email protected];Tel.:/Fax: +55-11-969783425.Received: 3 December 2013; in revised form: 21 February 2014 / Accepted: 5 March 2014 /Published: 7 March 2014Abstract: Conditional independence tests have received special attention lately in machinelearning and computational intelligence related literature as an important indicator of therelationship among the variables used by their models. In the field of probabilistic graphicalmodels, which includes Bayesian network models, conditional independence tests areespecially important for the task of learning the probabilistic graphical model structure fromdata. In this paper, we propose the full Bayesian significance test for tests of conditionalindependence for discrete datasets. The full Bayesian significance test is a powerfulBayesian test for precise hypothesis, as an alternative to the frequentist’s significance tests(characterized by the calculation of the p-value).Keywords: hypothesis testing; probabilistic graphical models1. IntroductionBarlow and Pereira [1] discussed a graphical approach to conditional independence. A probabilisticinfluence diagram is a directed acyclic graph (DAG) that helps model statistical problems. The graph iscomposed of a set of nodes or vertices, which represent the variables, and a set of arcs joining the nodes,which represent the dependence relationships shared by these variables.The construction of this model helps us understand the problem and gives a good representation ofthe interdependence of the implicated variables. The joint probability of these variables can be written asa product of their conditional distributions, based on their independence and conditional independence.