  In this paper we present a method ofcomputing the posterior probability
ofconditional independence of two or morecontinuous variables from
data,examined at several resolutions. Ourapproach is motivated by
theobservation that the appearance ofcontinuous data varies widely atvarious
resolutions, producing verydifferent independence estimatesbetween the
variablesinvolved. Therefore, it is difficultto ascertain independence
withoutexamining data at several carefullyselected resolutions. In our paper,
weaccomplish this using the exactcomputation of the posteriorprobability of
independence, calculatedanalytically given a resolution. Ateach examined
resolution, we assume amultinomial distribution with Dirichletpriors for the
discretized tableparameters, and compute the posteriorusing Bayesian
integration. Acrossresolutions, we use a search procedureto approximate the
Bayesian integral ofprobability over an exponential numberof possible
histograms. Our methodgeneralizes to an arbitrary numbervariables in a
straightforward manner.The test is suitable for Bayesiannetwork learning
algorithms that useindependence tests to infer the networkstructure, in domains
that contain anymix of continuous, ordinal andcategorical variables.
