A Bayesian Multiresolution Independence Test for Continuous Variables

In this paper we present a method of computing the posterior probability of conditional independence of two or more continuous variables from data, examined at several resolutions. Our approach is motivated by the observation that the appearance of continuous data varies widely at various resolutions, producing very different independence estimates between the variables involved. Therefore, it is difficult to ascertain independence without examining data at several carefully selected resolutions. In our paper, we accomplish this using the exact computation of the posterior probability of independence, calculated analytically given a resolution. At each examined resolution, we assume a multinomial distribution with Dirichlet priors for the discretized table parameters, and compute the posterior using Bayesian integration. Across resolutions, we use a search procedure to approximate the Bayesian integral of probability over an exponential number of possible histograms. Our method generalizes to an arbitrary number variables in a straightforward manner. The test is suitable for Bayesian network learning algorithms that use independence tests to infer the network structure, in domains that contain any mix of continuous, ordinal and categorical variables.