Abstract. Three simple and explicit procedures for testing the independence of two multi-dimensional random variables are described. Two of the associated test statistics (L1, log-likelihood) are defined when the empirical distribution of the variables is restricted to finite partitions. A third test statistic is defined as a kernel-based independence measure. Two kinds of tests are provided. Distribution-free strong consistent tests are derived on the basis of large deviation bounds on the test statistcs: these tests make almost surely no Type I or Type II error after a random sample size. Asymptotically α-level tests are obtained from the limiting distribution of the test statistics. For the latter tests, the Type I error converges to a fixed non-zero value α, and the Type II error drops to zero, for increasing sample size. All tests reject the null hypothesis of independence if the test statistics become large. The performance of the tests is evaluated experimentally on benchmark data. 1

1. Introduction. A dependence statistic, the Brownian Distance Covariance, has been proposed for use in dependence measurement and independence testing: we refer to this contribution henceforth as SR [we also note the earlier work on this topic of Székely, Rizzo and Bakirov (2007)]. Some advantages of the authors’ approach are that the random variables X and Y being tested may have arbitrary dimension

In this paper, we present two classes of Bayesian approaches to the twosample problem. Our first class of methods extends the Bayesian t-test to include all parametric models in the exponential family and their conjugate priors. Our second class of methods uses Dirichlet process mixtures (DPM) of such conjugate-exponential distributions as flexible nonparametric priors over the unknown distributions. 1