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Optimal kernel choice for kernel hypothesis testing

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We consider two nonparametric hypothesis testing problems: (1) Given samples from distributions p and q, a two-sample test determines whether to reject the null hypothesis p=q; and (2) Given a joint distribution p_xy over random variables x and y, an independence test determines whether to reject the null hypothesis of independence, p_xy = p_x p_y. In testing whether two distributions are identical, or whether two random variables are independent, we require a test statistic which is a measure of distance between probability distributions. One choice of test statistic is the maximum mean discrepancy (MMD), a distance between embeddings of the probability distributions in a reproducing kernel Hilbert space. The kernel used in obtaining these embeddings is critical in ensuring the test has high power, and correctly distinguishes unlike distributions with high probability.

In this talk, I will provide a tutorial overview of kernel distances on probabilities, and show how these may be used in two-sample and independence testing. I will then describe a strategy for optimal kernel choice, and compare it with earlier heuristics (including other multiple kernel learning approaches).

Joint work with: Bharath Sriperumbudur, Dino Sejdinovic, Heiko Strathmann, Sivaraman Balakrishnan, Massimiliano Pontil, Kenji Fukumizu

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