Maximum Likelihood and the Information Bottleneck

Part of Advances in Neural Information Processing Systems 15 (NIPS 2002)

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Noam Slonim, Yair Weiss


that defines partitions over the values of

The information bottleneck (IB) method is an information-theoretic formulation , this method constructs for clustering problems. Given a joint distribution a new variable that are informative . Maximum likelihood (ML) of mixture models is a standard statistical about approach to clustering problems. In this paper, we ask: how are the two methods related ? We define a simple mapping between the IB problem and the ML prob- lem for the multinomial mixture model. We show that under this mapping the problems are strongly related. In fact, for uniform input distribution over or for large sample size, the problems are mathematically equivalent. Specifically, in these cases, every fixed point of the IB-functional defines a fixed point of the (log) likelihood and vice versa. Moreover, the values of the functionals at the fixed points are equal under simple transformations. As a result, in these cases, every algorithm that solves one of the problems, induces a solution for the other.