Modified Dirichlet Distribution: Allowing Negative Parameters to Induce Stronger Sparsity

The Dirichlet distribution (Dir) is one of the most widely used prior distributions in statistical approaches to natural language processing. The parameters of Dir are required to be positive, which significantly limits its strength as a sparsity prior. In this paper, we propose a simple modification to the Dirichlet distribution that allows the parameters to be negative. Our modified Dirichlet distribution (mDir) not only induces much stronger sparsity, but also simultaneously performs smoothing. mDir is still conjugate to the multinomial distribution, which simplifies posterior inference. We introduce two simple and efficient algorithms for finding the mode of mDir. Our experiments on learning Gaussian mixtures and unsupervised dependency parsing demonstrate the advantage of mDir over Dir. 1 Dirichlet Distribution The Dirichlet distribution (Dir) is defined over probability vectors x = 〈x1, . . . , xn〉 with positive parameter vector α = 〈α1, . . . , αn〉: