The Matsumoto-Yor property on trees

The Matsumoto–Yor property on trees

Viewing the Matsumoto–Yor property as a bivariate property with respect to the simple tree with two vertices and one edge, we extend it to a p-variate property with respect to any tree with p vertices. The converse of the Matsumoto–Yor property, which characterizes the product of a gamma and a generalized inverse Gaussian distribution, is extended to characterize the product of a gamma and p 1 ...

Kummer and gamma laws through independences on trees - Another parallel with the Matsumoto-Yor property

Pitman-Yor Diffusion Trees

We introduce the Pitman Yor Diffusion Tree (PYDT) for hierarchical clustering, a generalization of the Dirichlet Diffusion Tree (Neal, 2001) which removes the restriction to binary branching structure. The generative process is described and shown to result in an exchangeable distribution over data points. We prove some theoretical properties of the model and then present two inference methods:...

Bayesian Modeling of Dependency Trees Using Hierarchical Pitman-Yor Priors

Recent work in hierarchical priors for language modeling [MacKay and Peto, 1994, Teh, 2006, Goldwater et al., 2006] has shown significant advantages to Bayesian methods in NLP. But the issue of sparse conditioning contexts is ubiquitous in NLP, and these smoothing ideas can be applied more broadly to extend the reach of Bayesian modeling in natural language. For example, a useful representation...

On an Optimality Property of Ternary Trees

In the book of Knuth (1972) an extensive discussion can be found on some problems concerning rooted trees. Given a prescribed number of end-vertices, one may ask for the binary (or ternary) tree with minimal external path length (i.e. minimal sum of path lengths from the root to end-vertices), or minimal weighted external path length (here the end-vertices have given weights). In the first case...