Le"vy discovered that the fraction of time a standard one-dimensional Brownian motion B spends positive before time t has arcsine distribution, both for / a fixed time when B, #0 almost surely, and for / an inverse local time, when B, = 0 almost surely. This identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of B away from 0. Similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator, for instance a Bessel process of dimension d for 1.

We propose an unbounded-depth, hierarchical, Bayesian nonparametric model for discrete sequence data. This model can be estimated from a single training sequence, yet shares statistical strength between subsequent symbol predictive distributions in such a way that predictive performance generalizes well. The model builds on a specific parameterization of an unbounded-depth hierarchical Pitman-Yor process. We introduce analytic marginalization steps (using coagulation operators) to reduce this model to one that can be represented in time and space linear in the length of the training sequence. We show how to perform inference in such a model without truncation approximation and introduce fragmentation operators necessary to do predictive inference. We demonstrate the sequence memoizer by using it as a language model, achieving state-of-the-art results. 1.

The Indian buffet process (IBP) is an exchangeable distribution over binary matrices used in Bayesian nonparametric featural models. In this paper we propose a three-parameter generalization of the IBP exhibiting power-law behavior. We achieve this by generalizing the beta process (the de Finetti measure of the IBP) to the stable-beta process and deriving the IBP corresponding to it. We find interesting relationships between the stable-beta process and the Pitman-Yor process (another stochastic process used in Bayesian nonparametric models with interesting power-law properties). We derive a stick-breaking construction for the stable-beta process, and find that our power-law IBP is a good model for word occurrences in document corpora. 1