Understanding Hierarchical Processes

Hierarchical stochastic processes, such as the hierarchical Dirichlet process, hold an important position as a modelling tool in statistical machine learning, and are even used in deep neural networks. They allow, for instance, networks of probability vectors to be used in general statistical modelling, intrinsically supporting information sharing through the network. This paper presents a general theory of hierarchical stochastic processes and illustrates its use on the gamma process and the generalised gamma process. In general, most of the convenient properties of hierarchical Dirichlet processes extend to the broader family. The main construction for this corresponds to estimating the moments of an infinitely divisible distribution based on its cumulants. Various equivalences and relationships can then be applied to networks of hierarchical processes. Examples given demonstrate the duplication in non-parametric research, and presents plots of the Pitman-Yor distribution.

More for less: predicting and maximizing genomic variant discovery via Bayesian nonparametrics

While the cost of sequencing genomes has decreased dramatically in recent years, this expense often remains nontrivial. Under a fixed budget, scientists face a natural trade-off between quantity and quality: spending resources to sequence a greater number of genomes or spending resources to sequence genomes with increased accuracy. Our goal is to find the optimal allocation of resources between quantity and quality. Optimizing resource allocation promises to reveal as many new variations in the genome as possible. We introduce a Bayesian nonparametric methodology to predict the number of new variants in a follow-up study based on a pilot study. When experimental conditions are kept constant between the pilot and follow-up, we find that our prediction is competitive with the best existing methods. Unlike current methods, though, our new method allows practitioners to change experimental conditions between the pilot and the follow-up. We demonstrate how this distinction allows our method to be used for more realistic predictions and for optimal allocation of a fixed budget between quality and quantity. We validate our method on cancer and human genomics data.