Tree-Structured Stick Breaking Processes for Hierarchical Data

Many data are naturally modeled by an unobserved hierarchical structure. In this paper we propose a flexible nonparametric prior over unknown data hierarchies. The approach uses nested stick-breaking processes to allow for trees of unbounded width and depth, where data can live at any node and are infinitely exchangeable. One can view our model as providing infinite mixtures where the components have a dependency structure corresponding to an evolutionary diffusion down a tree. By using a stick-breaking approach, we can apply Markov chain Monte Carlo methods based on slice sampling to perform Bayesian inference and simulate from the posterior distribution on trees. We apply our method to hierarchical clustering of images and topic modeling of text data.

A Hierarchical Model for Ordinal Matrix Factorization

This paper proposes a hierarchical probabilistic model for ordinal matrix factorization. Unlike previous approaches, we model the ordinal nature of the data and take a principled approach to incorporating priors for the hidden variables. Two algorithms are presented for inference, one based on Gibbs sampling and one based on variational Bayes. Importantly, these algorithms may be implemented in the factorization of very large matrices with missing entries. The model is evaluated on a collaborative filtering task, where users have rated a collection of movies and the system is asked to predict their ratings for other movies. The Netflix data set is used for evaluation, which consists of around 100 million ratings. Using root mean-squared error (RMSE) as an evaluation metric, results show that the suggested model outperforms alternative factorization techniques. Results also show how Gibbs sampling outperforms variational Bayes on this task, despite the large number of ratings and model parameters. Matlab implementations of the proposed algorithms are available from cogsys.imm.dtu.dk/ordinalmatrixfactorization.

Elliptical slice sampling

Many probabilistic models introduce strong dependencies between variables using a latent multivariate Gaussian distribution or a Gaussian process. We present a new Markov chain Monte Carlo algorithm for performing inference in models with multivariate Gaussian priors. Its key properties are: 1) it has simple, generic code applicable to many models, 2) it has no free parameters, 3) it works well for a variety of Gaussian process based models. These properties make our method ideal for use while model building, removing the need to spend time deriving and tuning updates for more complex algorithms.

Fixed-lag blind equalization and sequence estimation in digital communications systems using sequential importance sampling

We present methods for fixed-lag smoothing using Sequential Importance sampling (SIS) on a discrete non-linear, non-Gaussian state space system with unknown parameters. Our particular application is in the field of digital communication systems. Each input data point is taken from a finite set of symbols. We represent transmission media as a fixed filter with a finite impulse response (FIR), hence a discrete state-space system is formed. Conventional Markov chain Monte Carlo (MCMC) techniques such as the Gibbs sampler are unsuitable for this task because they can only perform processing on a batch of data. Data arrives sequentially, so it would seem sensible to process it in this way. In addition, many communication systems are interactive, so there is a maximum level of latency that can be tolerated before a symbol is decoded. We will demonstrate this method by simulation and compare its performance to existing techniques.

Bayesian model selection for time series using Markov chain Monte Carlo

We present a stochastic simulation technique for subset selection in time series models, based on the use of indicator variables with the Gibbs sampler within a hierarchical Bayesian framework. As an example, the method is applied to the selection of subset linear AR models, in which only significant lags are included. Joint sampling of the indicators and parameters is found to speed convergence. We discuss the possibility of model mixing where the model is not well determined by the data, and the extension of the approach to include non-linear model terms.

Hierarchical approach to timing verification in CMOS VLSI design

This paper describes a novel hierarchical approach to timing verification. Four types of relationship existing among signal paths are distinguished, based on a classification of the degree of interdependency in the circuit. In this way, irrelevant path delays can be excluded through consideration of the interaction between critical paths and others. Furthermore, under suitable conditions, bounded delay values for large hierarchical systems can be deduced using bounded delays determined for their constituent cells. Finally, we discuss the impact on design strategy of the hierarchical delay model presented in this paper.