arXiv subjects

Daniel M. Roy

Publications and source records attributed to Daniel M. Roy.

79 records · Page 5Linked to original sources

arXiv2015

Particle Gibbs for Bayesian Additive Regression Trees

Additive regression trees are flexible non-parametric models and popular off-the-shelf tools for real-world non-linear regression. In application domains, such as bioinformatics, where there is also demand for probabilistic predictions with measures of uncertainty, the Bayesian additive regression trees (BART) model, introduced by Chipman et al. (2010), is increasingly popular. As data sets have grown in size, however, the standard Metropolis-Hastings algorithms used to perform inference in BART are proving inadequate. In particular, these Markov chains make local changes to the trees and suffer from slow mixing when the data are high-dimensional or the best fitting trees are more than a few layers deep. We present a novel sampler for BART based on the Particle Gibbs (PG) algorithm (Andrieu et al., 2010) and a top-down particle filtering algorithm for Bayesian decision trees (Lakshminarayanan et al., 2013). Rather than making local changes to individual trees, the PG sampler proposes a complete tree to fit the residual. Experiments show that the PG sampler outperforms existing samplers in many settings.

stat.ML↗

arXiv2015

Bayesian Models of Graphs, Arrays and Other Exchangeable Random Structures

The natural habitat of most Bayesian methods is data represented by exchangeable sequences of observations, for which de Finetti's theorem provides the theoretical foundation. Dirichlet process clustering, Gaussian process regression, and many other parametric and nonparametric Bayesian models fall within the remit of this framework; many problems arising in modern data analysis do not. This article provides an introduction to Bayesian models of graphs, matrices, and other data that can be modeled by random structures. We describe results in probability theory that generalize de Finetti's theorem to such data and discuss their relevance to nonparametric Bayesian modeling. With the basic ideas in place, we survey example models available in the literature; applications of such models include collaborative filtering, link prediction, and graph and network analysis. We also highlight connections to recent developments in graph theory and probability, and sketch the more general mathematical foundation of Bayesian methods for other types of data beyond sequences and arrays.

math.ST↗

arXiv2014

The continuum-of-urns scheme, generalized beta and Indian buffet processes, and hierarchies thereof

We describe the combinatorial stochastic process underlying a sequence of conditionally independent Bernoulli processes with a shared beta process hazard measure. As shown by Thibaux and Jordan [TJ07], in the special case when the underlying beta process has a constant concentration function and a finite and nonatomic mean, the combinatorial structure is that of the Indian buffet process (IBP) introduced by Griffiths and Ghahramani [GG05]. By reinterpreting the beta process introduced by Hjort [Hjo90] as a measurable family of Dirichlet processes, we obtain a simple predictive rule for the general case, which can be thought of as a continuum of Blackwell-MacQueen urn schemes (or equivalently, one-parameter Hoppe urn schemes). The corresponding measurable family of Perman-Pitman-Yor processes leads to a continuum of two-parameter Hoppe urn schemes, whose ordinary component is the three-parameter IBP introduced by Teh and Görür [TG09], which exhibits power-law behavior, as further studied by Broderick, Jordan, and Pitman [BJP12]. The idea extends to arbitrary measurable families of exchangeable partition probability functions and gives rise to generalizations of the beta process with matching buffet processes. Finally, in the same way that hierarchies of Dirichlet processes were given Chinese restaurant franchise representations by Teh, Jordan, Beal, and Blei [Teh+06], one can construct representations of sequences of Bernoulli processes directed by hierarchies of beta processes (and their generalizations) using the stochastic process we uncover.

math.PR↗

arXiv2014

Church: a language for generative models

We introduce Church, a universal language for describing stochastic generative processes. Church is based on the Lisp model of lambda calculus, containing a pure Lisp as its deterministic subset. The semantics of Church is defined in terms of evaluation histories and conditional distributions on such histories. Church also includes a novel language construct, the stochastic memoizer, which enables simple description of many complex non-parametric models. We illustrate language features through several examples, including: a generalized Bayes net in which parameters cluster over trials, infinite PCFGs, planning by inference, and various non-parametric clustering models. Finally, we show how to implement query on any Church program, exactly and approximately, using Monte Carlo techniques.

cs.PL↗

arXiv2013

Towards common-sense reasoning via conditional simulation: legacies of Turing in Artificial Intelligence

The problem of replicating the flexibility of human common-sense reasoning has captured the imagination of computer scientists since the early days of Alan Turing's foundational work on computation and the philosophy of artificial intelligence. In the intervening years, the idea of cognition as computation has emerged as a fundamental tenet of Artificial Intelligence (AI) and cognitive science. But what kind of computation is cognition? We describe a computational formalism centered around a probabilistic Turing machine called QUERY, which captures the operation of probabilistic conditioning via conditional simulation. Through several examples and analyses, we demonstrate how the QUERY abstraction can be used to cast common-sense reasoning as probabilistic inference in a statistical model of our observations and the uncertain structure of the world that generated that experience. This formulation is a recent synthesis of several research programs in AI and cognitive science, but it also represents a surprising convergence of several of Turing's pioneering insights in AI, the foundations of computation, and statistics.

cs.AI↗

arXiv2013

Top-down particle filtering for Bayesian decision trees

Decision tree learning is a popular approach for classification and regression in machine learning and statistics, and Bayesian formulations---which introduce a prior distribution over decision trees, and formulate learning as posterior inference given data---have been shown to produce competitive performance. Unlike classic decision tree learning algorithms like ID3, C4.5 and CART, which work in a top-down manner, existing Bayesian algorithms produce an approximation to the posterior distribution by evolving a complete tree (or collection thereof) iteratively via local Monte Carlo modifications to the structure of the tree, e.g., using Markov chain Monte Carlo (MCMC). We present a sequential Monte Carlo (SMC) algorithm that instead works in a top-down manner, mimicking the behavior and speed of classic algorithms. We demonstrate empirically that our approach delivers accuracy comparable to the most popular MCMC method, but operates more than an order of magnitude faster, and thus represents a better computation-accuracy tradeoff.

stat.ML↗

arXiv2011

Computable de Finetti measures

We prove a computable version of de Finetti's theorem on exchangeable sequences of real random variables. As a consequence, exchangeable stochastic processes expressed in probabilistic functional programming languages can be automatically rewritten as procedures that do not modify non-local state. Along the way, we prove that a distribution on the unit interval is computable if and only if its moments are uniformly computable.

math.LO↗