arXiv subjects

Peter D. Hoff

Publications and source records attributed to Peter D. Hoff.

48 records · Page 3Linked to original sources

arXiv2013

Bayesian analysis of matrix data with rstiefel

We illustrate the use of the R-package "rstiefel" for matrix-variate data analysis in the context of two examples. The first example considers estimation of a reduced-rank mean matrix in the presence of normally distributed noise. The second example considers the modeling of a social network of friendships among teenagers. Bayesian estimation for these models requires the ability to simulate from the matrix-variate von Mises-Fisher distributions and the matrix-variate Bingham distributions on the Stiefel manifold.

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arXiv2013

Comment on "Bayesian Nonparametric Inference - Why and How" by Mueller and Mitra

Due to their great flexibility, nonparametric Bayes methods have proven to be a valuable tool for discovering complicated patterns in data. The term "nonparametric Bayes" suggests that these methods inherit model-free operating characteristics of classical nonparametric methods, as well as coherent uncertainty assessments provided by Bayesian procedures. However, as the authors say in the conclusion to their article, nonparametric Bayesian methods may be more aptly described as "massively parametric." Furthermore, I argue that many of the default nonparametric Bayes procedures are only Bayesian in the weakest sense of the term, and cannot be assumed to provide honest assessments of uncertainty merely because they carry the Bayesian label. However useful such procedures may be, we should be cautious about advertising default nonparametric Bayes procedures as either being "assumption free" or providing descriptions of our uncertainty. If we want our nonparametric Bayes procedures to have a Bayesian interpretation, we should modify default NP Bayes methods to accommodate real prior information, or at the very least, carefully evaluate the effects of hyperparameters on posterior quantities of interest.

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arXiv2012

Small-Sample Behavior of Novel Phase I Cancer Trial Designs

Novel dose-finding designs, using estimation to assign the best estimated maximum- tolerated-dose (MTD) at each point in the experiment, most commonly via Bayesian techniques, have recently entered large-scale implementation in Phase I cancer clinical trials. We examine the small-sample behavior of these "Bayesian Phase I" (BP1) designs, and also of non-Bayesian designs sharing the same main "long-memory" traits (hereafter: LMP1s). For all LMP1s examined, the number of cohorts treated at the true MTD (denoted here as n*) was highly variable between numerical runs drawn from the same toxicity-threshold distribution, especially when compared with "up-and-down" (U&D) short-memory designs. Further investigation using the same set of thresholds in permuted order, produced a nearly-identical magnitude of variability in n*. Therefore, this LMP1 behavior is driven by a strong sensitivity to the order in which toxicity thresholds appear in the experiment. We suggest that the sensitivity is related to LMP1's tendency to "settle" early on a specific dose level - a tendency caused by the repeated likelihood-based "winner-takes-all" dose assignment rule, which grants the early cohorts a disproportionately large influence upon experimental trajectories. Presently, U&D designs offer a simpler and more stable alternative, with roughly equivalent MTD estimation performance. A promising direction for combining the two approaches is briefly discussed (note: the '3+3' protocol is not a U&D design).

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arXiv2012

Marginally Specified Priors for Nonparametric Bayesian Estimation

Prior specification for nonparametric Bayesian inference involves the difficult task of quantifying prior knowledge about a parameter of high, often infinite, dimension. Realistically, a statistician is unlikely to have informed opinions about all aspects of such a parameter, but may have real information about functionals of the parameter, such the population mean or variance. This article proposes a new framework for nonparametric Bayes inference in which the prior distribution for a possibly infinite-dimensional parameter is decomposed into two parts: an informative prior on a finite set of functionals, and a nonparametric conditional prior for the parameter given the functionals. Such priors can be easily constructed from standard nonparametric prior distributions in common use, and inherit the large support of the standard priors upon which they are based. Additionally, posterior approximations under these informative priors can generally be made via minor adjustments to existing Markov chain approximation algorithms for standard nonparametric prior distributions. We illustrate the use of such priors in the context of multivariate density estimation using Dirichlet process mixture models, and in the modeling of high-dimensional sparse contingency tables.

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arXiv2011

A mixed effects model for longitudinal relational and network data, with applications to international trade and conflict

The focus of this paper is an approach to the modeling of longitudinal social network or relational data. Such data arise from measurements on pairs of objects or actors made at regular temporal intervals, resulting in a social network for each point in time. In this article we represent the network and temporal dependencies with a random effects model, resulting in a stochastic process defined by a set of stationary covariance matrices. Our approach builds upon the social relations models of Warner, Kenny and Stoto [Journal of Personality and Social Psychology 37 (1979) 1742--1757] and Gill and Swartz [Canad. J. Statist. 29 (2001) 321--331] and allows for an intra- and inter-temporal representation of network structures. We apply the methodology to two longitudinal data sets: international trade (continuous response) and militarized interstate disputes (binary response).

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arXiv2011

A covariance regression model

Classical regression analysis relates the expectation of a response variable to a linear combination of explanatory variables. In this article, we propose a covariance regression model that parameterizes the covariance matrix of a multivariate response vector as a parsimonious quadratic function of explanatory variables. The approach is analogous to the mean regression model, and is similar to a factor analysis model in which the factor loadings depend on the explanatory variables. Using a random-effects representation, parameter estimation for the model is straightforward using either an EM-algorithm or an MCMC approximation via Gibbs sampling. The proposed methodology provides a simple but flexible representation of heteroscedasticity across the levels of an explanatory variable, improves estimation of the mean function and gives better calibrated prediction regions when compared to a homoscedastic model.

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arXiv2010

A Statistical View of Learning in the Centipede Game

In this article we evaluate the statistical evidence that a population of students learn about the sub-game perfect Nash equilibrium of the centipede game via repeated play of the game. This is done by formulating a model in which a player's error in assessing the utility of decisions changes as they gain experience with the game. We first estimate parameters in a statistical model where the probabilities of choices of the players are given by a Quantal Response Equilibrium (QRE) (McKelvey and Palfrey, 1995, 1996, 1998), but are allowed to change with repeated play. This model gives a better fit to the data than similar models previously considered. However, substantial correlation of outcomes of games having a common player suggests that a statistical model that captures within-subject correlation is more appropriate. Thus we then estimate parameters in a model which allows for within-player correlation of decisions and rates of learning. Through out the paper we also consider and compare the use of randomization tests and posterior predictive tests in the context of exploratory and confirmatory data analyses.

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arXiv2010

Separable covariance arrays via the Tucker product, with applications to multivariate relational data

Modern datasets are often in the form of matrices or arrays,potentially having correlations along each set of data indices. For example, data involving repeated measurements of several variables over time may exhibit temporal correlation as well as correlation among the variables. A possible model for matrix-valued data is the class of matrix normal distributions, which is parametrized by two covariance matrices, one for each index set of the data. In this article we describe an extension of the matrix normal model to accommodate multidimensional data arrays, or tensors. We generate a class of array normal distributions by applying a group of multilinear transformations to an array of independent standard normal random variables. The covariance structures of the resulting class take the form of outer products of dimension-specific covariance matrices. We derive some properties of these covariance structures and the corresponding array normal distributions, discuss maximum likelihood and Bayesian estimation of covariance parameters and illustrate the model in an analysis of multivariate longitudinal network data.

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arXiv2010

Convergence of Nonparametric Long-Memory Phase I Designs

We examine nonparametric dose-finding designs that use toxicity estimates based on all available data at each dose allocation decision. We prove that one such design family, called here "interval design", converges almost surely to the maximum tolerated dose (MTD), if the MTD is the only dose level whose toxicity rate falls within the pre-specified interval around the desired target rate. Another nonparametric family, called "point design", has a positive probability of not converging. In a numerical sensitivity study, a diverse sample of dose-toxicity scenarios was randomly generated. On this sample, the "interval design" convergence conditions are met far more often than the conditions for one-parameter design convergence (the Shen-O'Quigley conditions), suggesting that the interval-design conditions are less restrictive. Implications of these theoretical and numerical results for small-sample behavior of the designs, and for future research, are discussed.

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arXiv2007

Modeling homophily and stochastic equivalence in symmetric relational data

This article discusses a latent variable model for inference and prediction of symmetric relational data. The model, based on the idea of the eigenvalue decomposition, represents the relationship between two nodes as the weighted inner-product of node-specific vectors of latent characteristics. This ``eigenmodel'' generalizes other popular latent variable models, such as latent class and distance models: It is shown mathematically that any latent class or distance model has a representation as an eigenmodel, but not vice-versa. The practical implications of this are examined in the context of three real datasets, for which the eigenmodel has as good or better out-of-sample predictive performance than the other two models.

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arXiv2007

Extending the rank likelihood for semiparametric copula estimation

Quantitative studies in many fields involve the analysis of multivariate data of diverse types, including measurements that we may consider binary, ordinal and continuous. One approach to the analysis of such mixed data is to use a copula model, in which the associations among the variables are parameterized separately from their univariate marginal distributions. The purpose of this article is to provide a method of semiparametric inference for copula models via the construction of what we call a marginal set likelihood function for the association parameters. The proposed method of inference can be viewed as a generalization of marginal likelihood estimation, in which inference for a parameter of interest is based on a summary statistic whose sampling distribution is not a function of any nuisance parameters. In the context of copula estimation, the marginal set likelihood is a function of the association parameters only and its applicability does not depend on any assumptions about the marginal distributions of the data, thus making it appropriate for the analysis of mixed continuous and discrete data with arbitrary margins. Estimation and inference for parameters of the Gaussian copula are available via a straightforward Markov chain Monte Carlo algorithm based on Gibbs sampling.

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arXiv2006

Model averaging and dimension selection for the singular value decomposition

Many multivariate data analysis techniques for an $m\times n$ matrix $\m Y$ are related to the model $\m Y = \m M +\m E$, where $\m Y$ is an $m\times n$ matrix of full rank and $\m M$ is an unobserved mean matrix of rank $K< (m\wedge n)$. Typically the rank of $\m M$ is estimated in a heuristic way and then the least-squares estimate of $\m M$ is obtained via the singular value decomposition of $\m Y$, yielding an estimate that can have a very high variance. In this paper we suggest a model-based alternative to the above approach by providing prior distributions and posterior estimation for the rank of $\m M$ and the components of its singular value decomposition. In addition to providing more accurate inference, such an approach has the advantage of being extendable to more general data-analysis situations, such as inference in the presence of missing data and estimation in a generalized linear modeling framework.

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