Abstract: The class of species sampling mixture models is introduced as an extension of semiparametric models based on the Dirichlet process to models based on the general class of species sampling priors, or equivalently the class of all exchangeable urn distributions. Using Fubini calculus in conjunction with Pitman (1995, 1996), we derive characterizations of the posterior distribution in terms of a posterior partition distribution that extend the results of Lo (1984) for the Dirichlet process. These results provide a better understanding of models and have both theoretical and practical applications. To facilitate the use of our models we generalize the work in Brunner, Chan, James and Lo (2001) by extending their weighted Chinese restaurant (WCR) Monte Carlo procedure, an i.i.d. sequential importance sampling (SIS) procedure for approximating posterior mean functionals based on the Dirichlet process, to the case of approximation of mean functionals and additionally their posterior laws in species sampling mixture models. We also discuss collapsed Gibbs sampling, Pólya urn Gibbs sampling and a Pólya urn SIS scheme. Our framework allows for numerous applications, including multiplicative counting process models subject to weighted gamma processes, as well as nonparametric and semiparametric hierarchical models based on the Dirichlet process, its two-parameter extension, the Pitman-Yor process and finite dimensional Dirichlet priors. Key words and phrases: Dirichlet process, exchangeable partition, finite dimensional Dirichlet prior, two-parameter Poisson-Dirichlet process, prediction rule, random probability measure, species sampling sequence.

We develop hierarchical, probabilistic models for objects, the parts composing them, and the visual scenes surrounding them. Our approach couples topic models originally developed for text analysis with spatial transformations, and thus consistently accounts for geometric constraints. By building integrated scene models, we may discover contextual relationships, and better exploit partially labeled training images. We first consider images of isolated objects, and show that sharing parts among object categories improves detection accuracy when learning from few examples. Turning to multiple object scenes, we propose nonparametric models which use Dirichlet processes to automatically learn the number of parts underlying each object category, and objects composing each scene. The resulting transformed Dirichlet process (TDP) leads to Monte Carlo algorithms which simultaneously segment and recognize objects in street and office scenes.

Many models for the study of point-referenced data explicitly introduce spatial random effects to capture residual spatial association. These spatial effects are customarily modelled as a zeromean stationary Gaussian process. The spatial Dirichlet process introduced by Gelfand et al. (2005) produces a random spatial process which is neither Gaussian nor stationary. Rather, it varies about a process that is assumed to be stationary and Gaussian. The spatial Dirichlet process arises as a probability-weighted collection of random surfaces. This can be limiting for modelling and inferential purposes since it insists that a process realization must be one of these surfaces. We introduce a random distribution for the spatial effects that allows different surface selection at different sites. Moreover, we can specify the model so that the marginal distribution of the effect at each site still comes from a Dirichlet process. The development is offered constructively, providing a multivariate extension of the stick-breaking representation of the weights. We then introduce mixing using this generalized spatial Dirichlet process. We illustrate with a simulated dataset of independent replications and note that we can embed the generalized process within a dynamic model specification to eliminate the independence assumption.

Abstract. The present paper provides exact expressions for the probability distribution of linear functionals of the two–parameter Poisson–Dirichlet process PD(α, θ). Distributional results that follow from the application of an inversion formula for a (generalized) Cauchy– Stieltjes transform are achieved. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean functional of a Poisson–Dirichlet process and the mean functional of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are illustrated. Our formulae are relevant to occupation time phenomena connected with Brownian motion and more general Bessel processes, as well as to models arising in Bayesian nonparametric statistics. 1. Introduction Let (Pi)i≥1, with P1> P2>...

Abstract — We consider the problem of state estimation for a dynamic system driven by unobserved, correlated inputs. We model these inputs via an uncertain set of temporally correlated dynamic models, where this uncertainty includes the number of modes, their associated statistics, and the rate of mode transitions. The dynamic system is formulated via two interacting graphs: a hidden Markov model (HMM) and a linear-Gaussian state space model. The HMM’s state space indexes system modes, while its outputs are the unobserved inputs to the linear dynamical system. This Markovian structure accounts for temporal persistence of input regimes, but avoids rigid assumptions about their detailed dynamics. Via a hierarchical Dirichlet process (HDP) prior, the complexity of our infinite state space robustly adapts to new observations. We present a learning algorithm and computational results that demonstrate the utility of the HDP for tracking, and show that it efficiently learns typical dynamics from noisy data.

Abstract: We focus on developing nonparametric Bayes methods for collections of dependent random functions, allowing individual curves to vary flexibly while adaptively borrowing information. A prior is proposed, which is expressed as a hierarchical mixture of weighted kernels placed at unknown locations. The induced prior for any individual function is shown to fall within a reproducing kernel Hilbert space. We allow flexible borrowing of information through the use of a hierarchical Dirichlet process prior for the random locations, along with a functional Dirich-let process for the weights. Theoretical properties are considered and an efficient MCMC algorithm is developed, relying on stick-breaking truncations. The meth-ods are illustrated using simulation examples and an application to reproductive hormone data.

The present paper provides exact expressions for the probability distributions of linear functionals of the two-parameter Poisson– Dirichlet process PD(α,θ). We obtain distributional results yielding exact forms for density functions of these functionals. Moreover, several interesting integral identities are obtained by exploiting a correspondence between the mean of a Poisson–Dirichlet process and the mean of a suitable Dirichlet process. Finally, some distributional characterizations in terms of mixture representations are proved. The usefulness of the results contained in the paper is demonstrated by means of some illustrative examples. Indeed, our formulae are relevant to occupation time phenomena connected with Brownian motion and more general Bessel processes, as well as to models arising in Bayesian nonparametric statistics.

The number of states in a hidden Markov model is an important parameter that has a critical impact on the inferred model. Bayesian approaches to addressing this issue include the nonparametric hierarchical Dirichlet process, which does not extend to a variational Bayesian solution. We present a fully conjugate, Bayesian approach to determining the number of states in a hidden Markov model, which does have a variational solution. The infinite-state hidden Markov model presented here utilizes a stick-breaking construction for each row of the state transition matrix, which allows for a sparse utilization of the same subset of observation parameters by all states. In addition to our variational solution, we discuss retrospective and collapsed Gibbs sampling methods for MCMC inference. We demonstrate our model on a music recommendation problem containing 2,250 pieces of music from the classical, jazz and rock genres.

We consider problems involving functional data where we have a collection of functions, each viewed as a process realization, e.g., a random curve or surface. For a particular process realization, we assume that the observation at a given location can be allocated to separate groups via a random allocation process, which we name the Dirichlet labeling process. We investigate properties of this process and its use as a prior in a mixture model. We develop exact and approximate representations for the labeling process, analyze the global and local clustering behavior, clarify model identifiability and posterior consistency, and develop efficient inference methods for models using such priors. Performance is demonstrated with synthetic data examples, a public-health application, and an image segmentation task.

A novel semiparametric regression model for censored data is proposed as an alternative to the widely used proportional hazards survival model. The proposed regression model for censored data turns out to be flexible and practically meaningful. Features include physical interpretation of the regression coefficients through the mean response time instead of the hazard functions, and a rigorous proof of consistency of the posterior distribution. It is shown that the regression model obtained by a mixture of parametric families, has a proportional mean structure (as in an accelerated failure time models). The statistical inference is based on a nonparametric Bayesian approach that uses a Dirichlet process prior for the mixing distribution. Consistency of the posterior distribution of the regression parameters in the Euclidean metric is established. Finite sample parameter estimates along with associated measure of uncertainties can be computed by a MCMC method. Simulation studies are presented to provide empirical validation of the new method. Some real data examples are provided to show the easy applicability of the proposed method.