Standard regression approaches assume that some finite number of the response distribution characteristics, such as location and scale, change as a (parametric or nonparametric) function of predictors. However, it is not always appropriate to assume a location/scale representation, where the error distribution has unchanging shape over the predictor space. In fact, it often happens in applied research that the distribution of responses under study changes with predictors in ways that cannot be reasonably represented by a finite dimensional functional form. This can seriously affect the answers to the scientific questions of interest, and therefore more general approaches are indeed needed. This gives rise to the study of fully nonparametric regression models. We review some of the main Bayesian approaches that have been employed to define probability models where the complete response distribution may vary flexibly with predictors. We focus on developments based on modifications of the Dirichlet process, historically termed dependent Dirichlet processes, and some of the extensions that have been proposed to tackle this general problem using nonparametric approaches.
We describe a procedure to introduce general dependence structures on a set of Dirichlet processes. Dependence can be in one direction to define a time series or in two directions to define spatial dependencies. More directions can also be considered. Dependence is induced via a set of latent processes and exploit the conjugacy property between the Dirichlet and the multinomial processes to ensure that the marginal law for each element of the set is a Dirichlet process. Dependence is characterised through the correlation between any two elements. Posterior distributions are obtained when we us…
In this article, we consider a non-parametric Bayesian approach to multivariate quantile regression. The collection of related conditional distributions of a response vector Y given a univariate covariate X is modeled using a Dependent Dirichlet Process (DDP) prior. The DDP is used to introduce dependence across x. As the realizations from a Dirichlet process prior are almost surely discrete, we need to convolve it with a kernel. To model the error distribution as flexibly as possible, we use a countable mixture of multidimensional normal distributions as our kernel. For posterior computations…
A Bayesian belief network is a model of a joint distribution over a finite set of variables, with a DAG structure representing immediate dependencies among the variables. For each node, a table of parameters (CPtable) represents local conditional probabilities, with rows indexed by conditioning events (assignments to parents). CP-table rows are usually modeled as independent random vectors, each assigned a Dirichlet prior distribution. The assumption that rows are independent permits a relatively simple analysis but may not reflect actual prior opinion about the parameters. Rows representing s…
Typical IRT rating-scale models assume that the rating category threshold parameters are the same over examinees. However, it can be argued that many rating data sets violate this assumption. To address this practical psychometric problem, we introduce a novel, Bayesian nonparametric IRT model for rating scale items. The model is an infinite-mixture of Rasch partial credit models, based on a localized Dependent Dirichlet process (DDP). The model treats the rating thresholds as the random parameters that are subject to the mixture, and has (stick-breaking) mixture weights that are covariate-dep…
A new approach on the joint estimation of partially exchangeable observations is presented by constructing pairwise dependence between $m$ random density functions, each of which is modeled as a mixture of geometric stick breaking processes. This approach is based on a new random central masses version of the Pairwise Dependent Dirichlet Process prior mixture model (PDDP) first introduced in Hatjispyros et al. (2011). The idea is to create pairwise dependence through random measures that are location-preserving-expectations of Dirichlet random measures. Our contention is that mixture modeling …
We study Dirichlet process-based models for sets of predictor-dependent probability distributions, where the domain and predictor space are general Polish spaces. We generalize the definition of dependent Dirichlet processes, originally constructed on Euclidean spaces, to more general Polish spaces. We provide sufficient conditions under which dependent Dirichlet processes have appealing properties regarding continuity (weak and strong), association structure, and support (under different topologies). We also provide sufficient conditions under which mixture models induced by dependent Dirichl…
In this paper we consider the problem of dynamic clustering, where cluster memberships may change over time and clusters may split and merge over time, thus creating new clusters and destroying existing ones. We propose a Bayesian nonparametric approach to dynamic clustering via mixture modeling. Our approach relies on a novel time-dependent nonparametric prior defined by combining: i) a copula-based transformation of a Gaussian autoregressive process; ii) the stick-breaking construction of the Dirichlet process. Posterior inference is performed through a particle Markov chain Monte Carlo algo…
East China Normal University（华东师范大学）；Columbia University（哥伦比亚大学）
Dirichlet processes (DP) are widely applied in Bayesian nonparametric modeling. However, in their basic form they do not directly integrate dependency information among data arising from space and time. In this paper, we propose location dependent Dirichlet processes (LDDP) which incorporate nonparametric Gaussian processes in the DP modeling framework to model such dependencies. We develop the LDDP in the context of mixture modeling, and develop a mean field variational inference algorithm for this mixture model. The effectiveness of the proposed modeling framework is shown on an image segmen…
Bayesian paradigm takes advantage of well fitting complicated survival models and feasible computing in survival analysis owing to the superiority in tackling the complex censoring scheme, compared with the frequentist paradigm. In this chapter, we aim to display the latest tendency in Bayesian computing, in the sense of automating the posterior sampling, through Bayesian analysis of survival modeling for multivariate survival outcomes with complicated data structure. Motivated by relaxing the strong assumption of proportionality and the restriction of a common baseline population, we propose …
We consider predictive inference using a class of temporally dependent Dirichlet processes driven by Fleming--Viot diffusions, which have a natural bearing in Bayesian nonparametrics and lend the resulting family of random probability measures to analytical posterior analysis. Formulating the implied statistical model as a hidden Markov model, we fully describe the predictive distribution induced by these Fleming--Viot-driven dependent Dirichlet processes, for a sequence of observations collected at a certain time given another set of draws collected at several previous times. This is identifi…
Time series data may exhibit clustering over time and, in a multiple time series context, the clustering behavior may differ across the series. This paper is motivated by the Bayesian non--parametric modeling of the dependence between the clustering structures and the distributions of different time series. We follow a Dirichlet process mixture approach and introduce a new class of multivariate dependent Dirichlet processes (DDP). The proposed DDP are represented in terms of vector of stick-breaking processes with dependent weights. The weights are beta random vectors that determine different …
MIT（麻省理工学院）；Duke University（杜克大学）；Ohio State University（俄亥俄州立大学）
This paper presents a novel algorithm, based upon the dependent Dirichlet process mixture model (DDPMM), for clustering batch-sequential data containing an unknown number of evolving clusters. The algorithm is derived via a low-variance asymptotic analysis of the Gibbs sampling algorithm for the DDPMM, and provides a hard clustering with convergence guarantees similar to those of the k-means algorithm. Empirical results from a synthetic test with moving Gaussian clusters and a test with real ADS-B aircraft trajectory data demonstrate that the algorithm requires orders of magnitude less computa…
Dirichlet Process Mixture (DPM) models have been increasingly employed to specify random partition models that take into account possible patterns within the covariates. Furthermore, to deal with large numbers of covariates, methods for selecting the most important covariates have been proposed. Commonly, the covariates are chosen either for their importance in determining the clustering of the observations or for their effect on the level of a response variable (when a regression model is specified). Typically both strategies involve the specification of latent indicators that regulate the in…
We extend classic characterisations of posterior distributions under Dirichlet process and gamma random measures priors to a dynamic framework. We consider the problem of learning, from indirect observations, two families of time-dependent processes of interest in Bayesian nonparametrics: the first is a dependent Dirichlet process driven by a Fleming-Viot model, and the data are random samples from the process state at discrete times; the second is a collection of dependent gamma random measures driven by a Dawson-Watanabe model, and the data are collected according to a Poisson point process …
In this article, using kernel convolution of order based dependent Dirichlet process (Griffin and Steel (2006)) we construct a nonstationary, nonseparable, nonparametric space-time process, which, as we show, satisfies desirable properties, and includes the stationary, separable, parametric processes as special cases. We also investigate the smoothness properties of our proposed model. Since our model entails an infinite random series, for Bayesian model fitting purpose we must either truncate the series or more appropriately consider a random number of summands, which renders the model dimens…
Semiparametric estimation for multivariate Hawkes processes using dependent Dirichlet processes: An application to order flow data in financial markets
Compositional data are met in many different fields, such as economics, archaeometry, ecology, geology and political sciences. Regression where the dependent variable is a composition is usually carried out via a log-ratio transformation of the composition or via the Dirichlet distribution. However, when there are zero values in the data these two ways are not readily applicable. Suggestions for this problem exist, but most of them rely on substituting the zero values. In this paper we adjust the Dirichlet distribution when covariates are present, in order to allow for zero values to be presen…
