Pitman-Yor Process Research Articles

Limit theorems associated with the Pitman–Yor process

Abstract The Pitman–Yor process is a random discrete measure. The random weights or masses follow the two-parameter Poisson–Dirichlet distribution with parameters 0 < α < 1, θ > -α. The parameters α and θ correspond to the stable and gamma components, respectively. The distribution of atoms is given by a probability η. In this paper we consider the limit theorems for the Pitman–Yor process and the two-parameter Poisson–Dirichlet distribution. These include the law of large numbers, fluctuations, and moderate or large deviation principles. The limiting procedures involve either α tending to 0 or 1. They arise naturally in genetics and physics such as the asymptotic coalescence time for explosive branching process and the approximation to the generalized random energy model for disordered systems.

A Marginal Sampler for σ-Stable Poisson–Kingman Mixture Models

ABSTRACTWe investigate the class of σ-stable Poisson–Kingman random probability measures (RPMs) in the context of Bayesian nonparametric mixture modeling. This is a large class of discrete RPMs, which encompasses most of the popular discrete RPMs used in Bayesian nonparametrics, such as the Dirichlet process, Pitman–Yor process, the normalized inverse Gaussian process, and the normalized generalized Gamma process. We show how certain sampling properties and marginal characterizations of σ-stable Poisson–Kingman RPMs can be usefully exploited for devising a Markov chain Monte Carlo (MCMC) algorithm for performing posterior inference with a Bayesian nonparametric mixture model. Specifically, we introduce a novel and efficient MCMC sampling scheme in an augmented space that has a small number of auxiliary variables per iteration. We apply our sampling scheme to a density estimation and clustering tasks with unidimensional and multidimensional datasets, and compare it against competing MCMC sampling schemes. Supplementary materials for this article are available online.

Hierarchical Back-off Modeling of Hiero Grammar based on Non-parametric Bayesian Model

In hierarchical phrase-based machine translation, a rule table is automatically learned by heuristically extracting synchronous rules from a parallel corpus. As a result, spuriously many rules are extracted which may be composed of various incorrect rules. The larger rule table incurs more run time for decoding and may result in lower translation quality. To resolve the problems, we propose a hierarchical back-off model for Hiero grammar, an instance of a synchronous context free grammar (SCFG), on the basis of the hierarchical Pitman-Yor process. The model can extract a compact rule and phrase table without resorting to any heuristics by hierarchically backing off to smaller phrases under SCFG. Inference is efficiently carried out using two-step synchronous parsing of Xiao et al., (2012) combined with slice sampling. In our experiments, the proposed model achieved higher or at least comparable translation quality against a previous Bayesian model on various language pairs; German/French/Spanish/JapaneseEnglish. When compared against heuristic models, our model achieved comparable translation quality on a full size GermanEnglish language pair in Europarl v7 corpus with significantly smaller grammar size; less than 10% of that for heuristic model.

Generalized póya urn for time-varying pitman-yor processes

This article introduces a class of first-order stationary time-varying Pitman-Yor processes. Subsuming our construction of time-varying Dirichlet processes presented in (Caron et al., 2007), these ...

A semiparametric Bayesian joint model for multiple mixed-type outcomes: an application to acute myocardial infarction

We propose a Bayesian semiparametric regression model to represent mixed-type multiple outcomes concerning patients affected by Acute Myocardial Infarction. Our approach is motivated by data coming from the ST-Elevation Myocardial Infarction (STEMI) Archive, a multi-center observational prospective clinical study planned as part of the Strategic Program of Lombardy, Italy. We specifically consider a joint model for a variable measuring treatment time and in-hospital and 60-day survival indicators. One of our main motivations is to understand how the various hospitals differ in terms of the variety of information collected as part of the study. To do so we postulate a semiparametric random effects model that incorporates dependence on a location indicator that is used to explicitly differentiate among hospitals in or outside the city of Milano. The model is based on the two parameter Poisson-Dirichlet prior, also known as the Pitman-Yor process prior. We discuss the resulting posterior inference, including sensitivity analysis, and a comparison with the particular sub-model arising when a Dirichlet process prior is assumed.

Nonparametric Bayesian topic modelling with the hierarchical Pitman–Yor processes

The Dirichlet process and its extension, the Pitman–Yor process, are stochastic processes that take probability distributions as a parameter. These processes can be stacked up to form a hierarchical nonparametric Bayesian model. In this article, we present efficient methods for the use of these processes in this hierarchical context, and apply them to latent variable models for text analytics. In particular, we propose a general framework for designing these Bayesian models, which are called topic models in the computer science community. We then propose a specific nonparametric Bayesian topic model for modelling text from social media. We focus on tweets (posts on Twitter) in this article due to their ease of access. We find that our nonparametric model performs better than existing parametric models in both goodness of fit and real world applications.

A menu-driven software package of Bayesian nonparametric (and parametric) mixed models for regression analysis and density estimation.

Most of applied statistics involves regression analysis of data. In practice, it is important to specify a regression model that has minimal assumptions which are not violated by data, to ensure that statistical inferences from the model are informative and not misleading. This paper presents a stand-alone and menu-driven software package, Bayesian Regression: Nonparametric and Parametric Models, constructed from MATLAB Compiler. Currently, this package gives the user a choice from 83 Bayesian models for data analysis. They include 47 Bayesian nonparametric (BNP) infinite-mixture regression models; 5 BNP infinite-mixture models for density estimation; and 31 normal random effects models (HLMs), including normal linear models. Each of the 78 regression models handles either a continuous, binary, or ordinal dependent variable, and can handle multi-level (grouped) data. All 83 Bayesian models can handle the analysis of weighted observations (e.g., for meta-analysis), and the analysis of left-censored, right-censored, and/or interval-censored data. Each BNP infinite-mixture model has a mixture distribution assigned one of various BNP prior distributions, including priors defined by either the Dirichlet process, Pitman-Yor process (including the normalized stable process), beta (two-parameter) process, normalized inverse-Gaussian process, geometric weights prior, dependent Dirichlet process, or the dependent infinite-probits prior. The software user can mouse-click to select a Bayesian model and perform data analysis via Markov chain Monte Carlo (MCMC) sampling. After the sampling completes, the software automatically opens text output that reports MCMC-based estimates of the model's posterior distribution and model predictive fit to the data. Additional text and/or graphical output can be generated by mouse-clicking other menu options. This includes output of MCMC convergence analyses, and estimates of the model's posterior predictive distribution, for selected functionals and values of covariates. The software is illustrated through the BNP regression analysis of real data.

Extracting Topical Phrases from Clinical Documents

In clinical documents, medical terms are often expressed in multi-word phrases. Traditional topic modelling approaches relying on the "bag-of-words" assumption are not effective in extracting topic themes from clinical documents. This paper proposes to first extract medical phrases using an off-the-shelf tool for medical concept mention extraction, and then train a topic model which takes a hierarchy of Pitman-Yor processes as prior for modelling the generation of phrases of arbitrary length. Experimental results on patients' discharge summaries show that the proposed approach outperforms the state-of-the-art topical phrase extraction model on both perplexity and topic coherence measure and finds more interpretable topics.

Online Learning of Hierarchical Pitman-Yor Process Mixture of Generalized Dirichlet Distributions With Feature Selection.

In this paper, a novel statistical generative model based on hierarchical Pitman-Yor process and generalized Dirichlet distributions (GDs) is presented. The proposed model allows us to perform joint clustering and feature selection thanks to the interesting properties of the GD distribution. We develop an online variational inference algorithm, formulated in terms of the minimization of a Kullback-Leibler divergence, of our resulting model that tackles the problem of learning from high-dimensional examples. This variational Bayes formulation allows simultaneously estimating the parameters, determining the model's complexity, and selecting the appropriate relevant features for the clustering structure. Moreover, the proposed online learning algorithm allows data instances to be processed in a sequential manner, which is critical for large-scale and real-time applications. Experiments conducted using challenging applications, namely, scene recognition and video segmentation, where our approach is viewed as an unsupervised technique for visual learning in high-dimensional spaces, showed that the proposed approach is suitable and promising.

Preordering using a Target-Language Parser via Cross-Language Syntactic Projection for Statistical Machine Translation

When translating between languages with widely different word orders, word reordering can present a major challenge. Although some word reordering methods do not employ source-language syntactic structures, such structures are inherently useful for word reordering. However, high-quality syntactic parsers are not available for many languages. We propose a preordering method using a target-language syntactic parser to process source-language syntactic structures without a source-language syntactic parser. To train our preordering model based on ITG, we produced syntactic constituent structures for source-language training sentences by (1) parsing target-language training sentences, (2) projecting constituent structures of the target-language sentences to the corresponding source-language sentences, (3) selecting parallel sentences with highly synchronized parallel structures, (4) producing probabilistic models for parsing using the projected partial structures and the Pitman-Yor process, and (5) parsing to produce full binary syntactic structures maximally synchronized with the corresponding target-language syntactic structures, using the constraints of the projected partial structures and the probabilistic models. Our ITG-based preordering model is trained using the produced binary syntactic structures and word alignments. The proposed method facilitates the learning of ITG by producing highly synchronized parallel syntactic structures based on cross-language syntactic projection and sentence selection. The preordering model jointly parses input sentences and identifies their reordered structures. Experiments with Japanese--English and Chinese--English patent translation indicate that our method outperforms existing methods, including string-to-tree syntax-based SMT, a preordering method that does not require a parser, and a preordering method that uses a source-language dependency parser.

Are Gibbs-Type Priors the Most Natural Generalization of the Dirichlet Process?

Discrete random probability measures and the exchangeable random partitions they induce are key tools for addressing a variety of estimation and prediction problems in Bayesian inference. Here we focus on the family of Gibbs-type priors, a recent elegant generalization of the Dirichlet and the Pitman-Yor process priors. These random probability measures share properties that are appealing both from a theoretical and an applied point of view: (i) they admit an intuitive predictive characterization justifying their use in terms of a precise assumption on the learning mechanism; (ii) they stand out in terms of mathematical tractability; (iii) they include several interesting special cases besides the Dirichlet and the Pitman-Yor processes. The goal of our paper is to provide a systematic and unified treatment of Gibbs-type priors and highlight their implications for Bayesian nonparametric inference. We deal with their distributional properties, the resulting estimators, frequentist asymptotic validation and the construction of time-dependent versions. Applications, mainly concerning mixture models and species sampling, serve to convey the main ideas. The intuition inherent to this class of priors and the neat results they lead to make one wonder whether it actually represents the most natural generalization of the Dirichlet process.

Latent IBP Compound Dirichlet Allocation.

We introduce the four-parameter IBP compound Dirichlet process (ICDP), a stochastic process that generates sparse non-negative vectors with potentially an unbounded number of entries. If we repeatedly sample from the ICDP we can generate sparse matrices with an infinite number of columns and power-law characteristics. We apply the four-parameter ICDP to sparse nonparametric topic modelling to account for the very large number of topics present in large text corpora and the power-law distribution of the vocabulary of natural languages. The model, which we call latent IBP compound Dirichlet allocation (LIDA), allows for power-law distributions, both, in the number of topics summarising the documents and in the number of words defining each topic. It can be interpreted as a sparse variant of the hierarchical Pitman-Yor process when applied to topic modelling. We derive an efficient and simple collapsed Gibbs sampler closely related to the collapsed Gibbs sampler of latent Dirichlet allocation (LDA), making the model applicable in a wide range of domains. Our nonparametric Bayesian topic model compares favourably to the widely used hierarchical Dirichlet process and its heavy tailed version, the hierarchical Pitman-Yor process, on benchmark corpora. Experiments demonstrate that accounting for the power-distribution of real data is beneficial and that sparsity provides more interpretable results.

Adaptive Bayesian Density Estimation in Lp-metrics with Pitman-Yor or Normalized Inverse-Gaussian Process Kernel Mixtures

We consider Bayesian nonparametric density estimation using a Pitman-Yor or a normalized inverse-Gaussian process convolution kernel mixture as the prior distribution for a density. The procedure is studied from a frequentist perspective. Using the stick-breaking representation of the Pitman-Yor process and the finite-dimensional distributions of the normalized inverse-Gaussian process, we prove that, when the data are independent replicates from a density with analytic or Sobolev smoothness, the posterior distribution concentrates on shrinking Lp-norm balls around the sampling density at a minimax-optimal rate, up to a logarithmic factor. The resulting hierarchical Bayesian procedure, with a fixed prior, is adaptive to the unknown smoothness of the sampling density.

Gaussian Process-Mixture Conditional Heteroscedasticity.

Generalized autoregressive conditional heteroscedasticity (GARCH) models have long been considered as one of the most successful families of approaches for volatility modeling in financial return series. In this paper, we propose an alternative approach based on methodologies widely used in the field of statistical machine learning. Specifically, we propose a novel nonparametric Bayesian mixture of Gaussian process regression models, each component of which models the noise variance process that contaminates the observed data as a separate latent Gaussian process driven by the observed data. This way, we essentially obtain a Gaussian process-mixture conditional heteroscedasticity (GPMCH) model for volatility modeling in financial return series. We impose a nonparametric prior with power-law nature over the distribution of the model mixture components, namely the Pitman-Yor process prior, to allow for better capturing modeled data distributions with heavy tails and skewness. Finally, we provide a copula-based approach for obtaining a predictive posterior for the covariances over the asset returns modeled by means of a postulated GPMCH model. We evaluate the efficacy of our approach in a number of benchmark scenarios, and compare its performance to state-of-the-art methodologies.

Genre-Based Music Language Modeling with Latent Hierarchical Pitman-Yor Process Allocation

In this work we present a new Bayesian topic model: latent hierarchical Pitman-Yor process allocation (LHPYA), which uses hierarchical Pitman-Yor process priors for both word and topic distributions, and generalizes a few of the existing topic models, including the latent Dirichlet allocation (LDA), the bigram topic model and the hierarchical Pitman-Yor topic model. Using such priors allows for integration of n-grams with a topic model, while smoothing them with the state-of-the-art method. Our model is evaluated by measuring its perplexity on a dataset of musical genre and harmony annotations 3 Genre Database (3GDB) and by measuring its ability to predict musical genre from chord sequences. In terms of perplexity, for a 262-chord dictionary we achieve a value of 2.74, compared to 18.05 for trigrams and 7.73 for a unigram topic model. In terms of genre prediction accuracy with 9 genres, the proposed approach performs about 33% better in relative terms than genre-dependent n-grams, achieving 60.4% of accuracy.

Beta-product dependent Pitman–Yor processes for Bayesian inference

Multiple time series data may exhibit clustering over time and the clustering effect may change across different series. This paper is motivated by the Bayesian non-parametric modelling of the dependence between clustering effects in multiple time series analysis. We follow a Dirichlet process mixture approach and define a new class of multivariate dependent Pitman–Yor processes (DPY). The proposed DPY are represented in terms of vectors of stick-breaking processes which determine dependent clustering structures in the time series. We follow a hierarchical specification of the DPY base measure to account for various degrees of information pooling across the series. We discuss some theoretical properties of the DPY and use them to define Bayesian non-parametric repeated measurement and vector autoregressive models. We provide efficient Monte Carlo Markov Chain algorithms for posterior computation of the proposed models and illustrate the effectiveness of the method with a simulation study and an application to the United States and the European Union business cycle.

Bayesian Constituent Context Model for Grammar Induction

Constituent Context Model (CCM) is an effective generative model for grammar induction, the aim of which is to induce hierarchical syntactic structure from natural text. The CCM simply defines the Multinomial distribution over constituents, which leads to a severe data sparse problem because long constituents are unlikely to appear in unseen data sets. This paper proposes a Bayesian method for constituent smoothing by defining two kinds of prior distributions over constituents: the Dirichlet prior and the Pitman-Yor Process prior. The Dirichlet prior functions as an additive smoothing method, and the PYP prior functions as a back-off smoothing method. Furthermore, a modified CCM is proposed to differentiate left constituents and right constituents in binary branching trees. Experiments show that both the proposed Bayesian smoothing method and the modified CCM are effective, and combining them attains or significantly improves the state-of-the-art performance of grammar induction evaluated on standard treebanks of various languages.

Autonomous tactile perception: A combined improved sensing and Bayesian nonparametric approach

In recent years, autonomous robots have increasingly been deployed in unknown environments and required to manipulate or categorize unknown objects. In order to cope with these unfamiliar situations, improvements must be made both in sensing technologies and in the capability to autonomously train perception models. In this paper, we explore this problem in the context of tactile surface identification and categorization. Using a highly-discriminant tactile probe based upon large bandwidth, triple axis accelerometer that is sensitive to surface texture and material properties, we demonstrate that unsupervised learning for surface identification with this tactile probe is feasible. To this end, we derived a Bayesian nonparametric approach based on Pitman–Yor processes to model power-law distributions, an extension of our previous work using Dirichlet processes Dallaire et al. (2011). When tested against a large collection of surfaces and without providing the actual number of surfaces, the tactile probe combined with our proposed approach demonstrated near-perfect recognition in many cases and achieved perfect recognition given the right conditions. We consider that our combined improvements demonstrate the feasibility of effective autonomous tactile perception systems.

A latent variable Gaussian process model with Pitman–Yor process priors for multiclass classification

Gaussian processes (GPs) constitute one of the most important Bayesian machine learning approaches. Several researchers have considered postulating mixtures of Gaussian processes as a means of dealing with non-stationary covariance functions, discontinuities, multi-modality, and overlapping output signals. In existing works, mixtures of Gaussian processes are based on the introduction of a gating function defined over the space of model input variables. This way, each postulated mixture component Gaussian process is effectively restricted in a limited subset of the input space. Additionally, the applicability of these models is limited to regression tasks. In this paper, for the first time in the literature, we devise a Gaussian process mixture model especially suitable for multiclass classification applications: We consider a GP classification scheme the prior distribution of which is a fully generative nonparametric Bayesian model with power-law behavior, generating Gaussian processes over the whole input space of the learned task. We provide an efficient algorithm for model inference, based on the variational Bayesian framework, and exhibit its efficacy using benchmark and real-world classification datasets.

Producing Power-Law Distributions and Damping Word Frequencies with Two-Stage Language Models

Standard statistical models of language fail to capture one of the most striking properties of natural languages: the power-law distribution in the frequencies of word tokens. We present a framework for developing statistical models that can generically produce power laws, breaking generative models into two stages. The first stage, the generator, can be any standard probabilistic model, while the second stage, the adaptor, transforms the word frequencies of this model to provide a closer match to natural language. We show that two commonly used Bayesian models, the Dirichlet-multinomial model and the Dirichlet process, can be viewed as special cases of our framework. We discuss two stochastic processes---the Chinese restaurant process and its two-parameter generalization based on the Pitman-Yor process---that can be used as adaptors in our framework to produce power-law distributions over word frequencies. We show that these adaptors justify common estimation procedures based on logarithmic or inverse-power transformations of empirical frequencies. In addition, taking the Pitman-Yor Chinese restaurant process as an adaptor justifies the appearance of type frequencies in formal analyses of natural language and improves the performance of a model for unsupervised learning of morphology.