Nonparametric Bayesian Statistics Research Articles

그리드 단체 위의 디리슐레 분포에서 마르코프 연쇄 몬테 칼로 표집

With the recent machine learning paradigm of using nonparametric Bayesian statistics and statistical inference based on random sampling, the Dirichlet distribution finds many uses in a variety of graphical models. It is a multivariate generalization of the gamma distribution and is defined on a continuous (K-1)-simplex. This paper presents a sampling method for a Dirichlet distribution for the problem of dividing an integer X into a sequence of K integers which sum to X. The target samples in our problem are all positive integer vectors when multiplied by a given X. They must be sampled from the correspondingly gridded simplex. In this paper we develop a Markov Chain Monte Carlo (MCMC) proposal distribution for the neighborhood grid points on the simplex and then present the complete algorithm based on the Metropolis-Hastings algorithm. The proposed algorithm can be used for the Markov model, HMM, and Semi-Markov model for accurate state-duration modeling. It can also be used for the Gamma-Dirichlet HMM to model q the global-local duration distributions.

Learning grounded finite-state representations from unstructured demonstrations

Robots exhibit flexible behavior largely in proportion to their degree of knowledge about the world. Such knowledge is often meticulously hand-coded for a narrow class of tasks, limiting the scope of possible robot competencies. Thus, the primary limiting factor of robot capabilities is often not the physical attributes of the robot, but the limited time and skill of expert programmers. One way to deal with the vast number of situations and environments that robots face outside the laboratory is to provide users with simple methods for programming robots that do not require the skill of an expert. For this reason, learning from demonstration (LfD) has become a popular alternative to traditional robot programming methods, aiming to provide a natural mechanism for quickly teaching robots. By simply showing a robot how to perform a task, users can easily demonstrate new tasks as needed, without any special knowledge about the robot. Unfortunately, LfD often yields little knowledge about the world, and thus lacks robust generalization capabilities, especially for complex, multi-step tasks. We present a series of algorithms that draw from recent advances in Bayesian non-parametric statistics and control theory to automatically detect and leverage repeated structure at multiple levels of abstraction in demonstration data. The discovery of repeated structure provides critical insights into task invariants, features of importance, high-level task structure, and appropriate skills for the task. This culminates in the discovery of a finite-state representation of the task, composed of grounded skills that are flexible and reusable, providing robust generalization and transfer in complex, multi-step robotic tasks. These algorithms are tested and evaluated using a PR2 mobile manipulator, showing success on several complex real-world tasks, such as furniture assembly.

Ensemble Registration of Multisensor Images by a Variational Bayesian Approach

In this paper, a novel image registration method, called ensemble image registration, is proposed. We use an infinite Gaussian mixture model (IGMM), which is based on a joint Gaussian mixture model and a Dirichlet process (DP), to model the joint intensity scatter plot (JISP) of the unregistered images. The DP is the cornerstone of nonparametric Bayesian statistics, and has capability of determining a proper number of mixing components. To simultaneously register a group of images, the cost function of reducing the dispersion in the JISP is preformed by a Bayesian method using IGMM. A variational Bayesian framework is developed to inference the posterior distribution of the parameters in the IGMM. To evaluate the performance of the proposed method, experiments of ensemble image registration are presented. The results show that the proposed method has improved performance compared with conventional methods.

On the topological support of species sampling priors

In Bayesian nonparametric statistics, it is crucial that the support of the prior is very large. Here, we consider species sampling priors. Such priors are widely used within mixture models and it has been shown in the literature that a large support for the mixing prior is essential to ensure the consistency of the posterior. In this paper, simple conditions are given that are necessary and sufficient for the support of a species sampling prior to be full. In particular, for proper species sampling priors, the condition is that the maximum size of the atoms of the corresponding process is small with positive probability. We apply this result to show that the main classes of species sampling priors known in literature have full support under mild conditions. Moreover, we find priors with a very simple construction still having full support.

Bayesian entropy estimation for countable discrete distributions

We consider the problem of estimating Shannon's entropy H from discrete data, in cases where the number of possible symbols is unknown or even countably infinite. The process, a generalization of Dirichlet process, provides a tractable prior distribution over the space of countably infinite discrete distributions, and has found major applications in Bayesian non-parametric statistics and machine learning. Here we show that it provides a natural family of priors for Bayesian entropy estimation, due to the fact that moments of the induced posterior distribution over H can be computed analytically. We derive formulas for the posterior mean (Bayes' least squares estimate) and variance under Dirichlet and process priors. Moreover, we show that a fixed Dirichlet or process prior implies a narrow prior distribution over H, meaning the prior strongly determines the entropy estimate in the under-sampled regime. We derive a family of continuous measures for mixing processes to produce an approximately flat prior over H. We show that the resulting Pitman-Yor Mixture (PYM) entropy estimator is consistent for a large class of distributions. Finally, we explore the theoretical properties of the resulting estimator, and show that it performs well both in simulation and in application to real data.

A Bayesian nonparametric goodness of fit test for right censored data based on approximate samples from the beta‐Stacy process

AbstractIn recent years, Bayesian nonparametric statistics has received extraordinary attention. The beta‐Stacy process, a generalization of the Dirichlet process, is a fundamental tool in studying Bayesian nonparametric statistics. In this article, we derive a simple, yet efficient, way to simulate the beta‐Stacy process. We compare the efficiency of the new approximation to several other well‐known approximations, and we demonstrate a significant improvement. Using the Kolmogorov distance and samples from the beta‐Stacy process, a Bayesian nonparametric goodness of fit test is proposed. The proposed test is very general in the sense that it can be applied to censored and non‐censored observations. Some illustrative examples are included. 41: 466–487; 2013 © 2013 Statistical Society of Canada

MCMC Methods for Functions: Modifying Old Algorithms to Make Them Faster

Many problems arising in applications result in the need to probe a probability distribution for functions. Examples include Bayesian nonparametric statistics and conditioned diffusion processes. Standard MCMC algorithms typically become arbitrarily slow under the mesh refinement dictated by nonparametric description of the unknown function. We describe an approach to modifying a whole range of MCMC methods, applicable whenever the target measure has density with respect to a Gaussian process or Gaussian random field reference measure, which ensures that their speed of convergence is robust under mesh refinement. Gaussian processes or random fields are fields whose marginal distributions, when evaluated at any finite set of $N$ points, are $\mathbb{R}^N$-valued Gaussians. The algorithmic approach that we describe is applicable not only when the desired probability measure has density with respect to a Gaussian process or Gaussian random field reference measure, but also to some useful non-Gaussian reference measures constructed through random truncation. In the applications of interest the data is often sparse and the prior specification is an essential part of the overall modelling strategy. These Gaussian-based reference measures are a very flexible modelling tool, finding wide-ranging application. Examples are shown in density estimation, data assimilation in fluid mechanics, subsurface geophysics and image registration. The key design principle is to formulate the MCMC method so that it is, in principle, applicable for functions; this may be achieved by use of proposals based on carefully chosen time-discretizations of stochastic dynamical systems which exactly preserve the Gaussian reference measure. Taking this approach leads to many new algorithms which can be implemented via minor modification of existing algorithms, yet which show enormous speed-up on a wide range of applied problems.

Gaussian Process Retrieval of Chlorophyll Content From Imaging Spectroscopy Data

Precise and spatially-explicit knowledge of leaf chlorophyll content <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$(Chl)$</tex></formula> is crucial to adequately interpret the chlorophyll fluorescence <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$(ChF)$</tex></formula> signal from space. Accompanying information about the reliability of the <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Chl$</tex></formula> estimation becomes more important than ever. Recently, a new statistical method was proposed within the family of nonparametric Bayesian statistics, namely Gaussian Processes regression (GPR). GPR is simpler and more robust than their machine learning family members while maintaining very good numerical performance and stability. Other features include: i) GPR requires a relatively small training data set and can adopt very flexible kernels, ii) GPR identifies the relevant bands and observations in establishing relationships with a variable, and finally iii) along with pixelwise estimations GPR provides accompanying confidence intervals. We used GPR to retrieve <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Chl$</tex> </formula> from hyperspectral reflectance data and evaluated the portability of the regression model to other images. Based on field <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex Notation="TeX">$Chl$</tex></formula> measurements from the SPARC dataset and corresponding spaceborne CHRIS spectra (acquired in 2003, Barrax, Spain), GPR developed a regression model that was excellently validated ( <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$r^{2}$</tex></formula> : 0.96, RMSE: 3.82 <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$\mu{\rm g/cm}^{2}$</tex> </formula> ). The SPARC-trained GPR model was subsequently applied to CHRIS images (Barrax, 2003, 2009) and airborne CASI flightlines (Barrax 2009) to generate <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Chl$</tex></formula> maps. The accompanying confidence maps provided insight in the robustness of the retrievals. Similar confidences were achieved by both sensors, which is encouraging for upscaling <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$Chl$</tex> </formula> estimates from field to landscape scale. Because of its robustness and ability to deliver confidence intervals, GPR is evaluated as a promising candidate for implementation into <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$ChF$</tex> </formula> processing chains.

A nonparametric Bayesian framework for constructing flexible feature representations.

Representations are a key explanatory device used by cognitive psychologists to account for human behavior. Understanding the effects of context and experience on the representations people use is essential, because if two people encode the same stimulus using different representations, their response to that stimulus may be different. We present a computational framework that can be used to define models that flexibly construct feature representations (where by a feature we mean a part of the image of an object) for a set of observed objects, based on nonparametric Bayesian statistics. Austerweil and Griffiths (2011) presented an initial model constructed in this framework that captures how the distribution of parts affects the features people use to represent a set of objects. We build on this work in three ways. First, although people use features that can be transformed on each observation (e.g., translate on the retinal image), many existing feature learning models can only recognize features that are not transformed (occur identically each time). Consequently, we extend the initial model to infer features that are invariant over a set of transformations, and learn different structures of dependence between feature transformations. Second, we compare two possible methods for capturing the manner that categorization affects feature representations. Finally, we present a model that learns features incrementally, capturing an effect of the order of object presentation on the features people learn. We conclude by considering the implications and limitations of our empirical and theoretical results.

Optimal scaling and diffusion limits for the Langevin algorithm in high dimensions

The Metropolis-adjusted Langevin (MALA) algorithm is a sampling algorithm which makes local moves by incorporating information about the gradient of the logarithm of the target density. In this paper we study the efficiency of MALA on a natural class of target measures supported on an infinite dimensional Hilbert space. These natural measures have density with respect to a Gaussian random field measure and arise in many applications such as Bayesian nonparametric statistics and the theory of conditioned diffusions. We prove that, started in stationarity, a suitably interpolated and scaled version of the Markov chain corresponding to MALA converges to an infinite dimensional diffusion process. Our results imply that, in stationarity, the MALA algorithm applied to an $N$-dimensional approximation of the target will take $\mathcal{O}(N^{1/3})$ steps to explore the invariant measure, comparing favorably with the Random Walk Metropolis which was recently shown to require $\mathcal{O}(N)$ steps when applied to the same class of problems. As a by-product of the diffusion limit, it also follows that the MALA algorithm is optimized at an average acceptance probability of $0.574$. Previous results were proved only for targets which are products of one-dimensional distributions, or for variants of this situation, limiting their applicability. The correlation in our target means that the rescaled MALA algorithm converges weakly to an infinite dimensional Hilbert space valued diffusion, and the limit cannot be described through analysis of scalar diffusions. The limit theorem is proved by showing that a drift-martingale decomposition of the Markov chain, suitably scaled, closely resembles a weak Euler–Maruyama discretization of the putative limit. An invariance principle is proved for the martingale, and a continuous mapping argument is used to complete the proof.

Frontiers in Nonparametric Statistics

The goal of this workshop was to discuss recent developments of nonparametric statistical inference. A particular focus was on high dimensional statistics, semiparametrics, adaptation, nonparametric bayesian statistics, shape constraint estimation and statistical inverse problems. The close interaction of these issues with optimization, machine learning and inverse problems has been addressed as well.

The infinite Student's t-factor mixture analyzer for robust clustering and classification

Recently, the Student's t-factor mixture analyzer (tFMA) has been proposed. Compared with the mixture of Student's t-factor analyzers (MtFA), the tFMA has better performance when processing high-dimensional data. Moreover, the factors estimated by the tFMA can be visualized in a low-dimensional latent space, which is not shared by the MtFA. However, as the tFMA belongs to finite mixtures and the related parameter estimation method is based on the maximum likelihood criterion, it could not automatically determine the appropriate model complexity according to the observed data, leading to overfitting. In this paper, we propose an infinite Student's t-factor mixture analyzer (itFMA) to handle this issue. The itFMA is based on the nonparametric Bayesian statistics which assumes infinite number of mixing components in advance, and automatically determines the proper number of components after observing the high-dimensional data. Moreover, we derive an efficient variational inference algorithm for the itFMA. The proposed itFMA and the related variational inference algorithm are used to cluster and classify high-dimensional data. Experimental results of some applications show that the itFMA has good generalization capacity, offering a more robust and powerful performance than other competing approaches.

A nonparametric Bayesian approach toward robot learning by demonstration

In the past years, many authors have considered application of machine learning methodologies to effect robot learning by demonstration. Gaussian mixture regression (GMR) is one of the most successful methodologies used for this purpose. A major limitation of GMR models concerns automatic selection of the proper number of model states, i.e., the number of model component densities. Existing methods, including likelihood- or entropy-based criteria, usually tend to yield noisy model size estimates while imposing heavy computational requirements. Recently, Dirichlet process (infinite) mixture models have emerged in the cornerstone of nonparametric Bayesian statistics as promising candidates for clustering applications where the number of clusters is unknown a priori. Under this motivation, to resolve the aforementioned issues of GMR-based methods for robot learning by demonstration, in this paper we introduce a nonparametric Bayesian formulation for the GMR model, the Dirichlet process GMR model. We derive an efficient variational Bayesian inference algorithm for the proposed model, and we experimentally investigate its efficacy as a robot learning by demonstration methodology, considering a number of demanding robot learning by demonstration scenarios.

Bayesian Learning of a Language Model from Continuous Speech

We propose a novel scheme to learn a language model (LM) for automatic speech recognition (ASR) directly from continuous speech. In the proposed method, we first generate phoneme lattices using an acoustic model with no linguistic constraints, then perform training over these phoneme lattices, simultaneously learning both lexical units and an LM. As a statistical framework for this learning problem, we use non-parametric Bayesian statistics, which make it possible to balance the learned model's complexity (such as the size of the learned vocabulary) and expressive power, and provide a principled learning algorithm through the use of Gibbs sampling. Implementation is performed using weighted finite state transducers (WFSTs), which allow for the simple handling of lattice input. Experimental results on natural, adult-directed speech demonstrate that LMs built using only continuous speech are able to significantly reduce ASR phoneme error rates. The proposed technique of joint Bayesian learning of lexical units and an LM over lattices is shown to significantly contribute to this improvement.

A rational model of the effects of distributional information on feature learning

Most psychological theories treat the features of objects as being fixed and immediately available to observers. However, novel objects have an infinite array of properties that could potentially be encoded as features, raising the question of how people learn which features to use in representing those objects. We focus on the effects of distributional information on feature learning, considering how a rational agent should use statistical information about the properties of objects in identifying features. Inspired by previous behavioral results on human feature learning, we present an ideal observer model based on nonparametric Bayesian statistics. This model balances the idea that objects have potentially infinitely many features with the goal of using a relatively small number of features to represent any finite set of objects. We then explore the predictions of this ideal observer model. In particular, we investigate whether people are sensitive to how parts co-vary over objects they observe. In a series of four behavioral experiments (three using visual stimuli, one using conceptual stimuli), we demonstrate that people infer different features to represent the same four objects depending on the distribution of parts over the objects they observe. Additionally in all four experiments, the features people infer have consequences for how they generalize properties to novel objects. We also show that simple models that use the raw sensory data as inputs and standard dimensionality reduction techniques (principal component analysis and independent component analysis) are insufficient to explain our results.

Bayesian nonparametric model with clustering individual co-exposure to pesticides found in the French diet

This work introduces a specific application of Bayesian nonparametric statistics to the food risk analysis framework. The goal was to determine the cocktails of pesticide residues to which the French population is simultaneously exposed through its current diet in order to study their possible combined effects on health through toxicological experiments. To do this, the joint distribution of exposures to a large number of pesticides, which we called the co-exposure distribution, was assessed from the available consumption data and food contamination analyses. We proposed modelling the co-exposure using a Dirichlet process mixture based on a multivariate Gaussian kernel so as to determine groups of individuals with similar co-exposure patterns. Posterior distributions and the optimal partition were computed through a Gibbs sampler based on stick-breaking priors. The study of the correlation matrix of the sub-population co-exposures will be used to define the cocktails of pesticides to which they are jointly exposed at high doses. To reduce the computational burden due to the high data dimensionality, a random-block sampling approach was used. In addition, we propose to account for the uncertainty of food contamination through the introduction of an additional level of hierarchy in the model. The results of both specifications are described and compared.

The Infinite Hidden Markov Random Field Model

Hidden Markov random field (HMRF) models are widely used for image segmentation, as they appear naturally in problems where a spatially constrained clustering scheme is asked for. A major limitation of HMRF models concerns the automatic selection of the proper number of their states, i.e., the number of region clusters derived by the image segmentation procedure. Existing methods, including likelihood- or entropy-based criteria, and reversible Markov chain Monte Carlo methods, usually tend to yield noisy model size estimates while imposing heavy computational requirements. Recently, Dirichlet process (DP, infinite) mixture models have emerged in the cornerstone of nonparametric Bayesian statistics as promising candidates for clustering applications where the number of clusters is unknown a priori; infinite mixture models based on the original DP or spatially constrained variants of it have been applied in unsupervised image segmentation applications showing promising results. Under this motivation, to resolve the aforementioned issues of HMRF models, in this paper, we introduce a nonparametric Bayesian formulation for the HMRF model, the infinite HMRF model, formulated on the basis of a joint Dirichlet process mixture (DPM) and Markov random field (MRF) construction. We derive an efficient variational Bayesian inference algorithm for the proposed model, and we experimentally demonstrate its advantages over competing methodologies.

On the posterior distribution of classes of random means

The study of properties of mean functionals of random probability measures is an important area of research in the theory of Bayesian nonparametric statistics. Many results are now known for random Dirichlet means, but little is known, especially in terms of posterior distributions, for classes of priors beyond the Dirichlet process. In this paper, we consider normalized random measures with independent increments (NRMI's) and mixtures of NRMI. In both cases, we are able to provide exact expressions for the posterior distribution of their means. These general results are then specialized, leading to distributional results for means of two important particular cases of NRMI's and also of the two-parameter Poisson--Dirichlet process.

Rational approximations to rational models: Alternative algorithms for category learning.

Rational models of cognition typically consider the abstract computational problems posed by the environment, assuming that people are capable of optimally solving those problems. This differs from more traditional formal models of cognition, which focus on the psychological processes responsible for behavior. A basic challenge for rational models is thus explaining how optimal solutions can be approximated by psychological processes. We outline a general strategy for answering this question, namely to explore the psychological plausibility of approximation algorithms developed in computer science and statistics. In particular, we argue that Monte Carlo methods provide a source of rational process models that connect optimal solutions to psychological processes. We support this argument through a detailed example, applying this approach to Anderson's (1990, 1991) rational model of categorization (RMC), which involves a particularly challenging computational problem. Drawing on a connection between the RMC and ideas from nonparametric Bayesian statistics, we propose 2 alternative algorithms for approximate inference in this model. The algorithms we consider include Gibbs sampling, a procedure appropriate when all stimuli are presented simultaneously, and particle filters, which sequentially approximate the posterior distribution with a small number of samples that are updated as new data become available. Applying these algorithms to several existing datasets shows that a particle filter with a single particle provides a good description of human inferences.

A Bernstein-Von Mises Theorem for discrete probability distributions

We investigate the asymptotic normality of the posterior distribution in the discrete setting, when model dimension increases with sample size. We consider a probability mass function $\theta_0$ on $\mathbbm{N}\setminus \{0\}$ and a sequence of truncation levels $(k_n)_n$ satisfying $k_n^3\leq n\inf_{i\leq k_n}\theta_0(i).$ Let $\hat{\theta}$ denote the maximum likelihood estimate of $(\theta_0(i))_{i\leq k_n}$ and let $\Delta_n(\theta_0)$ denote the $k_n$-dimensional vector which $i$-th coordinate is defined by \sqrt{n} (\hat{\theta}_n(i)-\theta_0(i)) for $1\leq i\leq k_n.$ We check that under mild conditions on $\theta_0$ and on the sequence of prior probabilities on the $k_n$-dimensional simplices, after centering and rescaling, the variation distance between the posterior distribution recentered around $\hat{\theta}_n$ and rescaled by $\sqrt{n}$ and the $k_n$-dimensional Gaussian distribution $\mathcal{N}(\Delta_n(\theta_0),I^{-1}(\theta_0))$ converges in probability to $0.$ This theorem can be used to prove the asymptotic normality of Bayesian estimators of Shannon and Renyi entropies. The proofs are based on concentration inequalities for centered and non-centered Chi-square (Pearson) statistics. The latter allow to establish posterior concentration rates with respect to Fisher distance rather than with respect to the Hellinger distance as it is commonplace in non-parametric Bayesian statistics.