Variational learning for Dirichlet process mixtures of Dirichlet distributions and applications

In this paper, we propose a Bayesian nonparametric approach for modeling and selection based on a mixture of Dirichlet processes with Dirichlet distributions, which can also be seen as an infinite Dirichlet mixture model. The proposed model uses a stick-breaking representation and is learned by a variational inference method. Due to the nature of Bayesian nonparametric approach, the problems of overfitting and underfitting are prevented. Moreover, the obstacle of estimating the correct number of clusters is sidestepped by assuming an infinite number of clusters. Compared to other approximation techniques, such as Markov chain Monte Carlo (MCMC), which require high computational cost and whose convergence is difficult to diagnose, the whole inference process in the proposed variational learning framework is analytically tractable with closed-form solutions. Additionally, the proposed infinite Dirichlet mixture model with variational learning requires only a modest amount of computational power which makes it suitable to large applications. The effectiveness of our model is experimentally investigated through both synthetic data sets and challenging real-life multimedia applications namely image spam filtering and human action videos categorization.     

Variational Bayesian inference for infinite generalized inverted Dirichlet mixtures with feature selection and its application to clustering

We developed a variational Bayesian learning framework for the infinite generalized Dirichlet mixture model (i.e. a weighted mixture of Dirichlet process priors based on the generalized inverted Dirichlet distribution) that has proven its capability to model complex multidimensional data. We also integrate a “feature selection” approach to highlight the features that are most informative in order to construct an appropriate model in terms of clustering accuracy. Experiments on synthetic data as well as real data generated from visual scenes and handwritten digits datasets illustrate and validate the proposed approach.     

Bayesian learning of inverted Dirichlet mixtures for SVM kernels generation

We describe approaches for positive data modeling and classification using both finite inverted Dirichlet mixture models and support vector machines (SVMs). Inverted Dirichlet mixture models are used to tackle an outstanding challenge in SVMs namely the generation of accurate kernels. The kernels generation approaches, grounded on ideas from information theory that we consider, allow the incorporation of data structure and its structural constraints. Inverted Dirichlet mixture models are learned within a principled Bayesian framework using both Gibbs sampler and Metropolis-Hastings for parameter estimation and Bayes factor for model selection (i.e., determining the number of mixture’s components). Our Bayesian learning approach uses priors, which we derive by showing that the inverted Dirichlet distribution belongs to the family of exponential distributions, over the model parameters, and then combines these priors with information from the data to build posterior distributions. We illustrate the merits and the effectiveness of the proposed method with two real-world challenging applications namely object detection and visual scenes analysis and classification.     

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.     

Infinite Liouville mixture models with application to text and texture categorization

This paper addresses the problem of proportional data modeling and clustering using mixture models, a problem of great interest and of importance for many practical pattern recognition, image processing, data mining and computer vision applications. Finite mixture models are broadly applicable to clustering problems. But, they involve the challenging problem of the selection of the number of clusters which requires a certain trade-off. The number of clusters must be sufficient to provide the discriminating capability between clusters required for a given application. Indeed, if too many clusters are employed overfitting problems may occur and if few are used we have a problem of underfitting. Here we approach the problem of modeling and clustering proportional data using infinite mixtures which have been shown to be an efficient alternative to finite mixtures by overcoming the concern regarding the selection of the optimal number of mixture components. In particular, we propose and discuss the consideration of infinite Liouville mixture model whose parameter values are fitted to the data through a principled Bayesian algorithm that we have developed and which allows uncertainty in the number of mixture components. Our experimental evaluation involves two challenging applications namely text classification and texture discrimination, and suggests that the proposed approach can be an excellent choice for proportional data modeling.     

Non-Gaussian Data Clustering via Expectation Propagation Learning of Finite Dirichlet Mixture Models and Applications

Learning appropriate statistical models is a fundamental data analysis task which has been the topic of continuing interest. Recently, finite Dirichlet mixture models have proved to be an effective and flexible model learning technique in several machine learning and data mining applications. In this article, the problem of learning and selecting finite Dirichlet mixture models is addressed using an expectation propagation (EP) inference framework. Within the proposed EP learning method, for finite mixture models, all the involved parameters and the model complexity (i.e. the number of mixture components), can be evaluated simultaneously in a single optimization framework. Extensive simulations using synthetic data along with two challenging real-world applications involving automatic image annotation and human action videos categorization demonstrate that our approach is able to achieve better results than comparable techniques.     

Dirichlet Process Gaussian Mixture Models: Choice of the Base Distribution

In the Bayesian mixture modeling framework it is possible to infer the necessary number of components to model the data and therefore it is unnecessary to explicitly restrict the number of components. Nonparametric mixture models sidestep the problem of finding the “correct” number of mixture components by assuming infinitely many components. In this paper Dirichlet process mixture (DPM) models are cast as infinite mixture models and inference using Markov chain Monte Carlo is described. The specification of the priors on the model parameters is often guided by mathematical and practical convenience. The primary goal of this paper is to compare the choice of conjugate and non-conjugate base distributions on a particular class of DPM models which is widely used in applications, the Dirichlet process Gaussian mixture model (DPGMM). We compare computational efficiency and modeling performance of DPGMM defined using a conjugate and a conditionally conjugate base distribution. We show that better density models can result from using a wider class of priors with no or only a modest increase in computational effort.     

High-dimensional unsupervised selection and estimation of a finite generalized Dirichlet mixture model based on minimum message length

We consider the problem of determining the structure of high-dimensional data, without prior knowledge of the number of clusters. Data are represented by a finite mixture model based on the generalized Dirichlet distribution. The generalized Dirichlet distribution has a more general covariance structure than the Dirichlet distribution and offers high flexibility and ease of use for the approximation of both symmetric and asymmetric distributions. This makes the generalized Dirichlet distribution more practical and useful. An important problem in mixture modeling is the determination of the number of clusters. Indeed, a mixture with too many or too few components may not be appropriate to approximate the true model. Here, we consider the application of the minimum message length (MML) principle to determine the number of clusters. The MML is derived so as to choose the number of clusters in the mixture model which best describes the data. A comparison with other selection criteria is performed. The validation involves synthetic data, real data clustering, and two interesting real applications: classification of web pages, and texture database summarization for efficient retrieval.     

Bayesian learning of finite generalized inverted Dirichlet mixtures: Application to object classification and forgery detection

The advent of mixture models has opened the possibility of flexible models which are practical to work with. A common assumption is that practitioners typically expect that data are generated from a Gaussian mixture. The inverted Dirichlet mixture has been shown to be a better alternative to the Gaussian mixture and to be of significant value in a variety of applications involving positive data. The inverted Dirichlet is, however, usually undesirable, since it forces an assumption of positive correlation. Our focus here is to develop a Bayesian alternative to both the Gaussian and the inverted Dirichlet mixtures when dealing with positive data. The alternative that we propose is based on the generalized inverted Dirichlet distribution which offers high flexibility and ease of use, as we show in this paper. Moreover, it has a more general covariance structure than the inverted Dirichlet. The proposed mixture model is subjected to a fully Bayesian analysis based on Markov Chain Monte Carlo (MCMC) simulation methods namely Gibbs sampling and Metropolis–Hastings used to compute the posterior distribution of the parameters, and on Bayesian information criterion (BIC) used for model selection. The adoption of this purely Bayesian learning choice is motivated by the fact that Bayesian inference allows to deal with uncertainty in a unified and consistent manner. We evaluate our approach on the basis of two challenging applications concerning object classification and forgery detection.     

分层Dirichlet过程及其应用综述

Dirichlet过程是一种应用于非参数贝叶斯模型中的随机过程, 特别是作为先验分布应用在概率图模型中. 与传统的参数模型相比, Dirichlet过程的应用更加广泛且模型更加灵活, 特别是应用于聚类问题时, 该过程能够自动确定聚类数目和生成聚类中心的分布参数. 因此, 近年来, 在理论和应用上均得到了迅速的发展, 引起越来越多的关注. 本文首先介绍Dirichlet过程, 而后描述了以Dirichlet过程为先验分布的Dirichlet过程混合模型及其应用, 接着概述分层Dirichlet过程及其在相关算法构造中的应用, 最后对分层Dirichlet过程的理论和应用进行了总结, 并对未来的发展方向作了探讨.     

Bayesian mixture of autoregressive models

An infinite mixture of autoregressive models is developed. The unknown parameters in the mixture autoregressive model follow a mixture distribution, which is governed by a Dirichlet process prior. One main feature of our approach is the generalization of a finite mixture model by having the number of components unspecified. A Bayesian sampling scheme based on a weighted Chinese restaurant process is proposed to generate partitions of observations. Using the partitions, Bayesian prediction, while accounting for possible model uncertainty, determining the most probable number of mixture components, clustering of time series and outlier detection in time series, can be done. Numerical results from simulated and real data are presented to illustrate the methodology.     

Bayesian mixture of autoregressive models

An infinite mixture of autoregressive models is developed. The unknown parameters in the mixture autoregressive model follow a mixture distribution, which is governed by a Dirichlet process prior. One main feature of our approach is the generalization of a finite mixture model by having the number of components unspecified. A Bayesian sampling scheme based on a weighted Chinese restaurant process is proposed to generate partitions of observations. Using the partitions, Bayesian prediction, while accounting for possible model uncertainty, determining the most probable number of mixture components, clustering of time series and outlier detection in time series, can be done. Numerical results from simulated and real data are presented to illustrate the methodology.     

Positive vectors clustering using inverted Dirichlet finite mixture models

In this work we present an unsupervised algorithm for learning finite mixture models from multivariate positive data. Indeed, this kind of data appears naturally in many applications, yet it has not been adequately addressed in the past. This mixture model is based on the inverted Dirichlet distribution, which offers a good representation and modeling of positive non-Gaussian data. The proposed approach for estimating the parameters of an inverted Dirichlet mixture is based on the maximum likelihood (ML) using Newton Raphson method. We also develop an approach, based on the minimum message length (MML) criterion, to select the optimal number of clusters to represent the data using such a mixture. Experimental results are presented using artificial histograms and real data sets. The challenging problem of software modules classification is investigated within the proposed statistical framework, also.     

A Model-Based Approach for Discrete Data Clustering and Feature Weighting Using MAP and Stochastic Complexity

In this paper, we consider the problem of unsupervised discrete feature selection/weighting. Indeed, discrete data are an important component in many data mining, machine learning, image processing, and computer vision applications. However, much of the published work on unsupervised feature selection has concentrated on continuous data. We propose a probabilistic approach that assigns relevance weights to discrete features that are considered as random variables modeled by finite discrete mixtures. The choice of finite mixture models is justified by its flexibility which has led to its widespread application in different domains. For the learning of the model, we consider both Bayesian and information-theoretic approaches through stochastic complexity. Experimental results are presented to illustrate the feasibility and merits of our approach on a difficult problem which is clustering and recognizing visual concepts in different image data. The proposed approach is successfully applied also for text clustering.     

Online clustering via finite mixtures of Dirichlet and minimum message length

This paper presents an online algorithm for mixture model-based clustering. Mixture modeling is the problem of identifying and modeling components in a given set of data. The online algorithm is based on unsupervised learning of finite Dirichlet mixtures and a stochastic approach for estimates updating. For the selection of the number of clusters, we use the minimum message length (MML) approach. The proposed method is validated by synthetic data and by an application concerning the dynamic summarization of image databases.     

Spatial color image segmentation based on finite non-Gaussian mixture models

Finite mixture models are one of the most widely and commonly used probabilistic techniques for image segmentation. Although the most well known and commonly used distribution when considering mixture models is the Gaussian, it is certainly not the best approximation for image segmentation and other related image processing problems. In this paper, we propose and investigate the use of several other mixture models based namely on Dirichlet, generalized Dirichlet and Beta–Liouville distributions, which offer more flexibility in data modeling, for image segmentation. A maximum likelihood (ML) based algorithm is applied for estimating the resulted segmentation model’s parameters. Spatial information is also employed for figuring out the number of regions in an image and several color spaces are investigated and compared. The experimental results show that the proposed segmentation framework yields good overall performance, on various color scenes, that is better than comparable techniques.     

Unsupervised selection of a finite Dirichlet mixture model: an MML-based approach

This paper proposes an unsupervised algorithm for learning a finite Dirichlet mixture model. An important part of the unsupervised learning problem is determining the number of clusters which best describe the data. We extend the minimum message length (MML) principle to determine the number of clusters in the case of Dirichlet mixtures. Parameter estimation is done by the expectation-maximization algorithm. The resulting method is validated for one-dimensional and multidimensional data. For the one-dimensional data, the experiments concern artificial and real SAP image histograms. The validation for multidimensional data involves synthetic data and two real applications: shadow detection in images and summarization of texture image databases for efficient retrieval. A comparison with results obtained for other selection criteria is provided.     

Discrete data clustering using finite mixture models

Finite mixture models have been applied for different computer vision, image processing and pattern recognition tasks. The majority of the work done concerning finite mixture models has focused on mixtures for continuous data. However, many applications involve and generate discrete data for which discrete mixtures are better suited. In this paper, we investigate the problem of discrete data modeling using finite mixture models. We propose a novel, well motivated mixture that we call the multinomial generalized Dirichlet mixture. The novel model is compared with other discrete mixtures. We designed experiments involving spatial color image databases modeling and summarization, and text classification to show the robustness, flexibility and merits of our approach.     

Bayesian feature and model selection for Gaussian mixture models

We present a Bayesian method for mixture model training that simultaneously treats the feature selection and the model selection problem. The method is based on the integration of a mixture model formulation that takes into account the saliency of the features and a Bayesian approach to mixture learning that can be used to estimate the number of mixture components. The proposed learning algorithm follows the variational framework and can simultaneously optimize over the number of components, the saliency of the features, and the parameters of the mixture model. Experimental results using high-dimensional artificial and real data illustrate the effectiveness of the method.     

Use of the dirichlet process for reliability analysis

Recently, many researchers apply the non-parametric Bayesian approach to predict the reliability of highly complex electronic systems. The Dirichlet process is the most common model for the non-parametric Bayesian analysis. The Kuo's simulation procedure [6] for Dirichlet process under a variance reduction techniques introduced in Chien and Kuo (1994) [2] is applied for a Weibull-distributed system. Optimal burn-in time is determined given the cost parameters. A model, the percentage of good items in a lot, is used to explain when the Dirichlet process is not a proper choice.