On a class of $$\sigma $$ σ -stable Poisson–Kingman models and an effective marginalized sampler

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.

We consider a Bayesian nonparametric models for spatial data of mixed category. Moreover, we adopt joint modeling strategy by assuming that responses and confounding variables are corresponding to continuous latent variables with multivariate Gaussian distribution. The model is built on a class of Gaussian Conditional Autoregressive (CAR) models, in combination with dependent sampling models (SSM) as well as probit stick-breaking process prior for accounting for complex interactions and high correlations of data. The key idea is to introducing spatial dependence by modeling the weights via probit transformation of Gaussian Markov random fields or discrete random probability measures of SSM. We illustrate the usefulness and effectiveness of the methodology through a real example from a China earthquake data set.

Mixture distributions are widely used to model data with distinct groups, providing a flexible approach to estimating density. However, Bayesian approaches for mixture models pose challenges, such as label switching in the Gibbs sampler output due to the non-identifiability of component parameters. We review advanced methods for Bayesian analysis, including the Markov chain Monte Carlo (MCMC) reversible jump algorithm and model comparison based on joint measures of fit and complexity. We also present a Bayesian regression model based on a two-component mixture model, implemented using the Gibbs sampler algorithm and applied to a dataset of time measurement differences between two clocks. Our theoretical investigation highlights the importance of latent variables in implementing the Bayesian normal mixture model with two components. When applied to the dataset, our model effectively assigned probabilities to the two states of the phenomenon under study and identified two processes with identical slopes, intercepts, and variances. Our findings demonstrate the power of Bayesian mixture models in uncovering hidden structures within complex datasets. In general, our review and application provide insight into the challenges and potential solutions for Bayesian mixture modeling and highlight the usefulness of these methods in various fields.

As a powerful nonparametric Bayesian model, the infinite mixture model has been successfully used in machine learning and computer vision. The success of the infinite mixture model owes to the capability clustering and density estimation. In this paper, we propose a nonparametric Bayesian model for single-image super-resolution. Specifically, we combine the Dirichlet process and Gaussian process regression for estimating the distribution of the training patches and modeling the relationship between the low-resolution and high-resolution patches: 1) the proposed method groups the training patches by utilizing the clustering property of Dirichlet process; 2) the proposed method relates the low-resolution and high-resolution patches by predicting the property of Gaussian process; and 3) the mentioned two points are not independent but jointly learned. Hence, the proposed method can make full use of the nonparametric Bayesian model. First, the Dirichlet process mixture model is used to obtain more accurate clusters for training patches. Second, Gaussian process regression is established on each cluster, and this directly reduces the computational complexity. Finally, the two procedures are learned simultaneously to gain the suitable clusters with the ability of prediction. The parameters can be inferred simply via the Gibbs sampling technique. Thorough super-resolution experiments on various images demonstrate that the proposed method is superior to some state-of-the-art methods.

We address the problem of X/gamma-ray spectra estimation in the fields of nuclear physics. Bayesian estimation of experimental backgrounds has been studied in [1] involving splines. Since Dirichlet processes (DP) sit on discrete measures, they provide an appealing prior for photopeaks. On the other hand, in order to tackle the complexity of experimental backgrounds, we consider a Polya Tree Mixture (PTM) - with suitable parameters yielding distribution continuity - for which predictive densities exhibit better smoothness properties than a single Polya Tree. Furthermore, it is easy to introduce some physical Compton line approximation formula (e.g. Klein-Nishina) in the base measure of the Polya Tree, or some physically driven local modifications of the PTM prior parameters. As backgrounds depend on photopeaks locations, we propose a hierarchical model where the PTM is conditioned on the DP. We use a beta prior for the mixing proportion between the DP and the PTM. Energies are not directly observed due to detection devices noises which introduce a convolution of both discrete and continuous measures by an assumed gaussian kernel whose variance is an unknown linear function of energy. Thus, the proposed semiparametric model for experimental data becomes a hierarchical Polya Tree-Dirichlet mixture of normal kernels. Besides, observed energies are binned in an histogram introducing additional quantification noise. The quantities of interest are usually posterior functionals of the DP mixing distribution. This implies an inverse problem which is carried out in the framework of finite stick-breaking representation. Thanks to conjugacy, draws from the posterior DP and PTM are easily obtained. The approach yields to a global peaks/background separation while offering spectrum resolution enhancement. The method is illustrated on experimental HPGe spectra.

In some applications of data mining, clustering analysis of directional data is often involved. In this case, conventional model-based clustering methods are not suitable for fitting the data of such type of structure. Therefore, a Dirichlet Process Mixture Model based on von Mises-Fisher distribution was proposed for the clustering analysis of directional data. The main motivation is that as a non-parametric Bayesian model, The Dirichlet process can automatically determine the complexity of the mixture model when the number of data categories is unknown. We use the accelerated Variational inference algorithm to quickly estimate the parameters involved in the model, which enables the method to be applied in applications with large data scale. The validity of the proposed model was verified by using different scale simulation data and clustering analysis of gene expression data.

The elicitation of an ordinal judgment on multiple alternatives is often required in many psychological and behavioral experiments to investigate preference/choice orientation of a specific population. The Plackett-Luce model is one of the most popular and frequently applied parametric distributions to analyze rankings of a finite set of items. The present work introduces a Bayesian finite mixture of Plackett-Luce models to account for unobserved sample heterogeneity of partially ranked data. We describe an efficient way to incorporate the latent group structure in the data augmentation approach and the derivation of existing maximum likelihood procedures as special instances of the proposed Bayesian method. Inference can be conducted with the combination of the Expectation-Maximization algorithm for maximum a posteriori estimation and the Gibbs sampling iterative procedure. We additionally investigate several Bayesian criteria for selecting the optimal mixture configuration and describe diagnostic tools for assessing the fitness of ranking distributions conditionally and unconditionally on the number of ranked items. The utility of the novel Bayesian parametric Plackett-Luce mixture for characterizing sample heterogeneity is illustrated with several applications to simulated and real preference ranked data. We compare our method with the frequentist approach and a Bayesian nonparametric mixture model both assuming the Plackett-Luce model as a mixture component. Our analysis on real datasets reveals the importance of an accurate diagnostic check for an appropriate in-depth understanding of the heterogenous nature of the partial ranking data.

The aim of this study was to implement a Dirichlet process mixture (DPM) model for automatic tumor edge identification on (18)F-fluorodeoxyglucose positron emission tomography ((18)F-FDG PET) images by optimizing the parameters on which the algorithm depends, to validate it experimentally, and to test its robustness. The DPM model belongs to the class of the Bayesian nonparametric models and uses the Dirichlet process prior for flexible nonparametric mixture modeling, without any preliminary choice of the number of mixture components. The DPM algorithm implemented in the statistical software package R was used in this work. The contouring accuracy was evaluated on several image data sets: on an IEC phantom (spherical inserts with diameter in the range 10-37 mm) acquired by a Philips Gemini Big Bore PET-CT scanner, using 9 different target-to-background ratios (TBRs) from 2.5 to 70; on a digital phantom simulating spherical/uniform lesions and tumors, irregular in shape and activity; and on 20 clinical cases (10 lung and 10 esophageal cancer patients). The influence of the DPM parameters on contour generation was studied in two steps. In the first one, only the IEC spheres having diameters of 22 and 37 mm and a sphere of the digital phantom (41.6 mm diameter) were studied by varying the main parameters until the diameter of the spheres was obtained within 0.2% of the true value. In the second step, the results obtained for this training set were applied to the entire data set to determine DPM based volumes of all available lesions. These volumes were compared to those obtained by applying already known algorithms (Gaussian mixture model and gradient-based) and to true values, when available. Only one parameter was found able to significantly influence segmentation accuracy (ANOVA test). This parameter was linearly connected to the uptake variance of the tested region of interest (ROI). In the first step of the study, a calibration curve was determined to automatically generate the optimal parameter from the variance of the ROI. This "calibration curve" was then applied to contour the whole data set. The accuracy (mean discrepancy between DPM model-based contours and reference contours) of volume estimation was below (1 ± 7)% on the whole data set (1 SD). The overlap between true and automatically segmented contours, measured by the Dice similarity coefficient, was 0.93 with a SD of 0.03. The proposed DPM model was able to accurately reproduce known volumes of FDG concentration, with high overlap between segmented and true volumes. For all the analyzed inserts of the IEC phantom, the algorithm proved to be robust to variations in radius and in TBR. The main advantage of this algorithm was that no setting of DPM parameters was required in advance, since the proper setting of the only parameter that could significantly influence the segmentation results was automatically related to the uptake variance of the chosen ROI. Furthermore, the algorithm did not need any preliminary choice of the optimum number of classes to describe the ROIs within PET images and no assumption about the shape of the lesion and the uptake heterogeneity of the tracer was required.

The mixture modeling framework is widely used in many applications. In this paper, we propose a component reduction technique, that collapses a Gaussian mixture model into a Gaussian mixture with fewer components. The EM (Expectation-Maximization) algorithm is usually used to fit a mixture model to data. Our algorithm is derived by extending mixture model learning using the EM-algorithm. In this extension, a difficulty arises from the fact that some crucial quantities cannot be evaluated analytically. We overcome this difficulty by introducing an effective approximation. The effectiveness of our algorithm is demonstrated by applying it to a simple synthetic component reduction task and a phoneme clustering problem.

Unsupervised outlier detection using random subspace and subsampling ensembles of Dirichlet process mixtures

Copula based factorization in Bayesian multivariate infinite mixture models

Asymptotic theory of tail index estimation has been studied extensively in the frequentist literature on extreme values, but rarely in the Bayesian context. We investigate whether popular Bayesian kernel mixture models are able to support heavy tailed distributions and consistently estimate the tail index. We show that posterior inconsistency in tail index is surprisingly common for both parametric and nonparametric mixture models. We then present a set of sufficient conditions under which posterior consistency in tail index can be achieved, and verify these conditions for Pareto mixture models under general mixing priors.

Tractable Bayesian density regression via logit stick-breaking priors

We propose a Bayesian method for multiple hypothesis testing in random effects models that uses Dirichlet process (DP) priors for a nonparametric treatment of the random effects distribution. We consider a general model formulation which accommodates a variety of multiple treatment conditions. A key feature of our method is the use of a product of spiked distributions, i.e., mixtures of a point-mass and continuous distributions, as the centering distribution for the DP prior. Adopting these spiked centering priors readily accommodates sharp null hypotheses and allows for the estimation of the posterior probabilities of such hypotheses. Dirichlet process mixture models naturally borrow information across objects through model-based clustering while inference on single hypotheses averages over clustering uncertainty. We demonstrate via a simulation study that our method yields increased sensitivity in multiple hypothesis testing and produces a lower proportion of false discoveries than other competitive methods. While our modeling framework is general, here we present an application in the context of gene expression from microarray experiments. In our application, the modeling framework allows simultaneous inference on the parameters governing differential expression and inference on the clustering of genes. We use experimental data on the transcriptional response to oxidative stress in mouse heart muscle and compare the results from our procedure with existing nonparametric Bayesian methods that provide only a ranking of the genes by their evidence for differential expression.

Recent work on Acoustic Unit Discovery (AUD) has led to the development of a non-parametric Bayesian phone-loop model where the prior over the probability of the phone-like units is assumed to be sampled from a Dirichlet Process (DP). In this work, we propose to improve this model by incorporating a Hierarchical Pitman-Yor based bigram Language Model on top of the units' transitions. This new model makes use of the phonotactic context information but assumes a fixed number of units. To remedy this limitation we first train a DP phone-loop model to infer the number of units, then, the bigram phone-loop is initialized from the DP phone-loop and trained until convergence of its parameters. Results show an absolute improvement of 1–2%on the Normalized Mutual Information (NMI) metric. Furthermore, we show that, combined with Multilingual Bottleneck (MBN) features the model yields a same or higher NMI as an English phone recogniser trained on TIMIT.