Within a Bayesian framework, a comprehensive investigation of mixtures of finite mixtures (MFMs), i.e., finite mixtures with a prior on the number of components, is performed. This model class has applications in model-based clustering as well as for semi-parametric density estimation and requires suitable prior specifications and inference methods to exploit its full potential. We contribute by considering a generalized class of MFMs where the hyperparameter $γ_K$ of a symmetric Dirichlet prior on the weight distribution depends on the number of components. We show that this model class may be regarded as a Bayesian non-parametric mixture outside the class of Gibbs-type priors. We emphasize the distinction between the number of components $K$ of a mixture and the number of clusters $K_+$, i.e., the number of filled components given the data. In the MFM model, $K_+$ is a random variable and its prior depends on the prior on $K$ and on the hyperparameter $γ_K$. We employ a flexible prior distribution for the number of components $K$ and derive the corresponding prior on the number of clusters $K_+$ for generalized MFMs. For posterior inference, we propose the novel telescoping sampler which allows Bayesian inference for mixtures with arbitrary component distributions without resorting to reversible jump Markov chain Monte Carlo (MCMC) methods. The telescoping sampler explicitly samples the number of components, but otherwise requires only the usual MCMC steps of a finite mixture model. The ease of its application using different component distributions is demonstrated on several data sets.
We study Bayesian estimation of mixture models and argue in favor of fitting the marginal posterior distribution over component assignments directly, rather than Gibbs sampling from the joint posterior on components and parameters as is commonly done. Some previous authors have found the former approach to have slow mixing, but we show that, implemented correctly, it can achieve excellent performance. In particular, we describe a new Monte Carlo algorithm for sampling from the marginal posterior of a general integrable mixture that makes use of rejection-free sampling from the prior over compo…
Ensembles of networks arise in various fields where multiple independent networks are observed on the same set of nodes, for example, a collection of brain networks constructed on the same brain regions for different individuals. However, there are few models that describe both the variations and characteristics of networks in an ensemble at the same time. In this paper, we propose to model the ensemble of networks using a Dirichlet Process Mixture of Exponential Random Graph Models (DPM-ERGMs), which divides the ensemble into different clusters and models each cluster of networks using a sepa…
Considering the creation of persistence landscape on a parametrized curve and structure of sampling, there exists a random process for which a finite mixture model of persistence landscape (FMMPL) can provide a better description for a given dataset. In this paper, a nonparametric approach for computing integrated mean of square error (IMSE) in persistence landscape has been presented. As a result, FMMPL is more accurate than the another way. Also, the sampling importance resampling (SIR) has been presented a better description of important landmark from parametrized curve. The result, provide…
The influence of Dirichlet process mixture is ubiquitous in the Bayesian nonparametrics literature. But sampling from its posterior distribution remains a challenge, despite the advent of various Markov chain Monte Carlo methods. The primary challenge is the infinite-dimensional setup, and even if the infinite-dimensional random measure is integrated out, high-dimensionality and discreteness still remain difficult issues to deal with. In this article, exploiting the key ideas proposed in Bhattacharya (2021b), we propose a novel methodology for drawing iid realizations from posteriors of Dirich…
We study the problem of sampling from the power posterior distribution in Bayesian Gaussian mixture models, a robust version of the classical posterior. This power posterior is known to be non-log-concave and multi-modal, which leads to exponential mixing times for some standard MCMC algorithms. We introduce and study the Reflected Metropolis-Hastings Random Walk (RMRW) algorithm for sampling. For symmetric two-component Gaussian mixtures, we prove that its mixing time is bounded as $d^{1.5}(d + \Vert θ_{0} \Vert^2)^{4.5}$ as long as the sample size $n$ is of the order $d (d + \Vert θ_{0} \Ver…
Practical tools for clustering streaming data must be fast enough to handle the arrival rate of the observations. Typically, they also must adapt on the fly to possible lack of stationarity; i.e., the data statistics may be time-dependent due to various forms of drifts, changes in the number of clusters, etc. The Dirichlet Process Mixture Model (DPMM), whose Bayesian nonparametric nature allows it to adapt its complexity to the data, seems a natural choice for the streaming-data case. In its classical formulation, however, the DPMM cannot capture common types of drifts in the data statistics. …
University of California, Irvine（加州大学欧文分校）；University of Algarve（阿尔加维大学）
Motivation: With the development of droplet based systems, massive single cell transcriptome data has become available, which enables analysis of cellular and molecular processes at single cell resolution and is instrumental to understanding many biological processes. While state-of-the-art clustering methods have been applied to the data, they face challenges in the following aspects: (1) the clustering quality still needs to be improved; (2) most models need prior knowledge on number of clusters, which is not always available; (3) there is a demand for faster computational speed. Results: We…
We propose Deep Residual Mixture Models (DRMMs), a novel deep generative model architecture. Compared to other deep models, DRMMs allow more flexible conditional sampling: The model can be trained once with all variables, and then used for sampling with arbitrary combinations of conditioning variables, Gaussian priors, and (in)equality constraints. This provides new opportunities for interactive and exploratory machine learning, where one should minimize the user waiting for retraining a model. We demonstrate DRMMs in constrained multi-limb inverse kinematics and controllable generation of ani…
The mixture cure model for analyzing survival data is characterized by the assumption that the population under study is divided into a group of subjects who will experience the event of interest over some finite time horizon and another group of cured subjects who will never experience the event irrespective of the duration of follow-up. When using the Bayesian paradigm for inference in survival models with a cure fraction, it is common practice to rely on Markov chain Monte Carlo (MCMC) methods to sample from posterior distributions. Although computationally feasible, the iterative nature of…
The Dirichlet Process Gaussian Mixture Model (DPGMM) is often used to cluster data when the number of clusters is unknown. One main DPGMM inference paradigm relies on sampling. Here we consider a known state-of-art sampler (proposed by Chang and Fisher III (2013) and improved by Dinari et al. (2019)), analyze its failure modes, and show how to improve it, often drastically. Concretely, in that sampler, whenever a new cluster is formed it is augmented with two subclusters whose labels are initialized at random. Upon their evolution, the subclusters serve to propose a split of the parent cluster…
This paper addresses challenges in flexibly modeling multimodal data that lie on constrained spaces. Such data are commonly found in spatial applications, such as climatology and criminology, where measurements are restricted to a geographical area. Other settings include domains where unsuitable recordings are discarded, such as flow-cytometry measurements. A simple approach to modeling such data is through the use of mixture models, especially nonparametric mixture models. Mixture models, while flexible and theoretically well-understood, are unsuitable for settings involving complicated cons…
A Copula Based Bayesian Approach for Paid-Incurred Claims Models for Non-Life Insurance Reserving
我们的文章认为，最近开发的随机模型，结合索赔支付和发生的损失信息到一个连贯的准备金方法。特别地，我们结合Hertig et al.（1985）和Gogol et al.（1993）的索赔准备金模型，发展了一个分层贝叶斯支付-发生-索赔模型族。在这个过程中，我们扩展了Merz等人（2010）的独立对数正态模型，通过使用数据增强混合Copula支付-发生索赔模型来合并不同的依赖结构。将支付和已发生的损失纳入未偿损失负债和由此产生的准备金的完整预测分布的估计中的效用和影响在以下情况下得到证明：（i）独立支付（P）数据模型;（ii）Merz等人（2010）的独立支付-已发生索赔（PIC）数据模型;（iii）一种新的相关滞后年伸缩块对角高斯Copula PIC数据模型，通过变换结合共轭;（iv）一种新的数据增强混合阿基米德Copula相关PIC数据模型。通过一类自适应马尔可夫链蒙特卡罗抽样算法，在这样的模型中的推理。这些方法结合了一个数据增强框架，用于有效地评估基于Copula的PIC模型在损失预留三角形中的可能性。自适应策略基于由对称矩阵的指数表示正定协方差矩阵，如Leonard等人（1992）所提出的。
Università degli Studi di Milano（米兰大学）；INFN Sezione di Milano（意大利国家核物理研究院米兰分部）；University of Lausanne（洛桑大学）
We show that the visible sector probability density function of the Riemann-Theta Boltzmann machine corresponds to a gaussian mixture model consisting of an infinite number of component multi-variate gaussians. The weights of the mixture are given by a discrete multi-variate gaussian over the hidden state space. This allows us to sample the visible sector density function in a straight-forward manner. Furthermore, we show that the visible sector probability density function possesses an affine transform property, similar to the multi-variate gaussian density.
In the realm of unsupervised learning, Bayesian nonparametric mixture models, exemplified by the Dirichlet Process Mixture Model (DPMM), provide a principled approach for adapting the complexity of the model to the data. Such models are particularly useful in clustering tasks where the number of clusters is unknown. Despite their potential and mathematical elegance, however, DPMMs have yet to become a mainstream tool widely adopted by practitioners. This is arguably due to a misconception that these models scale poorly as well as the lack of high-performance (and user-friendly) software tools …
