An important question in health services research is the estimation of the proportion of medical expenditures that exceed a given threshold. Typically, medical expenditures present highly skewed, heavy tailed distributions, for which (a) simple variable transformations are insufficient to achieve a tractable low-dimensional parametric form and (b) nonparametric methods are not efficient in estimating exceedance probabilities for large thresholds. Motivated by this context, in this paper we propose a general Bayesian approach for the estimation of tail probabilities of heavy-tailed distributions, based on a mixture of gamma distributions in which the mixing occurs over the shape parameter. This family provides a flexible and novel approach for modeling heavy-tailed distributions, it is computationally efficient, and it only requires to specify a prior distribution for a single parameter. By carrying out simulation studies, we compare our approach with commonly used methods, such as the log-normal model and nonparametric alternatives. We found that the mixture-gamma model significantly improves predictive performance in estimating tail probabilities, compared to these alternatives. We also applied our method to the Medical Current Beneficiary Survey (MCBS), for which we estimate the probability of exceeding a given hospitalization cost for smoking attributable diseases. We have implemented the method in the open source GSM package, available from the Comprehensive R Archive Network.
This paper deals with the extreme value analysis for the triangular arrays, which appear when some parameters of the mixture model vary as the number of observations grow. When the mixing parameter is small, it is natural to associate one of the components with "an impurity" (in case of regularly varying distribution, "heavy-tailed impurity"), which "pollutes" another component. We show that the set of possible limit distributions is much more diverse than in the classical Fisher-Tippett-Gnedenko theorem, and provide the numerical examples showing the efficiency of the proposed model for study…
In this paper we present a novel methodology to perform Bayesian model selection in linear models with heavy-tailed distributions. We consider a finite mixture of distributions to model a latent variable where each component of the mixture corresponds to one possible model within the symmetrical class of normal independent distributions. Naturally, the Gaussian model is one of the possibilities. This allows for a simultaneous analysis based on the posterior probability of each model. Inference is performed via Markov chain Monte Carlo - a Gibbs sampler with Metropolis-Hastings steps for a clas…
Purdue University（普渡大学）；North Carolina State University（北卡罗来纳州立大学）
Estimating parameters of mixture model has wide applications ranging from classification problems to estimating of complex distributions. Most of the current literature on estimating the parameters of the mixture densities are based on iterative Expectation Maximization (EM) type algorithms which require the use of either taking expectations over the latent label variables or generating samples from the conditional distribution of such latent labels using the Bayes rule. Moreover, when the number of components is unknown, the problem becomes computationally more demanding due to well-known lab…
In general insurance, risks from different categories are often modeled independently and their sum is regarded as the total risk the insurer takes on in exchange for a premium. The dependence from multiple risks is generally neglected even when correlation could exist, for example a single car accident may result in claims from multiple risk categories. It is desirable to take the covariance of different categories into consideration in modeling in order to better predict future claims and hence allow greater accuracy in ratemaking. In this work multivariate severity models are investigated u…
Continuous mixtures of distributions are widely employed in the statistical literature as models for phenomena with highly divergent outcomes; in particular, many familiar heavy-tailed distributions arise naturally as mixtures of light-tailed distributions (e.g., Gaussians), and play an important role in applications as diverse as modeling of extreme values and robust inference. In the case of social networks, continuous mixtures of graph distributions can likewise be employed to model social processes with heterogeneous outcomes, or as robust priors for network inference. Here, we introduce s…
Bayesian spatial modeling of heavy-tailed distributions has become increasingly popular in various areas of science in recent decades. We propose a Weibull regression model with spatial random effects for analyzing extreme economic loss. Model estimation is facilitated by a computationally efficient Bayesian sampling algorithm utilizing the multivariate Log-Gamma distribution. Simulation studies are carried out to demonstrate better empirical performances of the proposed model than the generalized linear mixed effects model. An earthquake data obtained from Yunnan Seismological Bureau, China i…
There are various metrics for financial risk, such as value at risk (VaR), expected shortfall, expected/unexpected loss, etc. When estimating these metrics, it was very common to assume Gaussian distribution for the asset returns, which may underestimate the real risk of the market, especially during the financial crisis. In this paper, we propose a series of time-varying mixture models for risk analysis and management. These mixture models contain two components: one component with Gaussian distribution, and the other one with a fat-tailed Cauchy distribution. We allow the distribution parame…
In this paper we propose a model with a Dirichlet process mixture of gamma densities in the bulk part below threshold and a generalized Pareto density in the tail for extreme value estimation. The proposed model is simple and flexible allowing us posterior density estimation and posterior inference for high quantiles. The model works well even for small sample sizes and in the absence of prior information. We evaluate the performance of the proposed model through a simulation study. Finally, the proposed model is applied to a real environmental data.
The sparse structure of the solution for an inverse problem can be modelled using different sparsity enforcing priors when the Bayesian approach is considered. Analytical expression for the unknowns of the model can be obtained by building hierarchical models based on sparsity enforcing distributions expressed via conjugate priors. We consider heavy tailed distributions with this property: the Student-t distribution, which is expressed as a Normal scale mixture, with the mixing distribution the Inverse Gamma distribution, the Laplace distribution, which can also be expressed as a Normal scale …
Mixture models with Gamma and or inverse-Gamma distributed mixture components are useful for medical image tissue segmentation or as post-hoc models for regression coefficients obtained from linear regression within a Generalised Linear Modeling framework (GLM), used in this case to separate stochastic (Gaussian) noise from some kind of positive or negative "activation" (modeled as Gamma or inverse-Gamma distributed). To date, the most common choice in this context it is Gaussian/Gamma mixture models learned through a maximum likelihood (ML) approach; we recently extended such algorithm for mi…
We consider the problem of Bayesian density estimation on the positive semiline for possibly unbounded densities. We propose a hierarchical Bayesian estimator based on the gamma mixture prior which can be viewed as a location mixture. We study convergence rates of Bayesian density estimators based on such mixtures. We construct approximations of the local Hölder densities, and of their extension to unbounded densities, to be continuous mixtures of gamma distributions, leading to approximations of such densities by finite mixtures. These results are then used to derive posterior concentration r…
Mixture models are useful in a wide array of applications to identify subpopulations in noisy overlapping distributions. For example, in multiplexed immunofluorescence (mIF), cell image intensities represent expression levels and the cell populations are a noisy mixture of expressed and unexpressed cells. Among mixture models, the gamma mixture model has the strength of being flexible in fitting skewed strictly positive data that occur in many biological measurements. However, the current estimation method uses numerical optimization within the expectation maximization algorithm and is computa…
Heavy-tailed distributions are widely used in robust mixture modelling due to possessing thick tails. As a computationally tractable subclass of the stable distributions, sub-Gaussian $α$-stable distribution received much interest in the literature. Here, we introduce a type of expectation maximization algorithm that estimates parameters of a mixture of sub-Gaussian stable distributions. A comparative study, in the presence of some well-known mixture models, is performed to show the robustness and performance of the mixture of sub-Gaussian $α$-stable distributions for modelling, simulated, syn…
RISE Research Institutes of Sweden AB（瑞典RISE研究院）；Elekta（医科达）
We present the elliptical processes -- a family of non-parametric probabilistic models that subsumes the Gaussian process and the Student-t process. This generalization includes a range of new fat-tailed behaviors yet retains computational tractability. We base the elliptical processes on a representation of elliptical distributions as a continuous mixture of Gaussian distributions and derive closed-form expressions for the marginal and conditional distributions. We perform numerical experiments on robust regression using an elliptical process defined by a piecewise constant mixing distributio…
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 model…
For extreme value estimation we propose to use a model with a Dirichlet process mixture of gamma densities in the center and generalized Pareto densities for the tails. Due to the randomness in the center and a heavy tailed density in the tails density estimation and posterior inference for high quantiles are possible. The approach can be used in a "default" manner on the positive reals because it works when prior information is unavailable. The proposed model can be easy to implement and a sensitivity analysis is provided. We applied the proposed model for simulated and real data sets.
We study modeling and inference with the Elliptical Gamma Distribution (EGD). We consider maximum likelihood (ML) estimation for EGD scatter matrices, a task for which we develop new fixed-point algorithms. Our algorithms are efficient and converge to global optima despite nonconvexity. Moreover, they turn out to be much faster than both a well-known iterative algorithm of Kent & Tyler (1991) and sophisticated manifold optimization algorithms. Subsequently, we invoke our ML algorithms as subroutines for estimating parameters of a mixture of EGDs. We illustrate our methods by applying them to m…
The finite Gamma mixture model is often used to describe randomness in income data, insurance data, and data from other applications. The popular likelihood approach, however, does not work for this model because the likelihood function is unbounded, and the maximum likelihood estimator is therefore not well defined. There has been much research into ways to ensure the consistent estimation of the mixing distribution, including placing an upper bound on the shape parameter or adding a penalty to the log-likelihood function. In this paper, we show that if the shape parameter in the finite Gamma…
University of Pretoria（比勒陀利亚大学）；Center for Artificial Intelligence Research (CAIR)（人工智能研究中心（CAIR））
Most topic models are constructed under the assumption that documents follow a multinomial distribution. The Poisson distribution is an alternative distribution to describe the probability of count data. For topic modelling, the Poisson distribution describes the number of occurrences of a word in documents of fixed length. The Poisson distribution has been successfully applied in text classification, but its application to topic modelling is not well documented, specifically in the context of a generative probabilistic model. Furthermore, the few Poisson topic models in literature are admixtu…
The contaminated Gaussian distribution represents a simple heavy-tailed elliptical generalization of the Gaussian distribution; unlike the often-considered t-distribution, it also allows for automatic detection of mild outlying or "bad" points in the same way that observations are typically assigned to the groups in the finite mixture model context. Starting from this distribution, we propose the contaminated factor analysis model as a method for dimensionality reduction and detection of bad points in higher dimensions. A mixture of contaminated Gaussian factor analyzers (MCGFA) model follows …
Kim Jaechul Graduate School of AI, KAIST（韩国科学技术院金在哲人工智能研究生院）；School of Computing, KAIST（韩国科学技术院计算机学院）；Institute for Basic Science (IBS)（基础科学研究院）；AITRICS
Recent works have revealed that infinitely-wide feed-forward or recurrent neural networks of any architecture correspond to Gaussian processes referred to as Neural Network Gaussian Processes (NNGPs). While these works have extended the class of neural networks converging to Gaussian processes significantly, however, there has been little focus on broadening the class of stochastic processes that such neural networks converge to. In this work, inspired by the scale mixture of Gaussian random variables, we propose the scale mixture of NNGPs for which we introduce a prior distribution on the sca…
In many large-scale inverse problems, such as computed tomography and image deblurring, characterization of sharp edges in the solution is desired. Within the Bayesian approach to inverse problems, edge-preservation is often achieved using Markov random field priors based on heavy-tailed distributions. Another strategy, popular in statistics, is the application of hierarchical shrinkage priors. An advantage of this formulation lies in expressing the prior as a conditionally Gaussian distribution depending of global and local hyperparameters which are endowed with heavy-tailed hyperpriors. In t…
In this work, we develop a constructive modeling framework for extreme threshold exceedances in repeated observations of spatial fields, based on general product mixtures of random fields possessing light or heavy-tailed margins and various spatial dependence characteristics, which are suitably designed to provide high flexibility in the tail and at sub-asymptotic levels. Our proposed model is akin to a recently proposed Gamma-Gamma model using a ratio of processes with Gamma marginal distributions, but it possesses a higher degree of flexibility in its joint tail structure, capturing strong d…
We introduce a mixture-model of beta distributions to identify significant correlations among $P$ predictors when $P$ is large. The method relies on theorems in convex geometry, which we use to show how to control the error rate of edge detection in graphical models. Our `betaMix' method does not require any assumptions about the network structure, nor does it assume that the network is sparse. The results in this article hold for a wide class of data generating distributions that include light-tailed and heavy-tailed spherically symmetric distributions.
Heavy-tailed random variables have been used in insurance research to model both loss frequencies and loss severities, with substantially more emphasis on the latter. In the present work, we take a step toward addressing this imbalance by exploring the class of heavy-tailed frequency models formed by continuous mixtures of Negative Binomial and Poisson random variables. We begin by defining the concept of a calibrative family of mixing distributions (each member of which is identifiable from its associated Negative Binomial mixture), and show how to construct such families from only a single m…
This paper provides a framework for estimating the mean and variance of a high-dimensional normal density. The main setting considered is a fixed number of vector following a high-dimensional normal distribution with unknown mean and diagonal covariance matrix. The diagonal covariance matrix can be known or unknown. If the covariance matrix is unknown, the sample size can be as small as $2$. The proposed estimator is based on the idea that the unobserved pairs of mean and variance for each dimension are drawn from an unknown bivariate distribution, which we model as a mixture of normal-inverse…
We extend classic characterisations of posterior distributions under Dirichlet process and gamma random measures priors to a dynamic framework. We consider the problem of learning, from indirect observations, two families of time-dependent processes of interest in Bayesian nonparametrics: the first is a dependent Dirichlet process driven by a Fleming-Viot model, and the data are random samples from the process state at discrete times; the second is a collection of dependent gamma random measures driven by a Dawson-Watanabe model, and the data are collected according to a Poisson point process …
Discrete mixture models provide a well-known basis for effective clustering algorithms, although technical challenges have limited their scope. In the context of gene-expression data analysis, a model is presented that mixes over a finite catalog of structures, each one representing equality and inequality constraints among latent expected values. Computations depend on the probability that independent gamma-distributed variables attain each of their possible orderings. Each ordering event is equivalent to an event in independent negative-binomial random variables, and this finding guides a dy…
Inter-event times of various human behavior are apparently non-Poissonian and obey long-tailed distributions as opposed to exponential distributions, which correspond to Poisson processes. It has been suggested that human individuals may switch between different states in each of which they are regarded to generate events obeying a Poisson process. If this is the case, inter-event times should approximately obey a mixture of exponential distributions with different parameter values. In the present study, we introduce the minimum description length principle to compare mixtures of exponential d…
Finite mixture and Markov-switching models generalize and, therefore, nest specifications featuring only one component. While specifying priors in the two: the general (mixture) model and its special (single-component) case, it may be desirable to ensure that the prior assumptions introduced into both structures are coherent in the sense that the prior distribution in the nested model amounts to the conditional prior in the mixture model under relevant parametric restriction. The study provides the rudiments of setting coherent priors in Bayesian univariate finite mixture and Markov-switching …
The seemingly disjoint problems of count and mixture modeling are united under the negative binomial (NB) process. A gamma process is employed to model the rate measure of a Poisson process, whose normalization provides a random probability measure for mixture modeling and whose marginalization leads to an NB process for count modeling. A draw from the NB process consists of a Poisson distributed finite number of distinct atoms, each of which is associated with a logarithmic distributed number of data samples. We reveal relationships between various count- and mixture-modeling distributions an…
The estimation of asset return distributions is crucial for determining optimal trading strategies. In this paper we describe the constrained mixture model, based on a mixture of Gamma and Gaussian distributions, to provide an accurate description of price trends as being clearly positive, negative or ranging while accounting for heavy tails and high kurtosis. The model is estimated in the Expectation Maximisation framework and model order estimation also respects the model's constraints.
Linear regression with the classical normality assumption for the error distribution may lead to an undesirable posterior inference of regression coefficients due to the potential outliers. This paper considers the finite mixture of two components with thin and heavy tails as the error distribution, which has been routinely employed in applied statistics. For the heavily-tailed component, we introduce the novel class of distributions; their densities are log-regularly varying and have heavier tails than those of Cauchy distribution, yet they are expressed as a scale mixture of normal distribut…
the University of Tokyo（东京大学）；Rikkyo University（立教大学）；National Institute of Informatics（国立信息学研究所）；The Canon Institute for Global Studies（佳能全球研究所）
Block modeling is widely used in studies on complex networks. The cornerstone model is the stochastic block model (SBM), widely used over the past decades. However, the SBM is limited in analyzing complex networks as the model is, in essence, a random graph model that cannot reproduce the basic properties of many complex networks, such as sparsity and heavy-tailed degree distribution. In this paper, we provide an edge exchangeable block model that incorporates such basic features and simultaneously infers the latent block structure of a given complex network. Our model is a Bayesian nonparamet…
For two vast families of mixture distributions and a given prior, we provide unified representations of posterior and predictive distributions. Model applications presented include bivariate mixtures of Gamma distributions labelled as Kibble-type, non-central Chi-square and F distributions, the distribution of $R^2$ in multiple regression, variance mixture of normal distributions, and mixtures of location-scale exponential distributions including the multivariate Lomax distribution. An emphasis is also placed on analytical representations and the relationships with a host of existing distribut…
The family of location and scale mixtures of Gaussians has the ability to generate a number of flexible distributional forms. It nests as particular cases several important asymmetric distributions like the Generalised Hyperbolic distribution. The Generalised Hyperbolic distribution in turn nests many other well known distributions such as the Normal Inverse Gaussian (NIG) whose practical relevance has been widely documented in the literature. In a multivariate setting, we propose to extend the standard location and scale mixture concept into a so called multiple scaled framework which has the…
Catastrophic loss data are known to be heavy-tailed. Practitioners then need models that are able to capture both tail and modal parts of claim data. To this purpose, a new parametric family of loss distributions is proposed as a gamma mixture of the generalized log-Moyal distribution from Bhati and Ravi (2018), termed the generalized log-Moyal gamma distribution (GLMGA). We discuss the probabilistic characteristics of the GLMGA, and statistical estimation of the parameters through maximum likelihood. While the GLMGA distribution is a special case of the GB2 distribution, we show that this sim…
