We develop a Bayesian approach for selecting the model which is the most supported by the data within a class of marginal models for categorical variables formulated through equality and/or inequality constraints on generalised logits (local, global, continuation or reverse continuation), generalised log-odds ratios and similar higher-order interactions. For each constrained model, the prior distribution of the model parameters is formulated following the encompassing prior approach. Then, model selection is performed by using Bayes factors which are estimated by an importance sampling method. The approach is illustrated through three applications involving some datasets, which also include explanatory variables. In connection with one of these examples, a sensitivity analysis to the prior specification is also considered.
We consider marginal log-linear models for parameterizing distributions on multidimensional contingency tables. These models generalize ordinary log-linear and multivariate logistic models, besides several others. First, we obtain some characteristic properties of marginal log-linear parameters. Then we define collapsibility and strict collapsibility of these parameters in a general sense. Several necessary and sufficient conditions for collapsibility and strict collapsibility are derived based on simple functions of only the cell probabilities, which are easily verifiable. These include resul…
Multivariate categorical data are common in many fields. We are motivated by election polls studies assessing evidence of changes in voters opinions with their candidates preferences in the 2016 United States Presidential primaries or caucuses. Similar goals arise routinely in several applications, but current literature lacks a general methodology which combines flexibility, efficiency, and tractability in testing for group differences in multivariate categorical data at different---potentially complex---scales. We address this goal by leveraging a Bayesian representation which factorizes the…
Learning the structure of dependence relations between variables is a pervasive issue in the statistical literature. A directed acyclic graph (DAG) can represent a set of conditional independences, but different DAGs may encode the same set of relations and are indistinguishable using observational data. Equivalent DAGs can be collected into classes, each represented by a partially directed graph known as essential graph (EG). Structure learning directly conducted on the EG space, rather than on the allied space of DAGs, leads to theoretical and computational benefits. Still, the majority of e…
Bayesian analysis methods often use some form of iterative simulation such as Monte Carlo computation. Models that involve discrete variables can sometime pose a challenge, either because the methods used do not support such variables (e.g. Hamiltonian Monte Carlo) or because the presence of such variables can slow down the computation. A common workaround is to marginalise the discrete variables out of the model. While it is reasonable to expect that such marginalisation would also lead to more time-efficient computations, to our knowledge this has not been demonstrated beyond a few specialis…
We propose a novel method for multiple clustering that assumes a co-clustering structure (partitions in both rows and columns of the data matrix) in each view. The new method is applicable to high-dimensional data. It is based on a nonparametric Bayesian approach in which the number of views and the number of feature-/subject clusters are inferred in a data-driven manner. We simultaneously model different distribution families, such as Gaussian, Poisson, and multinomial distributions in each cluster block. This makes our method applicable to datasets consisting of both numerical and categorica…
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…
We study a class of missingness mechanisms, called sequentially additive nonignorable, for modeling multivariate data with item nonresponse. These mechanisms explicitly allow the probability of nonresponse for each variable to depend on the value of that variable, thereby representing nonignorable missingness mechanisms. These missing data models are identified by making use of auxiliary information on marginal distributions, such as marginal probabilities for multivariate categorical variables or moments for numeric variables. We present theory proving identification results, and illustrate t…
Motivated by genome-wide association screening studies (GWAS), we study high-dimensional marginal screenings of categorical variables where test statistics have approximate chi-square distributions. We characterize four new phase transitions in high-dimensional chi-square models, and derive the signal sizes necessary and sufficient for statistical procedures to simultaneously control false discovery (in terms of family-wise error rate or false discovery rate) and missed detection (in terms of family-wise non-discovery rate or false non-discovery rate) in large dimensions. Remarkably, degrees o…
We discuss two parameterizations of models for marginal independencies for discrete distributions which are representable by bi-directed graph models, under the global Markov property. Such models are useful data analytic tools especially if used in combination with other graphical models. The first parameterization, in the saturated case, is also known as the multivariate logistic transformation, the second is a variant that allows, in some (but not all) cases, variation independent parameters. An algorithm for maximum likelihood fitting is proposed, based on an extension of the Aitchison and…
Tumor cell populations can be thought of as being composed of homogeneous cell subpopulations, with each subpopulation being characterized by overlapping sets of single nucleotide variants (SNVs). Such subpopulations are known as subclones and are an important target for precision medicine. Reconstructing such subclones from next-generation sequencing (NGS) data is one of the major challenges in precision medicine. We present PairClone as a new tool to implement this reconstruction. The main idea of PairClone is to model short reads mapped to pairs of proximal SNVs. In contrast, most existing …
Gaussian factor models have proven widely useful for parsimoniously characterizing dependence in multivariate data. There is a rich literature on their extension to mixed categorical and continuous variables, using latent Gaussian variables or through generalized latent trait models acommodating measurements in the exponential family. However, when generalizing to non-Gaussian measured variables the latent variables typically influence both the dependence structure and the form of the marginal distributions, complicating interpretation and introducing artifacts. To address this problem we prop…
This article proposes a powerful scheme to monitor a large number of categorical data streams with heterogeneous parameters or nature. The data streams considered may be either nominal with a number of attribute levels or ordinal with some natural order among their attribute levels, such as good, marginal, and bad. For an ordinal data stream, it is assumed that there is a corresponding latent continuous data stream determining it. Furthermore, different data streams may have different number of attribute levels and different values of level probabilities. Due to high dimensionality, traditiona…
Raking is widely used in categorical data modeling and survey practice but faced with methodological and computational challenges. We develop a Bayesian paradigm for raking by incorporating the marginal constraints as a prior distribution via two main strategies: 1) constructing the solution subspaces via basis functions or projection matrix and 2) modeling soft constraints. The proposed Bayes-raking estimation integrates the models for the margins, the sample selection and response mechanism, and the outcome, with the capability to propagate all sources of uncertainty. Computation is done via…
Bayesian model comparison requires the specification of a prior distribution on the parameter space of each candidate model. In this connection two concerns arise: on the one hand the elicitation task rapidly becomes prohibitive as the number of models increases; on the other hand numerous prior specifications can only exacerbate the well-known sensitivity to prior assignments, thus producing less dependable conclusions. Within the subjective framework, both difficulties can be counteracted by linking priors across models in order to achieve simplification and compatibility; we discuss links w…
Latent class analysis is used to perform model based clustering for multivariate categorical responses. Selection of the variables most relevant for clustering is an important task which can affect the quality of clustering considerably. This work considers a Bayesian approach for selecting the number of clusters and the best clustering variables. The main idea is to reformulate the problem of group and variable selection as a probabilistically driven search over a large discrete space using Markov chain Monte Carlo (MCMC) methods. This approach results in estimates of degree of relevance of e…
Istituto Dalle Molle di Studi sull’Intelligenza Artificiale (IDSIA)（达莱莫莱人工智能研究所（IDSIA））；USI（意大利语区大学）；SUPSI（瑞士南部应用科学与艺术大学）
Score functions for learning the structure of Bayesian networks in the literature assume that data are a homogeneous set of observations; whereas it is often the case that they comprise different related, but not homogeneous, data sets collected in different ways. In this paper we propose a new Bayesian Dirichlet score, which we call Bayesian Hierarchical Dirichlet (BHD). The proposed score is based on a hierarchical model that pools information across data sets to learn a single encompassing network structure, while taking into account the differences in their probabilistic structures. We der…
Trinity College（都柏林圣三一学院）；Dalle Molle Institute for Artificial Intelligence Research (IDSIA) - USI/SUPSI（达勒莫尔人工智能研究所（IDSIA）- 瑞士意大利语区应用艺术与科学大学/瑞士南部应用科学与艺术大学）
Preferential Bayesian optimisation (PBO) deals with optimisation problems where the objective function can only be accessed via preference judgments, such as "this is better than that" between two candidate solutions (like in A/B tests or recommender systems). The state-of-the-art approach to PBO uses a Gaussian process to model the preference function and a Bernoulli likelihood to model the observed pairwise comparisons. Laplace's method is then employed to compute posterior inferences and, in particular, to build an appropriate acquisition function. In this paper, we prove that the true post…
