Bayesian hierarchical models are commonly employed for inference in count datasets, as they account for multiple levels of variation by incorporating prior distributions for parameters at different levels. Examples include Beta-Binomial, Negative-Binomial (NB), Dirichlet-Multinomial (DM) distributions. In this paper, we address two crucial challenges that arise in various Bayesian count models: inference for the concentration parameter in the ratio of Gamma functions and the inability of these models to effectively handle excessive zeros and small nonzero counts. We propose a novel class of prior distributions that facilitates conjugate updating of the concentration parameter in Gamma ratios, enabling full Bayesian inference for the aforementioned count distributions. We use DM models as our running examples. Our methodology leverages fast residue computation and admits closed-form posterior moments. Additionally, we recommend a default horseshoe type prior which has a heavy tail and substantial mass around zero. It admits continuous shrinkage, making the posterior highly adaptable to sparsity or quasi-sparsity in the data. Furthermore, we offer insights and potential generalizations to other count models facing the two challenges. We demonstrate the usefulness of our approach on both simulated examples and on real-world applications. Finally, we conclude with directions for future research.
The University of Electro-Communications（电气通信大学）；National Police Agency Info-Communications Bureau（警察厅信息通信局）
In this paper, we propose a Bayesian Graphical LASSO for correlated countable data and apply it to spatial crime data. In the proposed model, we assume a Gaussian Graphical Model for the latent variables which dominate the potential risks of crimes. To evaluate the proposed model, we determine optimal hyperparameters which represent samples better. We apply the proposed model for estimation of the sparse inverse covariance of the latent variable and evaluate the partial correlation coefficients. Finally, we illustrate the results on crime spots data and consider the estimated latent variables …
MD Anderson Cancer Center（MD安德森癌症中心）；University of Pittsburgh（匹兹堡大学）；University of Maryland（马里兰大学）
Clustering with variable selection is a challenging yet critical task for modern small-n-large-p data. Existing methods based on sparse Gaussian mixture models or sparse K-means provide solutions to continuous data. With the prevalence of RNA-seq technology and lack of count data modeling for clustering, the current practice is to normalize count expression data into continuous measures and apply existing models with Gaussian assumption. In this paper, we develop a negative binomial mixture model with lasso or fused lasso gene regularization to cluster samples (small n) with high-dimensional g…
Non-negative tensor factorization models enable predictive analysis on count data. Among them, Bayesian Poisson-Gamma models can derive full posterior distributions of latent factors and are less sensitive to sparse count data. However, current inference methods for these Bayesian models adopt restricted update rules for the posterior parameters. They also fail to share the update information to better cope with the data sparsity. Moreover, these models are not endowed with a component that handles the imbalance in count data values. In this paper, we propose a novel variational auto-encoder f…
The field of retail analytics has been transformed by the availability of rich data which can be used to perform tasks such as demand forecasting and inventory management. However, one task which has proved more challenging is the forecasting of demand for products which exhibit very few sales. The sparsity of the resulting data limits the degree to which traditional analytics can be deployed. To combat this, we represent sales data as a structured sparse multivariate point process which allows for features such as auto-correlation, cross-correlation, and temporal clustering, known to be prese…
Accurate estimates of subnational populations are important for policy formulation and monitoring population health indicators. For example, estimates of the number of women of reproductive age are important to understand the population at risk to maternal mortality and unmet need for contraception. However, in many low-income countries, data on population counts and components of population change are limited, and so levels and trends subnationally are unclear. We present a Bayesian constrained cohort component model for the estimation and projection of subnational populations. The model buil…
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；Columbia University（哥伦比亚大学）；Microsoft Research（微软研究院）
We present a Bayesian tensor factorization model for inferring latent group structures from dynamic pairwise interaction patterns. For decades, political scientists have collected and analyzed records of the form "country $i$ took action $a$ toward country $j$ at time $t$"---known as dyadic events---in order to form and test theories of international relations. We represent these event data as a tensor of counts and develop Bayesian Poisson tensor factorization to infer a low-dimensional, interpretable representation of their salient patterns. We demonstrate that our model's predictive perform…
Understanding the underlying causes of maternal death across all regions of the world is essential to inform policies and resource allocation to reduce the mortality burden. However, in many countries there exists very little data on the causes of maternal death, and data that do exist do not capture the entire population at risk. In this paper, we present a Bayesian hierarchical multinomial model to estimate maternal cause of death distributions globally, regionally, and for all countries worldwide. The framework combines data from various sources to inform estimates, including data from civi…
University Carlos III in Madrid（马德里卡洛斯三世大学）；Macedonian Academy of Sciences and Arts（马其顿科学与艺术学院）
This paper presents a Bayesian nonparametric latent feature model specially suitable for exploratory analysis of high-dimensional count data. We perform a non-negative doubly sparse matrix factorization that has two main advantages: not only we are able to better approximate the row input distributions, but the inferred topics are also easier to interpret. By combining the three-parameter and restricted Indian buffet processes into a single prior, we increase the model flexibility, allowing for a full spectrum of sparse solutions in the latent space. We demonstrate the usefulness of our approa…
The SparseStep algorithm is presented for the estimation of a sparse parameter vector in the linear regression problem. The algorithm works by adding an approximation of the exact counting norm as a constraint on the model parameters and iteratively strengthening this approximation to arrive at a sparse solution. Theoretical analysis of the penalty function shows that the estimator yields unbiased estimates of the parameter vector. An iterative majorization algorithm is derived which has a straightforward implementation reminiscent of ridge regression. In addition, the SparseStep algorithm is …
In a variety of application areas, there is a growing interest in analyzing high dimensional sparse count data, with sparsity exhibited by an over-abundance of zeros and small non-zero counts. Existing approaches for analyzing multivariate count data via Poisson or negative binomial log-linear hierarchical models with zero-inflation cannot flexibly adapt to the level and nature of sparsity in the data. We develop a new class of continuous local-global shrinkage priors tailored for sparse counts. Theoretical properties are assessed, including posterior concentration, stronger control on false d…
In this paper, we consider testing the homogeneity of risk differences in independent binomial distributions especially when data are sparse. We point out some drawback of existing tests in either controlling a nominal size or obtaining powers through theoretical and numerical studies. The proposed test is designed to avoid such drawback of existing tests. We present the asymptotic null distributions and asymptotic powers for our proposed test. We also provide numerical studies including simulations and real data examples showing the proposed test has reliable results compared to existing test…
Modeling data with multivariate count responses is a challenging problem due to the discrete nature of the responses. Existing methods for univariate count responses cannot be easily extended to the multivariate case since the dependency among multiple responses needs to be properly accommodated. In this paper, we propose a multivariate Poisson log-normal regression model for multivariate data with count responses. By simultaneously estimating the regression coefficients and inverse covariance matrix over the latent variables with an efficient Monte Carlo EM algorithm, the proposed regression …
Canonical correlation analysis (CCA) is a classical and important multivariate technique for exploring the relationship between two sets of continuous variables. CCA has applications in many fields, such as genomics and neuroimaging. It can extract meaningful features as well as use these features for subsequent analysis. Although some sparse CCA methods have been developed to deal with high-dimensional problems, they are designed specifically for continuous data and do not consider the integer-valued data from next-generation sequencing platforms that exhibit very low counts for some importan…
This paper discusses predictive densities under the Kullback--Leibler loss for high-dimensional Poisson sequence models under sparsity constraints. Sparsity in count data implies zero-inflation. We present a class of Bayes predictive densities that attain asymptotic minimaxity in sparse Poisson sequence models. We also show that our class with an estimator of unknown sparsity level plugged-in is adaptive in the asymptotically minimax sense. For application, we extend our results to settings with quasi-sparsity and with missing-completely-at-random observations. The simulation studies as well a…
We propose a new approach to the problem of high-dimensional multivariate ANOVA via bootstrapping max statistics that involve the differences of sample mean vectors. The proposed method proceeds via the construction of simultaneous confidence regions for the differences of population mean vectors. It is suited to simultaneously test the equality of several pairs of mean vectors of potentially more than two populations. By exploiting the variance decay property that is a natural feature in relevant applications, we are able to provide dimension-free and nearly-parametric convergence rates for G…
Large-scale multiple testing problems require the simultaneous assessment of many p-values. This paper compares several methods to assess the evidence in multiple binomial counts of p-values: the maximum of the binomial counts after standardization (the `higher-criticism statistic'), the maximum of the binomial counts after a log-likelihood ratio transformation (the `Berk-Jones statistic'), and a newly introduced average of the binomial counts after a likelihood ratio transformation. Simulations show that the higher criticism statistic has a superior performance to the Berk-Jones statistic in …
In multivariate statistics, the question of finding direct interactions can be formulated as a problem of network inference - or network reconstruction - for which the Gaussian graphical model (GGM) provides a canonical framework. Unfortunately, the Gaussian assumption does not apply to count data which are encountered in domains such as genomics, social sciences or ecology. To circumvent this limitation, state-of-the-art approaches use two-step strategies that first transform counts to pseudo Gaussian observations and then apply a (partial) correlation-based approach from the abundant literat…
Subgraph counting estimation for the $β$-model in sparse networks
University of Science and Technology of China（中国科学技术大学）；Carnegie Mellon University（卡内基梅隆大学）；Southeast University（东南大学）；Central China Normal University（华中师范大学）
Gender disparities in rehospitalisations after coronary artery bypass grafting: evidence from a sparse functional causal mediation analysis of the MIMIC-IV data
Sparse departures from independence in two-way tables: a heteroscedasticity profile and detection boundary, an adaptive higher-criticism gate, and an assumption-lean exact anchor
