Bayesian multivariate mixed-scale density estimation

Canale, Antonio; Banjerdpongchai, David

doi:10.4310/sii.2015.v8.n2.a7

Cited by 16 publications

(19 citation statements)

References 19 publications

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“…Wu andGhosal (2008, 2010) showed consistency for a class of location-scale kernels and kernels with bounded support. Shen and Ghosal (2011) and Canale and Dunson (2011) showed consistency for DP location-scale mixtures based on multivariate Gaussian kernel. We extend on these results by analyzing a kernel composed of a copula density with location-scale marginals.…”

Section: Bayesian Nonparametric Copula Kernel Mixturementioning

confidence: 85%

“…The K-L property holds by the arguments of the previous section. However, the size condition does not hold automatically for the strong topology and one has to resort to the technique of truncating the parameter space, depending on the sample size (Canale and Dunson, 2011;Ghosal, 2010, Section 2.4).…”

Section: General Conditions For Strong Posterior Consistencymentioning

confidence: 99%

“…In the multivariate case, the metric entropy of the sieve components may quickly increase with increasing dimensions. Wu and Ghosal (2010) and Canale and Dunson (2011) argue that this substantially restricts the set of useable sieves in the multivariate setting: if the sieve is too small, this will restrict the prior; if it is too large, this may lead to an exploding entropy. This makes it difficult to provide general results on posterior consistency in multivariate settings.…”

Section: General Conditions For Strong Posterior Consistencymentioning

confidence: 99%

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Copula based factorization in Bayesian multivariate infinite mixture models

Burda

Prokhorov

2014

Journal of Multivariate Analysis

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Bayesian nonparametric models based on infinite mixtures of density kernels have been recently gaining in popularity due to their flexibility and feasibility of implementation even in complicated modeling scenarios. In economics, they have been particularly useful in estimating nonparametric distributions of latent variables. However, these models have been rarely applied in more than one dimension. Indeed, the multivariate case suffers from the curse of dimensionality, with a rapidly increasing number of parameters needed to jointly characterize each mixing component. In this paper, we propose a factorization scheme for nonparametric mixture models whereby each marginal dimension in the mixing parameter space is modeled separately, linked by a nonparametric random copula function. Specifically, we consider nonparametric univariate Gaussian mixtures for the marginals and a multivariate random Bernstein polynomial copula for the link function, under Dirichlet process priors. We show that this scheme leads to an improvement in the precision of a density estimate in finite samples, providing a suitable tool for applications in higher dimensions. We derive weak posterior consistency of the copula-based mixing scheme for general kernel types under high-level conditions, and strong posterior consistency for the specific Bernstein-Gaussian mixture model.

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Section: Bayesian Nonparametric Copula Kernel Mixturementioning

confidence: 85%

Section: General Conditions For Strong Posterior Consistencymentioning

confidence: 99%

Section: General Conditions For Strong Posterior Consistencymentioning

confidence: 99%

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Copula based factorization in Bayesian multivariate infinite mixture models

Burda

Prokhorov

2014

Journal of Multivariate Analysis

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“…The exposure cluster random effect formulation allows the estimation of profile mixture cluster effects while controlling for relevant fixed effects such as confounders. Note, the Gaussian assumption does not limit the form of the pollutants as the authors suggest the use of latent continuous variables for categorical pollutant measures [49, 50]. When pollutant measures are solely categorical, other random distributions have been proposed that can accommodate multivariate discrete data [51, 52, and 53 for sparse data].…”

Section: Resultsmentioning

confidence: 99%

Current approaches used in epidemiologic studies to examine short-term multipollutant air pollution exposures

Davalos

Luben

Herring

et al. 2017

Annals of Epidemiology

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Purpose Air pollution epidemiology traditionally focuses on the relationship between individual air pollutants and health outcomes (e.g., mortality). To account for potential copollutant confounding, individual pollutant associations are often estimated by adjusting or controlling for other pollutants in the mixture. Recently, the need to characterize the relationship between health outcomes and the larger multipollutant mixture has been emphasized in an attempt to better protect public health and inform more sustainable air quality management decisions. Methods New and innovative statistical methods to examine multipollutant exposures were identified through a broad literature search, with a specific focus on those statistical approaches currently used in epidemiologic studies of short-term exposures to criteria air pollutants (i.e., particulate matter, carbon monoxide, sulfur dioxide, nitrogen dioxide, and ozone). Results Five broad classes of statistical approaches were identified for examining associations between short-term multipollutant exposures and health outcomes, specifically Additive Main Effects, Effect Measure Modification, Unsupervised Dimension Reduction, Supervised Dimension Reduction, and Nonparametric methods. These approaches are characterized including advantages and limitations in different epidemiologic scenarios. Discussion By highlighting the characteristics of various studies in which multipollutant statistical methods have been employed, this review provides epidemiologists and biostatisticians with a resource to aid in the selection of the most optimal statistical method to use when examining multipollutant exposures.

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“…There is a growing literature on DPMMs that utilise latent variables and fixed cut-points. We note the work of Kottas, Müller & Quintana (2005) and DeYoreo & Kottas (2018) who focus on ordinal data, DeYoreo & Kottas (2015) who present regression models for binary outcomes, Canale & Dunson (2015) who treat the problem of mixed-scale density estimation from both a theoretical and an applied perspective, and Norets & Pelenis (2012) who also present theory and applications, but based on finite mixture models. However, models with fixed cut-points have yet to be extended to include covariates that are fixed by design, such as binary treatment allocation variables in clinical trial settings or the offset term in count regression settings.…”

Section: Introductionmentioning

confidence: 99%

Bayesian density regression for discrete outcomes

Παπαγεωργίου

2019

Aus NZ J of Statistics

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We develop Bayesian models for density regression with emphasis on discrete outcomes. The problem of density regression is approached by considering methods for multivariate density estimation of mixed scale variables, and obtaining conditional densities from the multivariate ones. The approach to multivariate mixed scale outcome density estimation that we describe represents discrete variables, either responses or covariates, as discretised versions of continuous latent variables. We present and compare several models for obtaining these thresholds in the challenging context of count data analysis where the response may be over-and/or under-dispersed in some of the regions of the covariate space. We utilise a nonparametric mixture of multivariate Gaussians to model the directly observed and the latent continuous variables. The paper presents a Markov chain Monte Carlo algorithm for posterior sampling, sufficient conditions for weak consistency, and illustrations on density, mean and quantile regression utilising simulated and real datasets.

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Bayesian multivariate mixed-scale density estimation

Cited by 16 publications

References 19 publications

Copula based factorization in Bayesian multivariate infinite mixture models

Copula based factorization in Bayesian multivariate infinite mixture models

Current approaches used in epidemiologic studies to examine short-term multipollutant air pollution exposures

Bayesian density regression for discrete outcomes

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