A Gradient Markov Chain Monte Carlo Algorithm for Computing Multivariate Maximum Likelihood Estimates and Posterior Distributions

Mixture Dose-Response Assessment

Ruochen Li, James Douglas Englehardt, Xiaoguang Li

Research output: Contribution to journalArticle

1 Citation (Scopus)

Abstract

Multivariate probability distributions, such as may be used for mixture dose-response assessment, are typically highly parameterized and difficult to fit to available data. However, such distributions may be useful in analyzing the large electronic data sets becoming available, such as dose-response biomarker and genetic information. In this article, a new two-stage computational approach is introduced for estimating multivariate distributions and addressing parameter uncertainty. The proposed first stage comprises agradient Markov chain Monte Carlo(GMCMC) technique to find Bayesian posterior mode estimates (PMEs) of parameters, equivalent to maximum likelihood estimates (MLEs) in the absence of subjective information. In the second stage, these estimates are used to initialize a Markov chain Monte Carlo (MCMC) simulation, replacing the conventional burn-in period to allow convergent simulation of the full joint Bayesian posterior distribution and the corresponding unconditional multivariate distribution (not conditional on uncertain parameter values). When the distribution of parameter uncertainty is such a Bayesian posterior, the unconditional distribution is termedpredictive. The method is demonstrated by finding conditional and unconditional versions of the recently proposed emergent dose-response function (DRF). Results are shown for the five-parameter common-mode and seven-parameter dissimilar-mode models, based on published data for eight benzene-toluene dose pairs. The common mode conditional DRF is obtained with a 21-fold reduction in data requirement versus MCMC. Example common-mode unconditional DRFs are then found using synthetic data, showing a 71% reduction in required data. The approach is further demonstrated for a PCB 126-PCB 153 mixture. Applicability is analyzed and discussed. Matlab ® computer programs are provided.

Original languageEnglish
Pages (from-to)345-359
Number of pages15
JournalRisk Analysis
Volume32
Issue number2
DOIs
StatePublished - Feb 1 2012

Fingerprint

Likelihood Functions
Markov Chains
Markov processes
Maximum likelihood
3,4,5,3',4'-pentachlorobiphenyl
2,4,5,2',4',5'-hexachlorobiphenyl
Polychlorinated biphenyls
Uncertainty
Toluene
Biomarkers
Benzene
Burns
Probability distributions
Computer program listings
Software

Keywords

  • Dose-response
  • Gradient
  • Markov chain Monte Carlo
  • Mixture
  • Predictive

ASJC Scopus subject areas

  • Physiology (medical)
  • Safety, Risk, Reliability and Quality

Cite this

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title = "A Gradient Markov Chain Monte Carlo Algorithm for Computing Multivariate Maximum Likelihood Estimates and Posterior Distributions: Mixture Dose-Response Assessment",
abstract = "Multivariate probability distributions, such as may be used for mixture dose-response assessment, are typically highly parameterized and difficult to fit to available data. However, such distributions may be useful in analyzing the large electronic data sets becoming available, such as dose-response biomarker and genetic information. In this article, a new two-stage computational approach is introduced for estimating multivariate distributions and addressing parameter uncertainty. The proposed first stage comprises agradient Markov chain Monte Carlo(GMCMC) technique to find Bayesian posterior mode estimates (PMEs) of parameters, equivalent to maximum likelihood estimates (MLEs) in the absence of subjective information. In the second stage, these estimates are used to initialize a Markov chain Monte Carlo (MCMC) simulation, replacing the conventional burn-in period to allow convergent simulation of the full joint Bayesian posterior distribution and the corresponding unconditional multivariate distribution (not conditional on uncertain parameter values). When the distribution of parameter uncertainty is such a Bayesian posterior, the unconditional distribution is termedpredictive. The method is demonstrated by finding conditional and unconditional versions of the recently proposed emergent dose-response function (DRF). Results are shown for the five-parameter common-mode and seven-parameter dissimilar-mode models, based on published data for eight benzene-toluene dose pairs. The common mode conditional DRF is obtained with a 21-fold reduction in data requirement versus MCMC. Example common-mode unconditional DRFs are then found using synthetic data, showing a 71{\%} reduction in required data. The approach is further demonstrated for a PCB 126-PCB 153 mixture. Applicability is analyzed and discussed. Matlab {\circledR} computer programs are provided.",
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