Calculation of marginal densities for parameters of multinomial distributions

Forster, Jonathan J.; Skene, Allan M.

doi:10.1007/bf00156751

Cited by 8 publications

(6 citation statements)

References 25 publications

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“…For example, MCMC methods have proved e ective for the e cient calculation of posterior quantities of interest in contingency table problems with just a single model under consideration. See Epstein and Fienberg (1991) and Forster and Skene (1994) for details. Green (1995) considered the general model determination problem introduced in Section 1.…”

Section: Markov Chain Monte Carlomentioning

confidence: 99%

“…They showed that the univariate conditional distributions required for sampling are log-concave, and hence adaptive rejection sampling (Gilks and Wild, 1992) is straightforward to implement. Forster and Skene (1994) used a Gibbs sampler to generate from posterior distributions arising from multinomial likelihoods and logistic normal prior distributions, which corresponds to the situation in the present example. They showed that the Gibbs sampler is extremely e cient when a particular parameterisation, where the parameters are approximately a posteriori independent, is adopted.…”

Section: Reversible Jump Mcmc For Log-linear Modelsmentioning

confidence: 99%

“…They recommended the use of the Gibbs sampler over a number of possible implementations of the Metropolis-Hastings algorithm, for multinomial applications. Although, in the current situation where a large numbers of di erent models are under consideration, it is impractical to use the parameterisation suggested by Forster and Skene (1994), the Gibbs sampler is still an e cient approach for generating from the posterior distribution of the parameters for a particular log-linear model. We therefore choose to use a Gibbs sampler for the null move.…”

Section: Reversible Jump Mcmc For Log-linear Modelsmentioning

confidence: 99%

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Markov chain Monte Carlo model determination for hierarchical and graphical log-linear models

1999

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The Bayesian approach to comparing models involves calculating the posterior probability of each plausible model. For high-dimensional contingency tables, the set of plausible models is very large. We focus attention on reversible jump Markov chain Monte Carlo (Green, 1995) and develop strategies for calculating posterior probabilities of hierarchical, graphical or decomposable log-linear models. Even for tables of moderate size, these sets of models may be very large. The choice of suitable prior distributions for model parameters is also discussed in detail, and two examples are presented. For the rst example, a 2 3 4 table, the model probabilities calculated using our reversible jump approach are compared with model probabilities calculated exactly or by using an alternative approximation. The second example is a 2 6 contingency table for which exact methods are infeasible, due to the large number of possible models.

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Section: Markov Chain Monte Carlomentioning

confidence: 99%

Section: Reversible Jump Mcmc For Log-linear Modelsmentioning

confidence: 99%

Section: Reversible Jump Mcmc For Log-linear Modelsmentioning

confidence: 99%

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Markov chain Monte Carlo model determination for hierarchical and graphical log-linear models

1999

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“…The performance of this strategy in the context of the air pollution data analysis is discussed in Section 4. Alternative MCMC strategies for generating samples from the Aitchison distribution are discussed by Forster and Skene (1994).…”

Section: Methodsmentioning

confidence: 99%

Combining Contingency Tables with Missing Dimensions

Dominici

2000

Biometrics

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We propose a methodology for estimating the cell probabilities in a multiway contingency table by combining partial information from a number of studies when not all of the variables are recorded in all studies. We jointly model the full set of categorical variables recorded in at least one of the studies, and we treat the variables that are not reported as missing dimensions of the study-specific contingency table. For example, we might be interested in combining several cohort studies in which the incidence in the exposed and nonexposed groups is not reported for all risk factors in all studies while the overall numbers of cases and cohort size is always available. To account for study-to-study variability, we adopt a Bayesian hierarchical model. At the first stage of the model, the observation stage, data are modeled by a multinomial distribution with fixed total number of observations. At the second stage, we use the logistic normal (LN) distribution to model variability in the study-specific cells' probabilities. Using this model and data augmentation techniques, we reconstruct the contingency table for each study regardless of which dimensions are missing, and we estimate population parameters of interest. Our hierarchical procedure borrows strength from all the studies and accounts for correlations among the cells' probabilities. The main difficulty in combining studies recording different variables is in maintaining a consistent interpretation of parameters across studies. The approach proposed here overcomes this difficulty and at the same time addresses the uncertainty arising from the missing dimensions. We apply our modeling strategy to analyze data on air pollution and mortality from 1987 to 1994 for six U.S. cities by combining six cross-classifications of low, medium, and high levels of mortality counts, particulate matter, ozone, and carbon monoxide with the complication that four of the six cities do not report all the air pollution variables. Our goals are to investigate the association between air pollution and mortality by reconstructing the tables with missing dimensions, to determine the most harmful pollutant combinations, and to make predictions about these key issues for a city other than the six sampled. We find that, for high levels of ozone and carbon monoxide, the number of cases with a high number of deaths increases as the levels of particulate matter, PM10, increases and that the most harmful combinations corresponds to high levels of PM10, confirming prior findings that levels of PM10 higher than the NAAQS standard are harmful.

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“…However, Markov chain Monte Carlo (MCMC) simulation techniques can overcome the computational problems, even for posterior densities of high dimensionatity or complexity. See, for example, Forster and Skene (1994) who estimate marginal posterior densities for multidimensional tables arising from an intractable prior family of distributions.…”

Section: Introductionmentioning

confidence: 99%