The Construction of Multivariate Distributions from Markov Random Fields

Kaiser, Mark J.; Cressie, Noel

doi:10.1006/jmva.1999.1878

Cited by 73 publications

(98 citation statements)

References 21 publications

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“…See for further details. Aggregation strategies based on Bayesian and classical geostatistical models as suggested by Handcock and Stein (1993), , Kaiser and Cressie (1993) and Cressie et al (1999) and Bayesian models for spatial interpolation (Le et al, 1997;Gaudard et al, 1999) are desirable in many contexts because they provide estimates of the error associated with exposure at any measured or unmeasured locations. However, they were not applicable to our data sets because of the limited number of monitoring stations that are available in the 20 counties.…”

Section: Description Of the Databasesmentioning

confidence: 99%

Combining Evidence on Air Pollution and Daily Mortality from the 20 Largest US Cities: A Hierarchical Modelling Strategy

Dominici

Samet

Zeger

2000

Journal of the Royal Statistical Society Series A: Statistics in Society

243

217

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Summary. Reports over the last decade of association between levels of particles in outdoor air and daily mortality counts have raised concern that air pollution shortens life, even at concentrations within current regulatory limits. Criticisms of these reports have focused on the statistical techniques that are used to estimate the pollution±mortality relationship and the inconsistency in ®ndings between cities. We have developed analytical methods that address these concerns and combine evidence from multiple locations to gain a uni®ed analysis of the data. The paper presents log-linear regression analyses of daily time series data from the largest 20 US cities and introduces hierarchical regression models for combining estimates of the pollution±mortality relationship across cities. We illustrate this method by focusing on mortality effects of PM 10 (particulate matter less than 10 m in aerodynamic diameter) and by performing univariate and bivariate analyses with PM 10 and ozone (O 3 ) level. In the ®rst stage of the hierarchical model, we estimate the relative mortality rate associated with PM 10 for each of the 20 cities by using semiparametric log-linear models. The second stage of the model describes between-city variation in the true relative rates as a function of selected city-speci®c covariates. We also ®t two variations of a spatial model with the goal of exploring the spatial correlation of the pollutant-speci®c coef®cients among cities. Finally, to explore the results of considering the two pollutants jointly, we ®t and compare univariate and bivariate models. All posterior distributions from the second stage are estimated by using Markov chain Monte Carlo techniques. In univariate analyses using concurrent day pollution values to predict mortality, we ®nd that an increase of 10 g m À3 in PM 10 on average in the USA is associated with a 0.48% increase in mortality (95% interval: 0.05, 0.92). With adjustment for the O 3 level the PM 10 -coef®cient is slightly higher. The results are largely insensitive to the speci®c choice of vague but proper prior distribution. The models and estimation methods are general and can be used for any number of locations and pollutant measurements and have potential applications to other environmental agents.

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Section: Description Of the Databasesmentioning

confidence: 99%

Combining Evidence on Air Pollution and Daily Mortality from the 20 Largest US Cities: A Hierarchical Modelling Strategy

Dominici

Samet

Zeger

2000

Journal of the Royal Statistical Society Series A: Statistics in Society

243

217

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“…For conditionals that are not derived from a known, though possibly unnormalised, joint distribution, compatibility must be checked; see for example Casella (1996) and Arnold et al (2001 (1996). Kaiser & Cressie (2000) consider the question of defining Markov random fields with arbitrary conditionals, and give necessary and sufficient conditions which such conditionals must fulfil. Their method, which relaxes the requirement of positivity, is based on checking for permutation invariance in the indices of clique associated terms of the Gibbs potential.…”

Section: Discussionmentioning

confidence: 99%

Efficient recursions for general factorisable models

Reeves¹,

Pettitt²

2004

Biometrika

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SummaryLet n S-valued categorical variables be jointly distributed according to a distribution known only up to an unknown normalising constant. For an unnormalised joint likelihood expressible as a product of factors, we give an algebraic recursion which can be used for computing the normalising constant and other summations. A saving in computation is achieved when each factor contains a lagged subset of the components combining in the joint distribution, with maximum computational efficiency as the subsets attain their minimum size. If each subset contains at most r + 1 of the n components in the joint distribution, we term this a lag-r model, whose normalising constant can be computed using a forward recursion in O(S r+1 ) computations, as opposed to O(S n ) for the direct computation. We show how a lag-r model represents a Markov random field and allows a neighbourhood structure to be related to the unnormalised joint likelihood. We illustrate the method by showing how the normalising constant of the Ising or autologistic model can be computed.

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“…This fundamental result is known as the HammersleyClifford theorem; see Besag [2,26] and Kaiser and Cressie [3] for more details and technical conditions on this MRF-Gibbs equivalence.…”

Section: Markov Random Fieldsmentioning

confidence: 99%

“…Caiser and Cressie [3] and Hughes et al [25] provide formulae for constructing the joint PMF (up to a normalizing constant) given the conditionals:…”

Section: Acknowledgmentsmentioning

confidence: 99%

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Better Autologistic Regression

Wolters

2017

Front. Appl. Math. Stat.

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Autologistic regression is an important probability model for dichotomous random variables observed along with covariate information. It has been used in various fields for analyzing binary data possessing spatial or network structure. The model can be viewed as an extension of the autologistic model (also known as the Ising model, quadratic exponential binary distribution, or Boltzmann machine) to include covariates. It can also be viewed as an extension of logistic regression to handle responses that are not independent. Not all authors use exactly the same form of the autologistic regression model. Variations of the model differ in two respects. First, the variable coding-the two numbers used to represent the two possible states of the variables-might differ. Common coding choices are (zero, one) and (minus one, plus one). Second, the model might appear in either of two algebraic forms: a standard form, or a recently proposed centered form. Little attention has been paid to the effect of these differences, and the literature shows ambiguity about their importance. It is shown here that changes to either coding or centering in fact produce distinct, non-nested probability models. Theoretical results, numerical studies, and analysis of an ecological data set all show that the differences among the models can be large and practically significant. Understanding the nature of the differences and making appropriate modeling choices can lead to significantly improved autologistic regression analyses. The results strongly suggest that the standard model with plus/minus coding, which we call the symmetric autologistic model, is the most natural choice among the autologistic variants.

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The Construction of Multivariate Distributions from Markov Random Fields

Cited by 73 publications

References 21 publications

Combining Evidence on Air Pollution and Daily Mortality from the 20 Largest US Cities: A Hierarchical Modelling Strategy

Combining Evidence on Air Pollution and Daily Mortality from the 20 Largest US Cities: A Hierarchical Modelling Strategy

Efficient recursions for general factorisable models

Better Autologistic Regression

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