Prior Model Probabilities Research Articles

MODEL WEIGHTS AND THE FOUNDATIONS OF MULTIMODEL INFERENCE

Statistical thinking in wildlife biology and ecology has been profoundly influenced by the introduction of AIC (Akaike's information criterion) as a tool for model selection and as a basis for model averaging. In this paper, we advocate the Bayesian paradigm as a broader framework for multimodel inference, one in which model averaging and model selection are naturally linked, and in which the performance of AIC-based tools is naturally evaluated. Prior model weights implicitly associated with the use of AIC are seen to highly favor complex models: in some cases, all but the most highly parameterized models in the model set are virtually ignored a priori. We suggest the usefulness of the weighted BIC (Bayesian information criterion) as a computationally simple alternative to AIC, based on explicit selection of prior model probabilities rather than acceptance of default priors associated with AIC. We note, however, that both procedures are only approximate to the use of exact Bayes factors. We discuss and illustrate technical difficulties associated with Bayes factors, and suggest approaches to avoiding these difficulties in the context of model selection for a logistic regression. Our example highlights the predisposition of AIC weighting to favor complex models and suggests a need for caution in using the BIC for computing approximate posterior model weights.

Sensitivity analysis and assessment of prior model probabilities in MLBMA with application to unsaturated fractured tuff

Previous application of maximum likelihood Bayesian model averaging (MLBMA, Neuman (2002, 2003)) to alternative variogram models of log air permeability data in fractured tuff has demonstrated its effectiveness in quantifying conceptual model uncertainty and enhancing predictive capability (Ye et al., 2004). A question remained how best to ascribe prior probabilities to competing models. In this paper we examine the extent to which lead statistics of posterior log permeability predictions are sensitive to prior probabilities of seven corresponding variogram models. We then explore the feasibility of quantifying prior model probabilities by (1) maximizing Shannon's entropy H (Shannon, 1948) subject to constraints reflecting a single analyst's (or a group of analysts') prior perception about how plausible each alternative model (or a group of models) is relative to others, and (2) selecting a posteriori the most likely among such maxima corresponding to alternative prior perceptions of various analysts or groups of analysts. Another way to select among alternative prior model probability sets, which, however, is not guaranteed to yield optimum predictive performance (though it did so in our example) and would therefore not be our preferred option, is a minimum‐maximum approach according to which one selects a priori the set corresponding to the smallest value of maximum entropy. Whereas maximizing H subject to the prior perception of a single analyst (or group) maximizes the potential for further information gain through conditioning, selecting the smallest among such maxima gives preference to the most informed prior perception among those of several analysts (or groups). We use the same variogram models and log permeability data as Ye et al. (2004) to demonstrate that our proposed approach yields the least amount of posterior entropy (residual uncertainty after conditioning) and enhances predictive model performance as compared to (1) the noninformative neutral case in which all prior model probabilities are set equal to each other and (2) an informed case that nevertheless violates the principle of parsimony.

Matrix models for childhood infections: a Bayesian approach with applications to rubella and mumps

Mathematical modelling is an established tool for planning and monitoring vaccination programmes. However, the matrices describing contact rates are based on subjective choices, which have a large impact on results. This paper reviews published models and obtains prior model probabilities based on publication frequency and expert opinion. Using serological survey data on rubella and mumps, Bayesian methods of model choice are applied to select the most plausible models. Estimates of the basic reproduction number R0 are derived, taking into account model uncertainty and individual heterogeneity in contact rates. Twenty-two models are documented, for which publication frequency and expert opinion are negatively correlated. Using the expert prior with individual heterogeneity, R0=6.1 [95% credible region (CR) 4.3-9.2] for rubella and R0=19.3 (95% CR 4.0-31.5) for mumps. The posterior modes are insensitive to the prior for rubella but not for mumps. Overall, assortative models with individual heterogeneity are recommended.

Bayesian model choice based on Monte Carlo estimates of posterior model probabilities

A range of approximate methods have been proposed for model choice based on Bayesian principles, given the problems involved in multiple integration in multi-parameter problems. Formal Bayesian model assessment is based on prior model probabilities P ( M = j ) and posterior model probabilities P ( M = j | Y ) after observing the data. An approach is outlined here that produces posterior model probabilities and hence Bayes factor estimates but not marginal likelihoods. It uses a Monte Carlo approximation based on independent MCMC sampling of two or more different models. While parallel sampling of the models is not necessary, such a form of sampling facilitates model averaging and assessing the impact of individual observations on the overall estimated Bayes factor. Three worked examples used before in model choice studies illustrate application of the method.

Bayesian correlation estimation

We propose prior probability models for variance-covariance matrices in order to address two important issues. First, the models allow a researcher to represent substantive prior information about the strength of correlations among a set of variables. Secondly, even in the absence of such information, the increased flexibility of the models mitigates dependence on strict parametric assumptions in standard prior models. For example, the model allows a posteriori different levels of uncertainty about correlations among different subsets of variables. We achieve this by including a clustering mechanism in the prior probability model. Clustering is with respect to variables and pairs of variables. Our approach leads to shrinkage towards a mixture structure implied by the clustering. We discuss appropriate posterior simulation schemes to implement posterior inference in the proposed models, including the evaluation of normalising constants that are functions of parameters of interest. The normalising constants result from the restriction that the correlation matrix be positive definite. We discuss examples based on simulated data, a stock return dataset and a population genetics dataset.

Application of breast cancer risk prediction models in clinical practice.

Breast cancer risk assessment provides an estimation of disease risk that can be used to guide management for women at all levels of risk. In addition, the likelihood that breast cancer risk is due to specific genetic susceptibility (such as BRCA1 or BRCA2 mutations) can be determined. Recent developments have reinforced the clinical importance of breast cancer risk assessment. Tamoxifen chemoprevention as well as prevention studies such as the Study of Tamoxifen and Raloxifene are available to women at increased risk of developing breast cancer. In addition, specific management strategies are now defined for BRCA1 and BRCA2 mutation carriers. Risk may be assessed as the likelihood of developing breast cancer (using risk assessment models) or as the likelihood of detecting a BRCA1 or BRCA2 mutation (using prior probability models). Each of the models has advantages and disadvantages, and all need to be interpreted in context. We review available risk assessment tools and discuss their application. As illustrated by clinical examples, optimal counseling may require the use of several models, as well as clinical judgment, to provide the most accurate and useful information to women and their families.

Prior elicitation for model selection and estimation in generalized linear mixed models

Generalized linear models serve as a useful class of regression models for discrete and continuous data. In applications such as longitudinal studies, observations are typically correlated. The correlation structure in the data is induced by introducing a random effect, leading to the generalized linear mixed model (GLMM). In this paper, we propose a class of informative prior distributions for the class of GLMMs and investigate their theoretical as well as computational properties. Specifically, we investigate conditions for propriety of the proposed priors for the class of GLMMs and show that they are proper under some very general conditions. In addition, we examine recent computational methods such as hierarchical centering and semi-hierarchical centering for performing Gibbs sampling for this class of models. One of the main applications of the proposed priors is variable subset selection. Novel computational tools are developed for sampling from the posterior distributions and computing the prior and posterior model probabilities. We demonstrate our methodology with two real longitudinal datasets.

Gibbs Variable Selection UsingBUGS

In this paper we discuss and present in detail the implementation of Gibbs variable selection as defined by Dellaportas et al. (2000, 2002) using the BUGS software (Spiegelhalter et al. , 1996a,b,c). The specification of the likelihood, prior and pseudo-prior distributions of the parameters as well as the prior term and model probabilities are described in detail. Guidance is also provided for the calculation of the posterior probabilities within BUGS environment when the number of models is limited. We illustrate the application of this methodology in a variety of problems including linear regression, log-linear and binomial response models.

Prior Elicitation, Variable Selection and Bayesian Computation for Logistic Regression Models

Summary Bayesian selection of variables is often difficult to carry out because of the challenge in specifying prior distributions for the regression parameters for all possible models, specifying a prior distribution on the model space and computations. We address these three issues for the logistic regression model. For the first, we propose an informative prior distribution for variable selection. Several theoretical and computational properties of the prior are derived and illustrated with several examples. For the second, we propose a method for specifying an informative prior on the model space, and for the third we propose novel methods for computing the marginal distribution of the data. The new computational algorithms only require Gibbs samples from the full model to facilitate the computation of the prior and posterior model probabilities for all possible models. Several properties of the algorithms are also derived. The prior specification for the first challenge focuses on the observables in that the elicitation is based on a prior prediction y 0 for the response vector and a quantity a 0 quantifying the uncertainty in y 0. Then, y 0 and a 0 are used to specify a prior for the regression coefficients semi-automatically. Examples using real data are given to demonstrate the methodology.

Predictive specification of prior model probabilities in variable selection

SUMMARY We examine the problem of specifying prior probabilities for all possible subset models in the context of variable selection in normal linear models. A solution is proposed that uses a prior prediction for the observable, an associated weight, and prior opinion regarding error precision as the only required input. Numerical examples are given to illustrate the method.

The Synthesis of Predictive Models in Marketing Research

A formal procedure for synthesizing the variety of predictive models usually available to the market research analyst is presented as a practical method. Two case studies indicate its feasibility. The method is Bayesian, involving the assignment of subjective probabilities to the various predictive models and the revision of these prior model probabilities according to a Dirichlet process if a set of performance data is available.

The mutual information principle and applications

A general theory of prior probability models is presented, valid for both discrete and continuous random variables, even when the prior information about them has been obtained with errors. An example is included as an illustration.