Eliciting prior information to enhance the predictive performance of bayesian graphical models

Both knowledge-based systems and statistical models are typically concerned with making predictions about future observables. Here we focus on assessment of predictive performance and provide two techniques for improving the predictive performance of Bayesian graphical models. First, we present Bayesian model averaging, a technique for accounting for model uncertainty.

Second, we describe a technique for eliciting a prior distribution for competing models from domain experts. We explore the predictive performance of both techniques in the context of a urological diagnostic problem.     

Sensitivity of a bayesian inference to prior assumptions

The sensitivity of-a Bayesian inference to prior assumptions is examined by Monte Carlo simulation for the beta-binomial conjugate family of distributions. Results for the effect on a Bayesian probability interval of the binomial parameter indicate that the Bayesian inference is for the most part quite sensitive to misspecification of the prior distribution. The magnitude of the sensitivity depends primarily on the difference of assigned means and variances from the respective means and variances of the actually-sampled prior distributions. The effect of a disparity in form between the assigned prior and actually-sampled distributions was less important for the cases tested.     

Image reconstruction by adaptive bayesian classification with a locally proportional prior

This paper concerns the problem of reconstructing images from noisy data by means of Bayesian classification methods. In Klein and Press, 1992, the authors presented a method for reconstructing images called Adaptive Bayesian Classification (ABC). The ABC procedure was shown to preform very well in simulation experiments. The ABC procedure was multistaged; moreover, it involved selecting a prior at Stage n that was the posterior at Stage n - 1. In this paper the authors show that we can improve upon ABC for some problems by modifying the way we take the prior at each stage. The new proposal is to take the prior for the pixel label at each stage as proportional to the number of pixels with that label in a small neighborhood of the pixel. The ABC procedure with a locally proportional prior (ABC/LPP) tends to improve upon the ABC procedure for some problems because the prior in the iterative portion of ABC/LPP is contextual, while that in ABC in non- contextual.     

Sample size re‐estimation incorporating prior information on a nuisance parameter

Prior information is often incorporated informally when planning a clinical trial. Here, we present an approach on how to incorporate prior information, such as data from historical clinical trials, into the nuisance parameter–based sample size re‐estimation in a design with an internal pilot study. We focus on trials with continuous endpoints in which the outcome variance is the nuisance parameter. For planning and analyzing the trial, frequentist methods are considered. Moreover, the external information on the variance is summarized by the Bayesian meta‐analytic‐predictive approach. To incorporate external information into the sample size re‐estimation, we propose to update the meta‐analytic‐predictive prior based on the results of the internal pilot study and to re‐estimate the sample size using an estimator from the posterior. By means of a simulation study, we compare the operating characteristics such as power and sample size distribution of the proposed procedure with the traditional sample size re‐estimation approach that uses the pooled variance estimator. The simulation study shows that, if no prior‐data conflict is present, incorporating external information into the sample size re‐estimation improves the operating characteristics compared to the traditional approach. In the case of a prior‐data conflict, that is, when the variance of the ongoing clinical trial is unequal to the prior location, the performance of the traditional sample size re‐estimation procedure is in general superior, even when the prior information is robustified. When considering to include prior information in sample size re‐estimation, the potential gains should be balanced against the risks.     

An information-theoretic approach to incorporating prior information in binomial sampling

The incorporation of prior information about θ, where θ is the success probability in a binomial sampling model, is an essential feature of Bayesian statistics. Methodology based on information-theoretic concepts is introduced which (a) quantifies the amount of information provided by the sample data relative to that provided by the prior distribution and (b) allows for a ranking of prior distributions with respect to conservativeness, where conservatism refers to restraint of extraneous information about θ which is embedded in any prior distribution. In effect, the most conservative prior distribution from a specified class (each member o f which carries the available prior information about θ) is that prior distribution within the class over which the likelihood function has the greatest average domination. The most conservative prior distributions from five different families of prior distributions over the interval (0,1) including the beta distribution are determined and compared for three situations: (1) no prior estimate of θ is available, (2) a prior point estimate or θ is available, and (3) a prior interval estimate of θ is available. The results of the comparisons not only advocate the use of the beta prior distribution in binomial sampling but also indicate which particular one to use in the three aforementioned situations.     

Bayesian analysis of correlated mixed categorical data by incorporating historical prior information

In this article, we develop statistical models for analysis of correlated mixed categorical (binary and ordinal) response data arising in medical and epidemi-ologic studies. There is evidence in the literature to suggest that models including correlation structure can lead to substantial improvement in precision of estimation or are more appropriate (accurate). We use a very rich class of scale mixture of multivariate normal (SMMVN) iink functions to accommodate heavy tailed distributions. In order to incorporate available historical information, we propose a unified prior elicitation scheme based on SMMVN-link models. Further, simulation-based techniques are developed to assess model adequacy. Finally, a real data example from prostate cancer studies is used to illustrate the proposed methodologies.     

Implementation of a robust bayesian method

In this work we study robustness in Bayesian models through a generalization of the Normal distribution. We show new appropriate techniques in order to deal with this distribution in Bayesian inference. Then we propose two approaches to decide, in some applications, if we should replace the usual Normal model by this generalization. First, we pose this dilemma as a model rejection problem, using diagnostic measures. In the second approach we evaluate the model's predictive efficiency. We illustrate those perspectives with a simulation study, a non linear model and a longitudinal data model.     

Comparison of proportions with asymmetric prior information

In this paper a Bayesian model is developed for comparing two binomial proportions. A two stage hierarchical prior distribution is used to represent prior dependence. Prior exchangeability and independence are shown to be but special cases. The relevant distributions have to be computed numerically and some examples are presented.     

Limited information bayesian analysis of a structural coefficient in a simultaneous equations system

An analytical expression is obtained for the marginal posterior density for a structural coefficient in a simultaneous equations system based on a limited information Bayesian analysis. A con- ditional posterior density is obtained given reduced form para- meters. This conditional posterior density is in univariate student t form. Numerical examples suggest that the conditional density hasa tighter distribution around the posterior mean than the unconditional density when the correlation between the endo- genous variables and the structural error term is high.     

Properties of a fourier bootstrap method for time series

A method for bootstrapping stationary Gaussian sequences is studied. The FFT is applied to the original data, randomized in the frequency domain, and the inverse FFT is applied. The result is a sequence whose second order properties are similar to those of the original sequence. Conditions under which the method is valid are given.     

Inferences for hierarchical models with partial prior information

Suppose items can be purchased from one of k-suppliers and it is required to purchase from the one with the smaller failure rate or equivalently from the one with the larger mean-time-to-failure. It is assumed that data $\mathit{d}$, in the form of the times-to-failure for $n_1,\dots ,n_k$ items from suppliers $1,\dots ,k$, respectively is available. There are two suggested selection criteria studied in this paper and when comparing only two suppliers they reduce to$P({\theta }_1<b{\theta }_2|d)\text{and}P(Y_1>{\mathit{cY}}_2|d)\text{,}$where b and c are prespecified practical constants; ${\theta }_1$ and ${\theta }_2$ are the respective mean failure rates; $Y_1$ and $Y_2$ are the predicted times to failure for individual items purchased from each supplier.In addition partial prior information about the $k$-suppliers collectively is assumed to have been elicited. This situation is modelled using the hierarchical Bayesian approach, which easily facilitates interpreting the elicited partial prior information as constraints on the hyperpriors, i.e. hyperpriors that are known only to be contained in families with specified properties. In this paper these properties are assumed to be in the form of specifying certain quantiles arising from the elicited information. Minimum and maximum values of the above selection criteria are obtained and are used to indicate whether or not the elicited prior information is useful. Specific examples are given for comparing two suppliers but generalisation to k-suppliers follows easily.     

Improved confidence intervals for a binomial parameter using the bayesian method

Constructing confidence intervals for a binomial proportion parameter using the Bayesian technique is considered. For an appropriate choice of priors, the proposed Bayes confidence intervals may, in frequentist performance, uniformly improve the traditional C-P (Clopper and Pearson, 1934) confidence intervals when the sample size is not large (n < 30).     

Unbiased ridge estimation with prior information and ridge trace

A procedure is illustrated to incorporate prior information in the ridge regression model. Unbiased ridge estimators with prior information are defined and a robust estimate of the ridge parameter k is proposed.     

Testing base load with non-sample prior information on process load

Inference about population parameters could be improved using non- sample prior information (NSPI) from reliable sources along with the available data. This paper studies the problem of testing the intercept parameter of a simple regression model when NSPI is available on the value of the slope. The information on the slope may have the three different scenarios: (i) unknown (unspecified), (ii) known (certain or specified), and (iii) uncertain if the suspected value is unsure, for which we define the unrestricted test (UT), restricted test (RT) and pre-test test (PTT) for the intercept parameter. The test statistics, their sampling distributions, and power functions are derived. Comparison of the power functions and size of the tests are used to search and recommend a best test. The study reveals that the PTT has a reasonable dominance over the UT and RT both in terms of achieving highest power and lowest size.     

A bayesian method in determining the order of a finite state markov chain

In this paper, we use the Bayesian method in the application of hypothesis testing and model selection to determine the order of a Markov chain. The criteria used are based on Bayes factors with noninformative priors. Com¬parisons with the commonly used AIC and BIC criteria are made through an example and computer simulations. The results show that the proposed method is better than the AIC and BIC criteria, especially for Markov chains with higher orders and larger state spaces.     

On forecasting with univariate autoregressive processes: a bayesian approach

Using a normal-gamma prior density for the parameters of a p-th order autoregressive process, the Bayesian predictive density of k future observations is derived and it is shown that it is the product of k univariate t densities. Our results are illustrated with one step ahead forecasts employing AR(1) and AR(2) models with a vague prior density for the parameters.     

Pooling means under uncertain prior information with application to discrete distributions

The problem of pooling means is considered based on two samples in presence of the uncertain prior information that these samples are taken from possibly identical populations. Two discrete models, Poisson and binomial are considered in particular. Three estimators, i.e. the unrestricted estimator, shrinkage restricted estimator and estimators based on preliminary test are proposed. Their asymptotic mean squared errors are derived and compared. It is demonstrated via asymptotic results that the range of the parameter space in which shrinkage preliminary test estimator dominates the unrestricted estimator is wider than that of the usual preliminary test estimator. A Monte Carlo study for Poisson model is presented to compare the performance of the estimators for small samples.     

Estimation of structural equation models with exact and stochastic prior information

Recently, structural equation models are widely used in assessing data in economical and behavioral researches. To give more freedom in defining the structures of the model and obtain more precise and meaningful interpretations to the data, prior informations about the unknown parameters are usually incorporated in the analysis. In this article, basic estimation theory of structural equation models with both exact and stochastic prior informations is developed via the generalized least squares approach. Asymptotic properties of the estimator are derived and an iterative algorithm is implemented to obtain the estimates. An illustrative example is reported.     

Behaviour of the posterior error rate with partial prior information in auditing

Most of the Bayesian literature on statistical techniques in auditing has focused on assessing appropriate prior density using parameters such as interest, error rate and the mean of the error amount. Frequently, prior beliefs and mathematical tractable reasons are jointly used to assess prior distributions. As a robust Bayesian approach, we propose to replace the prior distribution with a set of prior distributions compatible with auditor's beliefs. We show how an auditor may draw the behaviour of the posterior error rate, using only partial prior information (quartiles of the prior distribution for the error rate O and, very often, the prior distribution is assumed to be unimodal). An example is pursued in depth.