Conditioning can make imprecise probabilities uniformly more imprecise. We call this effect "dilation". In a previous paper (1993), Seidenfeld and Wasserman established some basic results about dilation. In this paper we further investigate dilation on several models. In particular, we consider conditions under which dilation persists under marginalization and we quantify the degree of dilation. We also show that dilation manifests itself asymptotically in certain robust Bayesian models and we characterize the rate at which dilation occurs.

We report two issues concerning diverging sets of Bayesian (conditional) probabilities-divergence of "posteriors"-that can result with increasing evidence. Consider a set P of probabilities typically, but not always, based on a set of Bayesian "priors." Fix E, an event of interest, and X, a random variable to be observed. With respect to P, when the set of conditional probabilities for E, given X, strictly contains the set of unconditional probabilities for E, for each possible outcome X = x, call this (...) phenomenon dilation of the set of probabilities (Seidenfeld and Wasserman 1993). Thus, dilation contrasts with the asymptotic merging of posterior probabilities reported by Savage (1954) and by Blackwell and Dubins (1962). (1) In a wide variety of models for Robust Bayesian inference the extent to which X dilates E is related to a model specific index of how far key elements of P are from a distribution that makes X and E independent. (2) At a fixed confidence level, (1-α), Classical interval estimates A n for, e.g., a Normal mean θ have length O(n -1/2 ) (for sample size n). Of course, the confidence level correctly reports the (prior) probability that θ ∈ A n ,P(A n )=1-α , independent of the prior for θ . However, as shown by Pericchi and Walley (1991), if an ε -contamination class is used for the prior on the parameter θ , there is asymptotic (posterior) dilation for the A n , given the data. If, however, the intervals A ′ n are chosen with length $O(\sqrt{\log (\text{n})/\text{n})}$ , then there is no asymptotic dilation. We discuss the asymptotic rates of dilation for ClassClassical and Bayesian interval estimates and relate these to Bayesian hypothesis testing. (shrink)

James M. Robins, Richard Scheines, Peter Spirtes, and Larry Wasserman. The Limits of Causal Knowledge.

DNA microarrays are perfectly suited for comparing gene expression in different populations of cells. An important application of microarray techniques is identifying genes which are activated by a particular drug of interest. This process will allow biologists to identify therapies targeted to particular diseases, and, eventually, to gain more knowledge about the biological processes in organisms. Such an application is described in this paper. It is focused on diabetes and obesity, which is a genetically heterogeneous disease, meaning that multiple defective (...) genes are responsible for the diseases. The paper is divided in three parts, each dealing with a different problem addressed to our study. First we validate the data from our microarray experiment. We identified significant systematic sources of variability which are potentially issues for other microarray datasets. Second, we applied multiple hypothesis testing to identify differentially expressed genes. We found a set of genes which appear to change in expression level over time in response to a drug treatment. Third, we tried to address the problem of identification of co-expressed genes using cluster analysis. This last problem is still under discussion. (shrink)

We present evidence of a potentially serious source of error intrinsic to all spotted cDNA microarrays that use IMAGE clones of expressed sequence tags (ESTs). We found that a high proportion of these EST sequences contain 5V-end poly(dT) sequences that are remnants from the oligo(dT)-primed reverse transcription of polyadenylated mRNA templates used to generate EST cDNA for sequence clone libraries. Analysis of expression data from two single-dye cDNA microarray experiments showed that ESTs whose sequences contain repeats of consecutive 5V- end (...) dT residues appeared to be strongly coexpressed, while expression data of all other sequences exhibited no such pattern. Our analysis suggests that expression data from sequences containing 5V poly(dT) tracts are more likely to be due to systematic cross-hybridization of these poly(dT) tracts than to true mRNA coexpression. This indicates that existing data generated by cDNA microarrays containing IMAGE clone ESTs should be filtered to remove expression data containing significant 5V poly(dT) tracts. D 2003 Elsevier Inc. All rights reserved. (shrink)