The supplementary materials and XPN Matlab code are publicly available at website: https://genome.unc.edu/xpn

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Discrete-state Markov random ®elds on regular arrays have played a signi®cant role in spatial statistics and image analysis. For example, they are used to represent objects against background in computer vision and pixel-based classi®cation of a region into different crop types in remote sensing. Convenience has generally favoured formulations that involve only pairwise interactions. Such models are in themselves unrealistic and, although they often perform surprisingly well in tasks such as the restoration of degraded images, they are unsatisfactory for many other purposes. In this paper, we consider particular forms of Markov random ®elds that involve higher-order interactions and therefore are better able to represent the large-scale properties of typical spatial scenes. Interpretations of the parameters are given and realizations from a variety of models are produced via Markov chain Monte Carlo. Potential applications are illustrated in two examples. The ®rst concerns Bayesian image analysis and con®rms that pairwise-interaction priors may perform very poorly for image functionals such as number of objects, even when restoration apparently works well. The second example describes a model for a geological dataset and obtains maximum-likelihood parameter estimates using Markov chain Monte Carlo. Despite the complexity of the formulation, realizations of the estimated model suggest that the representation is quite realistic.

Markov chain Monte Carlo algorithms generate samples from a target distribution by simulating a Markov chain. Large¯exibility exists in speci®cation of transition matrix of the chain. In practice, however, most algorithms used only allow small changes in the state vector in each iteration. This choice typically causes problems for multimodal distributions as moves between modes become rare and, in turn, results in slow convergence to the target distribution. In this paper we consider continuous distributions on R n and specify how optimization for local maxima of the target distribution can be incorporated in the speci®cation of the Markov chain. Thereby, we obtain a chain with frequent jumps between modes. We demonstrate the effectiveness of the approach in three examples. The ®rst considers a simple mixture of bivariate normal distributions, whereas the two last examples consider sampling from posterior distributions based on previously analysed data sets.

In this paper we define a hierarchical Bayesian model for microarray expression data collected from several studies and use it to identify genes that show differential expression between two conditions. Key features include shrinkage across both genes and studies; flexible modeling that allows for interactions between platforms and the estimated effect, and for both concordant and discordant differential expression across studies. We evaluated the performance of our model in a comprehensive fashion, using both artificial data, and a "split-sample" validation approach that provides an agnostic assessment of the model's behavior not only under the null hypothesis but also under a realistic alternative. The simulation results from the artificial data demonstrate the advantages of a Bayesian model. Compared to a more direct combination of t-or SAMstatistics, the 1 − AUC values for the Bayesian model is roughly half of the corresponding values for the t-and SAM-statistics. Furthermore, the simulations provide guidelines for when the Bayesian model is most likely to be useful. Most noticeably, in small studies the Bayesian model generally outperforms other methods when evaluated by AUC, FDR, and MDR across a range of simulation parameters, and this difference diminishes for larger sample sizes in the individual studies. The split-study validation illustrates appropriate shrinkage of the Bayesian model in the absence of platform-, sample-, and annotation-differences that otherwise complicate experimental data analyses. Finally, we fit our model to four breast cancer studies employing different technologies (cDNA and Affymetrix) to estimate differential expression in estrogen receptor positive tumors versus negative ones. Software and data for reproducing our analysis are publicly available.

ABSTRACT. We propose new control variates for variance reduction in estimation of mean values using the Metropolis-Hastings algorithm. Traditionally, states that are rejected in the Metropolis-Hastings algorithm are simply ignored, which intuitively seems to be a waste of information. We present a setting for construction of zero mean control variates for general target and proposal distributions and develop ideas for the standard Metropolis-Hastings and reversible jump algorithms. We give results for three simulation examples. We get best results for variates that are functions of the current state x and the proposal y, but we also consider variates that in addition are functions of the Metropolis-Hastings acceptance/rejection decision. The variance reduction achieved varies depending on the target distribution and proposal mechanisms used. In simulation experiments, we typically achieve relative variance reductions between 15% and 35%.

S U M M A R YInversion of seismic reflection coefficients is formulated in a Bayesian framework. Measured reflection coefficients and model parameters are assigned statistical distributions based on information known prior to the inversion, and together with the forward model uncertainties are propagated into the final result. This enables a quantification of the reliability of the inversion. Quadratic approximations to the Zoeppritz equations are used as the forward model. Compared with the linear approximations the bias is reduced and the uncertainty estimate is more reliable. The differences when using the quadratic approximations and the exact expressions are minor. The solution algorithm is sampling based, and because of the non-linear forward model, the Metropolis-Hastings algorithm is used. To achieve convergence it is important to keep strict control of the acceptance probability in the algorithm. Joint inversion using information from both reflected PP waves and converted PS waves yields smaller bias and reduced uncertainty compared to using only reflected PP waves.

No abstract