Håkon Tjelmeland

Merging two gene-expression studies via cross-platform normalization

MOTIVATION Gene-expression microarrays are currently being applied in a variety of biomedical applications. This article considers the problem of how to merge datasets arising from different gene-expression studies of a common organism and phenotype. Of particular interest is how to merge data from different technological platforms. RESULTS The article makes two contributions to the problem. The first is a simple cross-study normalization method, which is based on linked gene/sample clustering of the given datasets. The second is the introduction and description of several general validation measures that can be used to assess and compare cross-study normalization methods. The proposed normalization method is applied to three existing breast cancer datasets, and is compared to several competing normalization methods using the proposed validation measures. AVAILABILITY The supplementary materials and XPN Matlab code are publicly available at website: https://genome.unc.edu/xpn

Markov Random Fields* with Higher-order Interactions

Discrete-state Markov random fields on regular arrays have played a signifi- cant role in spatial statistics and image analysis. For example, they are used to represent objects against background in computer vision and pixel-based classification 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 fields that involve higher-order interactions and therefore are better able to represent the large-scale properties of typical spatial scenes. Interpretations of the para- meters are given and realizations from a variety of models are produced via Markov chain Monte Carlo. Potential applications are illustrated in two examples. The first concerns Bayesian image analysis and confirms 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 represen- tation is quite realistic.

Mode Jumping Proposals in MCMC

Markov chain Monte Carlo algorithms generate samples from a target distribution by simulating a Markov chain. Large flexibility exists in specification 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 multi‐modal 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 Rn and specify how optimization for local maxima of the target distribution can be incorporated in the specification 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 first considers a simple mixture of bivariate normal distributions, whereas the two last examples consider sampling from posterior distributions based on previously analysed data sets.

Bayesian CART: Prior Specification and Posterior Simulation

We present advances in Bayesian modeling and computation for CART (classification and regression tree) models. The modeling innovations include a formal prior distributional structure for tree generation—the pinball prior—that allows for the combination of an explicit specification of a distribution for both the tree size and the tree shape. The core computational innovations involve a novel Metropolis–Hastings method that can dramatically improve the convergence and mixing properties of MCMC methods of Bayesian CART analysis. Earlier MCMC methods have simulated Bayesian CART models using very local MCMC moves, proposing only small changes to a “current” CART model. Our new Metropolis–Hastings move makes large changes in the CART tree, but is at the same time local in that it leaves unchanged the partition of observations into terminal nodes. We evaluate the effectiveness of the proposed algorithm in two examples, one with a constructed data set and one concerning analysis of a published breast cancer dataset.

A Bayesian model for cross-study differential gene expression.

In this article 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, and flexible modeling that allows for interactions between platforms and the estimated effect, as well as concordant and discordant differential expression across studies. We evaluate the performance of our model in a comprehensive fashion, using both artificial data, and a “split-study” validation approach that provides an agnostic assessment of the model’s behavior under both the null hypothesis and a realistic alternative. The simulation results from the artificial data demonstrate the advantages of the Bayesian model. Furthermore, the simulations provide guidelines for when the Bayesian model is most likely to be useful. Most notably, in small studies the Bayesian model generally outperforms other methods when evaluated based on several performance measures across a range of simulation parameters, with the differences diminishing 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 using different technologies (cDNA and Affymetrix) to estimate differential expression in estrogen receptor– positive tumors versus estrogen receptor–negative tumors. Software and data for reproducing our analysis are available publicly.

Construction of Binary Multi-grid Markov Random Field Prior Models from Training Images

Bayesian modeling requires the specification of prior and likelihood models. In reservoir characterization, it is common practice to estimate the prior from a training image. This paper considers a multi-grid approach for the construction of prior models for binary variables. On each grid level we adopt a Markov random field (MRF) conditioned on values in previous levels. Parameter estimation in MRFs is complicated by a computationally intractable normalizing constant. To cope with this problem, we generate a partially ordered Markov model (POMM) approximation to the MRF and use this in the model fitting procedure. Approximate unconditional simulation from the fitted model can easily be done by again adopting the POMM approximation to the fitted MRF. Approximate conditional simulation, for a given and easy to compute likelihood function, can also be performed either by the Metropolis–Hastings algorithm based on an approximation to the fitted MRF or by constructing a new POMM approximation to this approximate conditional distribution. The proposed methods are illustrated using three frequently used binary training images.

Bayesian modelling of spatial compositional data

Compositional data are vectors of proportions, specifying fractions of a whole. Aitchison (1986) defines logistic normal distributions for compositional data by applying a logistic transformation and assuming the transformed data to be multi- normal distributed. In this paper we generalize this idea to spatially varying logistic data and thereby define logistic Gaussian fields. We consider the model in a Bayesian framework and discuss appropriate prior distributions. We consider both complete observations and observations of subcompositions or individual proportions, and discuss the resulting posterior distributions. In general, the posterior cannot be analytically handled, but the Gaussian base of the model allows us to define efficient Markov chain Monte Carlo algorithms. We use the model to analyse a data set of sediments in an Arctic lake. These data have previously been considered, but then without taking the spatial aspect into account.

Exact and Approximate Recursive Calculations for Binary Markov Random Fields Defined on Graphs

In this article, we propose computationally feasible approximations to binary Markov random fields (MRFs). The basis of the approximation is the forward-backward algorithm. The exact forward-backward algorithm is computationally feasible only for fields defined on small lattices. The forward part of the algorithm computes a series of joint marginal distributions by summing out each variable in turn. We represent these joint marginal distributions by interaction parameters of different orders. The approximation is defined by approximating to zero all interaction parameters that are sufficiently close to zero. In addition, an interaction parameter is approximated to zero whenever all associated lower-level interactions are (approximated to) zero. If sufficiently many interaction parameters are set to zero, the algorithm is computationally feasible both in terms of computation time and memory requirements. The resulting approximate forward part of the forward-backward algorithm defines an approximation to the intractable normalizing constant, and the corresponding backward part of the algorithm defines a computationally feasible approximation to the MRF. We present numerical examples demonstrating the quality of the approximation. The supplementary materials for this article, which are available online, include two appendices and R and C codes for the proposed recursive algorithms.

On directional Metropolis---Hastings algorithms

New Metropolis–Hastings algorithms using directional updates are introduced in this paper. Each iteration of a directional Metropolis–Hastings algorithm consists of three steps (i) generate a line by sampling an auxiliary variable, (ii) propose a new state along the line, and (iii) accept/reject according to the Metropolis–Hastings acceptance probability. We consider two classes of directional updates. The first uses a point in