Abstract. We propose a dynamic Bayesian network and nonparametric regression model for constructing a gene network from time series microarray gene expression data. The proposed method can overcome a shortcoming of the Bayesian network model in the sense of the construction of cyclic regulations. The proposed method can analyze the microarray data as continuous data and can capture even nonlinear relations among genes. It can be expected that this model will give a deeper insight into the complicated biological systems. We also derive a new criterion for evaluating an estimated network from Bayes approach. We demonstrate the effectiveness of our method by analyzing Saccharomyces cerevisiae gene expression data. 1

We propose a statistical method for estimating a gene network based on Bayesian networks from microarray gene expression data together with biological knowledge including protein-protein interactions, protein-DNA interactions, binding site information, existing literature and so on. Unfortunately, microarray data do not contain enough information for constructing gene networks accurately in many cases. Our method adds biological knowledge to the estimation method of gene networks under a Bayesian statistical framework, and also controls the trade-off between microarray information and biological knowledge automatically. We conduct Monte Carlo simulations to show the effectiveness of the proposed method. We analyze Saccharomyces cerevisiae gene expression data as an application. 1.

We propose a new statistical method for constructing a genetic network from microarray gene expression data by using a Bayesian network. An essential point of Bayesian network construction is in the estimation of the conditional distribution of each random variable. We consider fitting nonparametric regression models with heterogeneous error variances to the microarray gene expression data to capture the nonlinear structures between genes. A problem still remains to be solved in selecting an optimal graph, which gives the best representation of the system among genes. We theoretically derive a new graph selection criterion from Bayes approach in general situations. The proposed method includes previous methods based on Bayesian networks. We demonstrate the effectiveness of the proposed method through the analysis of Saccharomyces cerevisiae gene expression data newly obtained by disrupting 100 genes. 1.

liez's theorem. Directions for future development are indicated. Some key words: Bayesian inference; Conditional inference; Robustness; Score function. 1. INTRODUCTION An exact representation for the posterior mean, E(Oly), is given where y is a 1 x n vector of observations from a location model, f(x-0), and 0 has a prior density, p(0),that is a normal scale mixture. Let L(O) denote the likelihood function and let y = (, a) where 0 is the maximum likelihood estimator and a is the maximal ancillary. The representation makes use of two results: the conditional distribution of the maximum likelihood estimator, p([O, a) (Barndorff-Nielsen, 1983), and a result of Masreliez (1975). It is shown that, under a normaLprior , E(O[y) ca.n be represented as a linear transformation of the score function of p(O]a), where p(O[a)= p([O, a)p(O) dO. The representation can be viewed as a generalization of Masreliez's result that deals with the model, X = 0 + e, 0 N(m, 2) and represents the posterior m

this paper have implicity assumed a single homogeneous sample. However, they are also applicable in multi-sample problems, in which the parameters of the model are possibly different from one sample to another. Such problems lead to what are usually called empirical Bayes methods of analysis. In recent years it has become more common to solve such problems from a fully Bayesian point of view, using a hierarchical model structure to link together the parameters of the different subsamples. This is the point of view taken, for example, in the excellent recent monograph by Carlin and Louis (1996). Despite the very rapid growth of this field, there has been comparatively little study of the frequentist properties of Bayesian procedures in this setting. Berger and Strawderman (1996) established some admissibility results, which have the advantage of not relying on any kind of asymptotics, and which provide guidance on the choice of prior particularly where improper priors are concerned. On the other hand, the class of models to which their results apply is restrictive, and admissibility results do not necessarily help to pick out a prior distribution which has good properties under particular conditions. In contrast, the results of the present paper are asymptotic (letting sample size n !1 while the number of samples remains fixed) but they do allow explicit computations to be made under a veriety of circumstances. In the present section, these ideas are worked out in some detail for the simplest problem in this class: the case of p normal distributions with unknown means and known common variance. In the next section, a more complicated example is considered. Suppose there are p subgroups and the data in the j'th subgroup follow a N(` j ; 1) distribution. Here the vector ...

this paper is based on the Laplace approximation for integrals. Consider a smooth convex function h(\Delta) of a p-dimensional parameter u with a minimum at u, where

The authors show how saddlepoint techniques lead to highly accurate approximations for Bayesian predictive densities and cumulative distribution functions in stochastic model settings where the prior is tractable, but not necessarily the likelihood or the predictand distribution. They consider more specifically models involving predictions associated with waiting times for semi-Markov processes whose distributions are indexed by an unknown parameter #. Bayesian prediction for such processes when they are not stationary is also addressed and the inverse-Gaussian based saddlepoint approximation of Wood et al. (1993) is shown to accurately deal with the nonstationarity whereas the normal-based Lugannani & Rice (1980) approximation cannot. Their methods are illustrated by predicting various waiting times associated with M/M/q and M/G/1 queues. They also discuss modifications to the matrix renewal theory needed for computing the moment generating functions that are used in the saddlepoint m...

this paper, however, shows that this is too simple a conclusion. For many models, when assessed

Selection of significant genes via expression patterns is an important problem in microarray experiments. Owing to small sample size and the large number of variables (genes), the selection process can be unstable. This paper proposes a hierarchical Bayesian model for gene (variable) selection. We employ latent variables to specialize the model to a regression setting and uses a Bayesian mixture prior to perform the variable selection. We control the size of the model by assigning a prior distribution over the dimension (number of significant genes) of the model. The posterior distributions of the parameters are not in explicit form and we need to use a combination of truncated sampling and Markov Chain Monte Carlo (MCMC) based computation techniques to simulate the parameters from the posteriors. The Bayesian model is flexible enough to identify significant genes as well as to perform future predictions. The method is applied to cancer classification via cDNA microarrays where the genes BRCA1 and BRCA2 are associated with a hereditary disposition to breast cancer, and the method is used to identify a set of significant genes. The method is also applied successfully to the leukemia data.

We present a statistical method for estimating gene networks and detecting promoter elements simultaneously. When estimating a network from gene expression data alone, a common problem is that the number of microarrays is limited compared to the number of variables in the network model, making accurate estimation a difficult task. Our method overcomes this problem by integrating the microarray gene expression data and the DNA sequence information into a Bayesian network model. The basic idea of our method is that, if a parent gene is a transcription factor, its children may share a consensus motif in their promoter regions of the DNA sequences. Our method detects consensus motifs based on the structure of the estimated network, then re-estimates the network using the result of the motif detection. We continue this iteration until the network becomes stable. To show the effectiveness of our method, we conducted Monte Carlo simulations and applied our method to Saccharomyces cerevisiae data as areal application.