Journal of Biometrics & Biostatistics

This paper considers the problem of undertaking fully Bayesian inference for both the parameters and structure of a vector autoregressive model on the basis of time course data in the “p >> n scenario’’. The autoregressive matrix is assumed to be sparse, but of unknown structure. The resulting algorithm for dynamic Bayesian network inference is shown to be highly effective, and is applied to the problem of dynamic network inference from time course microarray data using a dataset concerned with the transient response of budding yeast to telomere damage. Research Article J o ur na l o f B iometrics & Bistatis t i c s ISSN: 2155-6180 Journal of Biometrics & Biostatistics Citation: Lei G, Boys RJ, Gillespie CS, Greenall A, Wilkinson DJ (2011) Bayesian Inference for Sparse VAR(1) Models, with Application to Time Course Microarray Data. J Biomet Biostat 2:127. doi:10.4172/2155-6180.1000127 Volume 2 • Issue 5 • 1000127 J Biomet Biostat ISSN:2155-6180 JBMBS, an open access journal Page 2 of 8 1 y 1 0 0 ( | , , ) = { ( | , , , )} A E A π τ π τ y y y μ μ . However, we seek a likelihood function which retains the simple structure given by the autoregressive model. Taking a marginal model which is uniform achieves this result. Other large sample arguments (T large) suggest that little information will be lost if the contribution of this marginal model is ignored, with this essentially conditioning the likelihood on the observed value of y1. In this paper, we will simplify the likelihood function by ignoring the contribution of this marginal model. Prior distribution We assume a priori independence between the parameters in the model, giving prior distribution π(A,μ,τ) = π(A)π(μ)π(τ), and where possible, the (marginal) prior distributions are chosen to yield semi-conjugate updates. The prior distribution for the autoregressive matrix A = (aij) is formed by assuming independence across its elements. Further, we assume that an edge is present from variable Yj to variable Yi (i.e aij ≠ 0) with probability p and that these non-zero elements follow a normal distribution with mean c and standard deviation d. We also take a normal N(m,q-1) prior distribution for the process mean μ which allows a priori correlation between variables and a gamma Γ(g,h) prior distribution for the precision of the error terms. This prior distribution places a binomial B(k2,p) distribution on the number of edges in the network and, as the network is believed to be sparse, we take p=1/k. In the subsequent analyses, we also take the prior for the non-zero elements of A to have mean c = 0 and standard deviation d = 0.2. We input weak prior information about the process mean by taking μ = 0 and q = 0.01Ik and for the error terms, by taking g = 1 and h = 0.01. Posterior distribution The complexity of the model requires that we adopt a Monte Carlo Markov chain (MCMC) strategy to obtaining the posterior distribution for the model parameters. A complicating feature is that variable dimension moves are required to add or delete edges within the A matrix. These moves have the effect of replacing a zero element aij with a non-zero value and vice versa. Although the MCMC scheme could be initialised by simulating a realisation of the autoregressive matrix A from its prior distribution, given the “known’’ sparsity of the network, we have chosen to initialise the scheme by taking A as the null matrix. Outline of MCMC scheme At each iteration of the MCMC algorithm, a fixed scan is performed of the following moves: a. update the autoregressive matrix A by proposing to add or delete an edge; b. update all non-zero elements of A in turn; c. update the process mean vector μ by using ( | , ) π ⋅ Y μ ; d. update the error precision τ by using ( | , ) π τ ⋅ Y . Move (a): In Move (a), an element of A is selected at random, say aij where i and j are independent discrete uniform numbers from {1,2,. ,k}. If this element is zero then an edge is proposed from variable Yj to variable Yi and a new value u proposed for aij from its full conditional distribution in the larger network, namely 2 ( , ) ij j u N C D  . Following the arguments in [8], this reversible jump move is accepted with probability min(Aa,1),