Helene Massam

Nanwei Wang, Laurent Briollais, Helene Massam

Recent advances in biological research have seen the emergence of high-throughput technologies with numerous applications that allow the study of biological mechanisms at an unprecedented depth and scale. A large amount of genomic data is now distributed through consortia like The Cancer Genome Atlas (TCGA), where specific types of biological information on specific type of tissue or cell are available. In cancer research, the challenge is now to perform integrative analyses of high-dimensional multi-omic data with the goal to better understand genomic processes that correlate with cancer outcomes, e.g. elucidate gene networks that discriminate a specific cancer subgroups (cancer sub-typing) or discovering gene networks that overlap across different cancer types (pan-cancer studies). In this paper, we propose a novel mixed graphical model approach to analyze multi-omic data of different types (continuous, discrete and count) and perform model selection by extending the Birth-Death MCMC (BDMCMC) algorithm initially proposed by \citet{stephens2000bayesian} and later developed by \cite{mohammadi2015bayesian}. We compare the performance of our method to the LASSO method and the standard BDMCMC method using simulations and find that our method is superior in terms of both computational efficiency and the accuracy of the model selection results. Finally, an application to the TCGA breast cancer data shows that integrating genomic information at different levels (mutation and expression data) leads to better subtyping of breast cancers.

Reza Mohammadi, Helene Massam, Gerard Letac

Gaussian graphical models are relevant tools to learn conditional independence structure between variables. In this class of models, Bayesian structure learning is often done by search algorithms over the graph space. The conjugate prior for the precision matrix satisfying graphical constraints is the well-known G-Wishart. With this prior, the transition probabilities in the search algorithms necessitate evaluating the ratios of the prior normalizing constants of G-Wishart. In moderate to high-dimensions, this ratio is often approximated using sampling-based methods as computationally expensive updates in the search algorithm. Calculating this ratio so far has been a major computational bottleneck. We overcome this issue by representing a search algorithm in which the ratio of normalizing constant is carried out by an explicit closed-form approximation. Using this approximation within our search algorithm yields significant improvement in the scalability of structure learning without sacrificing structure learning accuracy. We study the conditions under which the approximation is valid. We also evaluate the efficacy of our method with simulation studies. We show that the new search algorithm with our approximation outperforms state-of-the-art methods in both computational efficiency and accuracy. The implementation of our work is available in the R package BDgraph.

Helene Massam, Nanwei Wang

We consider two connected aspects of maximum likelihood estimation of the parameter for high-dimensional discrete graphical models: the existence of the maximum likelihood estimate (mle) and its computation. When the data is sparse, there are many zeros in the contingency table and the maximum likelihood estimate of the parameter may not exist. Fienberg and Rinaldo (2012) have shown that the mle does not exists iff the data vector belongs to a face of the so-called marginal cone spanned by the rows of the design matrix of the model. Identifying these faces in high-dimension is challenging. In this paper, we take a local approach : we show that one such face, albeit possibly not the smallest one, can be identified by looking at a collection of marginal graphical models generated by induced subgraphs $G_i,i=1,\ldots,k$ of $G$. This is our first contribution. Our second contribution concerns the composite maximum likelihood estimate. When the dimension of the problem is large, estimating the parameters of a given graphical model through maximum likelihood is challenging, if not impossible. The traditional approach to this problem has been local with the use of composite likelihood based on local conditional likelihoods. A more recent development is to have the components of the composite likelihood be marginal likelihoods centred around each $v$. We first show that the estimates obtained by consensus through local conditional and marginal likelihoods are identical. We then study the asymptotic properties of the composite maximum likelihood estimate when both the dimension of the model and the sample size $N$ go to infinity.

Helene Massam, Nanwei Wang

We consider discrete graphical models Markov with respect to a graph $G$ and propose two distributed marginal methods to estimate the maximum likelihood estimate of the canonical parameter of the model. Both methods are based on a relaxation of the marginal likelihood obtained by considering the density of the variables represented by a vertex $v$ of $G$ and a neighborhood. The two methods differ by the size of the neighborhood of $v$. We show that the estimates are consistent and that those obtained with the larger neighborhood have smaller asymptotic variance than the ones obtained through the smaller neighborhood.