Inference of genetic regulatory networks with unknown covariance structure

The major challenge in reverse-engineering genetic regulatory networks is the small number of (time) measurements or experiments compared to the number of genes, which makes the system under-determined and hence unidentifiable. The only way to overcome the identifiability problem is to incorporate prior knowledge about the system. It is often assumed that genetic networks are sparse. In addition, if the measurements, in each experiment, present an unknown correlation structure, then the estimation problem becomes even more challenging. Estimating the covariance structure will improve the estimation of the network connectivity but will also make the estimation of the already under-determined problem even more challenging. In this paper, we formulate reverse-engineering genetic networks as a multiple linear regression problem. We show that, if the number of experiments is smaller than the number of genes and if the measurements present an unknown covariance structure, then the likelihood function diverges, making the maximum likelihood estimator senseless. We subsequently propose a normalized likelihood function that guarantees convergence while keeping the form of the Gaussian distribution. The optimal connectivity matrix is approximated as the solution of a convex optimization problem. Our simulation results show that the proposed maximum normalized-likelihood estimator outperforms the classical regularized maximum likelihood estimator, which assumes a known covariance structure.