Gaussian graphical models are a popular choice for estimating biological networks, such as gene co-expression and protein-protein interaction networks. The data are assumed to follow a multivariate normal distribution, and the network is constructed from the precision matrix (inverse of the covariance matrix). The frequentist network estimation literature is vast and well-studied, with well-known and widely used methods, such as graphical LASSO. However, there is still room for new studies and methods in Bayesian literature, particularly in terms of accelerating computation. In this work, we propose a computationally efficient conditional maximization algorithm for Bayesian network estimation that utilizes the graphical horseshoe prior. We derive an approximation for the posterior edge probabilities, which are used to construct a sparse network estimate. We also introduce a simple, data-dimension-dependent formula for choosing the global scale parameter in the prior, which helps reduce computation time. With simulated and real-world examples, we show that the network estimates of our method are highly similar to those of the fully Bayesian Gibbs sampler with the graphical horseshoe prior. Still, our method is 50–100 times faster and arguably easier to use. We implemented the method in the R package GHSCM, available on GitHub at https://github.com/THautamaki/GHSCM .

Gaussian graphical models (GGM) are powerful tools to examine partial correlation structures in high-dimensional omics datasets. Partial correlation networks can explain complex relationships between genes or other biological variables. Bayesian implementations of GGMs have recently received more attention. Usually, the most demanding parts of GGM implementations are: (i) hyperparameter tuning, (ii) edge selection, (iii) scalability for large datasets, and (iv) the prior choice for Bayesian GGM. To address these limitations, we introduce a novel Bayesian GGM using a hierarchical matrix-F prior with a fast implementation. We show, with extensive simulations and biological example analyses, that this prior has competitive network recovery capabilities compared to state-of-the-art approaches and good properties for recovering meaningful networks. We present a new way of tuning the shrinkage hyperparameter by constraining the condition number of the estimated precision matrix. For edge selection, we propose using approximated credible intervals (CI) whose width is controlled by the false discovery rate. An optimal CI is selected by maximizing an estimated F1-score via permutations. In addition, a specific choice of hyperparameter can make the proposed prior better suited for clustering and community detection. Our method, with a generalized expectation-maximization algorithm, computationally outperforms existing Bayesian GGM approaches that use Markov chain Monte Carlo algorithms. The method is implemented in the R package HMFGraph, found on GitHub at https://github.com/AapoKorhonen/HMFGraph. All codes to reproduce the results are found on GitHub at https://github.com/AapoKorhonen/HMFGraph-Supplementary.

MOTIVATION: Due to advances in measuring technology, many new phenotype, gene expression, and other omics time-course datasets are now commonly available. Cluster analysis may provide useful information about the structure of such data. RESULTS: In this work, we propose BELMM (Bayesian Estimation of Latent Mixture Models): a flexible framework for analysing, clustering, and modelling time-series data in a Bayesian setting. The framework is built on mixture modelling: first, the mean curves of the mixture components are assumed to follow random walk smoothing priors. Second, we choose the most plausible model and the number of mixture components using the Reversible-jump Markov chain Monte Carlo. Last, we assign the individual time series into clusters based on the similarity to the cluster-specific trend curves determined by the latent random walk processes. We demonstrate the use of fast and slow implementations of our approach on both simulated and real time-series data using widely available software R, Stan, and CU-MSDSp. AVAILABILITY AND IMPLEMENTATION: The French mortality dataset is available at http://www.mortality.org, the Drosophila melanogaster embryogenesis gene expression data at https://www.ncbi.nlm.nih.gov/geo/query/acc.cgi?acc=GSE121160. Details on our simulated datasets are available in the Supplementary Material, and R scripts and a detailed tutorial on GitHub at https://github.com/ollisa/BELMM. The software CU-MSDSp is available on GitHub at https://github.com/jtchavisIII/CU-MSDSp.