Modeling heterogeneity in random graphs through latent space models: a selective review

We present a selective review on probabilistic modeling of heterogeneity in random graphs. We focus on latent space models and more particularly on stochastic block models and their extensions that have undergone major developments in the last five years

The Dependence of Routine Bayesian Model Selection Methods on Irrelevant Alternatives

Bayesian methods - either based on Bayes Factors or BIC - are now widely used for model selection. One property that might reasonably be demanded of any model selection method is that if a model ${M}_{1}$ is preferred to a model ${M}_{0}$, when these two models are expressed as members of one model class $\mathbb{M}$, this preference is preserved when they are embedded in a different class $\mathbb{M}'$. However, we illustrate in this paper that with the usual implementation of these common Bayesian procedures this property does not hold true even approximately. We therefore contend that to use these methods it is first necessary for there to exist a "natural" embedding class. We argue that in any context like the one illustrated in our running example of Bayesian model selection of binary phylogenetic trees there is no such embedding

Robust causal structure learning with some hidden variables

We introduce a new method to estimate the Markov equivalence class of a directed acyclic graph (DAG) in the presence of hidden variables, in settings where the underlying DAG among the observed variables is sparse, and there are a few hidden variables that have a direct effect on many of the observed ones. Building on the so-called low rank plus sparse framework, we suggest a two-stage approach which first removes the effect of the hidden variables, and then estimates the Markov equivalence class of the underlying DAG under the assumption that there are no remaining hidden variables. This approach is consistent in certain high-dimensional regimes and performs favourably when compared to the state of the art, both in terms of graphical structure recovery and total causal effect estimation