Identifiability of directed Gaussian graphical models with one latent source

We study parameter identifiability of directed Gaussian graphical models with one latent variable. In the scenario we consider, the latent variable is a confounder that forms a source node of the graph and is a parent to all other nodes, which correspond to the observed variables. We give a graphical condition that is sufficient for the Jacobian matrix of the parametrization map to be full rank, which entails that the parametrization is generically finite-to-one, a fact that is sometimes also referred to as local identifiability. We also derive a graphical condition that is necessary for such identifiability. Finally, we give a condition under which generic parameter identifiability can be determined from identifiability of a model associated with a subgraph. The power of these criteria is assessed via an exhaustive algebraic computational study on models with 4, 5, and 6 observable variables.

Testing independence in high dimensions with sums of rank correlations

We treat the problem of testing independence between m continuous variables when m can be larger than the available sample size n. We consider three types of test statistics that are constructed as sums or sums of squares of pairwise rank correlations. In the asymptotic regime where both m and n tend to infinity, a martingale central limit theorem is applied to show that the null distributions of these statistics converge to Gaussian limits, which are valid with no specific distributional or moment assumptions on the data. Using the framework of U-statistics, our result covers a variety of rank correlations including Kendalls tau and a dominating term of Spearmans rank correlation coefficient (rho), but also degenerate U-statistics such as Hoeffdings $D$, or the $tau^*$ of Bergsma and Dassios (2014). As in the classical theory for U-statistics, the test statistics need to be scaled differently when the rank correlations used to construct them are degenerate U-statistics. The power of the considered tests is explored in rate-optimality theory under Gaussian equicorrelation alternatives as well as in numerical experiments for specific cases of more general alternatives.

Order-invariant prior specification in Bayesian factor analysis

In (exploratory) factor analysis, the loading matrix is identified only up to orthogonal rotation. For identifiability, one thus often takes the loading matrix to be lower triangular with positive diagonal entries. In Bayesian inference, a standard practice is then to specify a prior under which the loadings are independent, the off-diagonal loadings are normally distributed, and the diagonal loadings follow a truncated normal distribution. This prior specification, however, depends in an important way on how the variables and associated rows of the loading matrix are ordered. We show how a minor modification of the approach allows one to compute with the identifiable lower triangular loading matrix but maintain invariance properties under reordering of the variables.