  Bayesian methodologies prioritising accurate associations above sparsity in
Gaussian graphical model (GGM) estimation remain relatively scarce in
scientific literature. It is well accepted that the $\ell_2$ penalty enjoys a
smaller computational footprint in GGM estimation, whilst the $\ell_1$ penalty
encourages sparsity in the estimand. The Bayesian adaptive graphical lasso
prior is used as a departure point in the formulation of a computationally
efficient graphical ridge-type prior for events where accurate associations are
prioritised over sparse representations. A novel block Gibbs sampler for
simulating precision matrices is constructed using a ridge-type penalisation.
The Bayesian graphical ridge-type prior is extended to a Bayesian adaptive
graphical ridge-type prior. Synthetic experiments indicate that the graphical
ridge-type estimators enjoy computational efficiency, in moderate dimensions,
and numerical performance, for relatively non-sparse precision matrices, when
compared to their lasso counterparts. The adaptive graphical ridge-type
estimator is applied to cell signaling data to infer key associations between
phosphorylated proteins in human T cell signalling. All computational workloads
are carried out using the baygel R package.
