  We present a method for simultaneous Bayesian learning of the correlation
matrix and graphical model of a multivariate dataset, along with uncertainties
in each, to subsequently compute distance between the learnt graphical models
of a pair of datasets, using a new metric that approximates an
uncertainty-normalised Hellinger distance between the posterior probabilities
of the graphical models given the respective dataset; correlation between the
pair of datasets is then computed as a corresponding affinity measure. We
achieve a closed-form likelihood of the between-columns correlation matrix by
marginalising over the between-row matrices. This between-columns correlation
is updated first, given the data, and the graph is then updated, given the
partial correlation matrix that is computed given the updated correlation,
allowing for learning of the 95$\%$ Highest Probability Density credible
regions of the correlation matrix and graphical model of the data. Difference
made to the learnt graphical model, by acknowledgement of measurement noise, is
demonstrated on a small simulated dataset, while the large human
disease-symptom network--with $>8,000$ nodes--is learnt using real data. Data
on vino-chemical attributes of Portuguese red and white wine samples are
employed to learn with-uncertainty graphical model of each dataset, and
subsequently, the distance between these learnt graphical models.
