The notion of multivariate total positivity has proved to be useful in finance
and psychology but may be too restrictive in other applications. In this
paper, we propose a concept of local association, where highly connected
components in a graphical model are positively associated and study its properties.
Our main motivation comes from gene expression data, where graphical
models have become a popular exploratory tool. The models are instances
of what we term mixed convex exponential families and we show that a mixed
dual likelihood estimator has simple exact properties for such families as
well as asymptotic properties similar to the maximum likelihood estimator.
We further relax the positivity assumption by penalizing negative partial correlations
in what we term the positive graphical lasso. Finally, we develop
a GOLAZO algorithm based on block-coordinate descent that applies to a
number of optimization procedures that arise in the context of graphical models,
including the estimation problems described above. We derive results on
existence of the optimum for such problems.