Tree-structured Ising models under mean parameterization

Abstract

In the risk modeling literature, the Ising model has emerged as a valuable framework for dependent Bernoulli random variables, as its underlying graphical structure captures complex dependence patterns. It is almost always employed in its exponential-family parameterization, which comes with some drawbacks such as the marginal distributions not being fixed and an intractable normalizing constant. We study tree-structured Ising models under their mean parameterization, particularly well suited to risk modeling applications, especially as it eliminates these drawbacks. We study tree-structured Ising models under their mean parameterization, which eliminates these drawbacks and is particularly attractive for risk modeling. The mean parameterization also facilitates the characterization of dependence structures and the distribution of aggregate risks. In particular, we derive an analytic expression for the joint probability generating function, yielding an efficient method for computing the aggregate-risk distribution. Similarly, we obtain the ordinary generating function of expected allocations allowing exact computations for risk allocation. The mean parameterization also allows for a stochastic representation of Ising models, yielding a direct sampling algorithm otherwise unavailable under the exponential-family parameterization. We show that Markov random fields with fixed Poisson marginal distributions provide an efficient and accurate approximation for tree-structured Ising models, in the spirit of Poisson approximations.