Perfect Tree-Like Markovian Distributions
This provides a theoretical foundation for understanding independence in tree-structured distributions, which is incremental but clarifies a key property in probabilistic graphical models.
The paper proves that for strictly positive binary joint distributions and multivariate normal distributions that factor according to a tree, vertex separation exactly captures all independence relations, using a new property of conditional independence.
We show that if a strictly positive joint probability distribution for a set of binary random variables factors according to a tree, then vertex separation represents all and only the independence relations enclosed in the distribution. The same result is shown to hold also for multivariate strictly positive normal distributions. Our proof uses a new property of conditional independence that holds for these two classes of probability distributions.