UCL PG Lecture 8

Joint Distribution that uses a graph structure for representat to encode conditional independence assumptions between random variables

The joint distribution in a PGM is the product of local factors, defined by the graph structure, that capture dependencies among the variables.

Describe how powerful PGMs coupled with marginals and conditionals computation really are, like you would to an ignorant friend.

PGMs model random variables as nodes, conditioning each variable only on the ones it directly depends on, as defined by the graph structure. This makes it straightforward to compute the conditional mean and variance in Multivariate Gaussian distributions.
If you assume the joint distribution of the big vector being Gaussian means that all the components of that vector are Gaussian. All the conditionals are still Gaussian and only thing you need to do is compute the conditional mean and variance which is really powerful. After that, you can uniquely determine the distribution correctly.
You can calculate the probability of anything as long as you know the conditional relation.