Joint Distribution Not Obvious Algebraically for Graphical Models

Question

I am having some trouble discerning joint distributions. The joint distribution below is given for $p(x_1,x_2,x_3,x_4)$ however by following the algebra, applying Bayes rule etc., I cannot reach the joint distribution.

Or the Markov chain here

In such cases (or all) do we always need to know what the graph looks like, otherwise how can we construct the joint distribution? Or can we build the graph from individual probabilities, and looking at that, we can apply the algebra (factoring in the conditional independence), allowing us to reach the joint distribution this way?

Thanks,

Answer 1

If you use the Bayes rule, you marginalize your joint density to some conditional densities. This is the most general picture that you can have. In case you have additional information, about the dependence/independence structure of random variables you can make further simplifications.

At this point a graphical model dictates such dependencies and makes the problem specific.