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?