I have a question about consistency in graphical models. It is often stated that when running the junction tree algorithm on a clique (cluster) tree, the marginals of all nodes are locally and globally "consistent", ie, "equal".
Imagine one clique with variables xa, xb and xc, and another with variables xc, xd and xe. The shared variable is xc. The marginals after taking out the shared variables, the way I see it, should be p(xa,xb) and p(xd,xe). These marginals do not look equal to me.
What is the explanation of this concept of consistency?
If needed, more detailed notes can be read as a refresher here:
http://www.cs.princeton.edu/courses/archive/spr09/cos513/scribe/lecture07.pdf
Your assumption is wrong. Imagine one clique with variables $x_a$, $x_b$ and $x_c$, and another with variables $x_c$, $x_d$ and $x_e$. The marginals (the one discussed on the local consistency) are the followings, not the ones you mentioned. $$ \int p(x_a, x_b, x_c) dx_a fx_b $$
$$ \int p(x_c, x_d, x_e) dx_d fx_e $$