On MRF factorization

Recently, I figured out I didn't understand two simple things about MRFs:

1. When we are talking about MRF factorization, in the formulation of the likelihood

   
   should we consider all the cliques in the network or only maximal ones?
2. Generally, why do we have a right to factorize the likelihood this way?

I asked some folks, and it turned out that nobody understood this basic property clearly (In spite of Dmitry Vetrov assured me he had explained that fact in his courses). Finally, I've found the answers in Christopher Bishop's book. If you are not familiar with Markov Random Fields or graphical models, I recommend you to read the book, it is relatively understandable.

So, let's start with the second question. For those of you who like names, it is exactly the necessity part of the Hammersley-Clifford theorem. As Bishop put it:

If we consider two nodes x_i and x_j that are not connected by a link, then these variables must be conditionally independent given all other nodes in the graph. This follows from the fact that there is no direct path between the two nodes, and all other paths pass through nodes that are observed, and hence those paths are blocked. This conditional independence property can be expressed as

where x_\{i,j} denotes the set x of all variables with x_i and x_j removed. The factorization of the joint distribution must therefore be such that x_i and x_j do not appear in the same factor in order for the conditional independence property to hold for all possible distributions belonging to the graph.

One could find the answer to the first question in Bishop's book as well. Normally, we should include all the cliques to the product. Actually, every function of variables X is also a function of any Y that is a superset of X. Based on that, a potential for a non-maximal clique could be merged into a potential for any maximal clique such that it is a supergraph of the first clique! Thus, both formulations (with all cliques and only maximal cliques) are equally expressive.

camouflage

there is a paint color by ben moore that i am totally in love with called camouflage (2143-40). it is a beautiful soft green without screaming "green". i used it on the exterior of a beach house along with the accent color, alexandria beige (HC-77). these images remind me of how pretty this color is.

*images courtesy of mod girl, martha stewart living, house and home, southern accents