Solved – What’s the difference between a Markov Random Field and a Conditional Random Field

Question

If I fix the values of the observed nodes of an MRF, does it become a CRF?

Answer 1

Ok, I found the answer myself:

Conditinal Random Fields (CRFs) are a special case of Markov Random Fields (MRFs).

1.5.4 Conditional Random Field

A Conditional Random Field (CRF) is a form of MRF that defines a posterior for variables x given data z, as with the hidden MRF above. Unlike the hidden MRF, however, the factorization into the data distribution P (x|z) and the prior P (x) is not made explicit [288]. This allows complex dependencies of x on z to be written directly in the posterior distribution, without the factorization being made explicit. (Given P (x|z), such factorizations always exist, however—infinitely many of them, in fact—so there is no suggestion that the CRF is more general than the hidden MRF, only that it may be more convenient to deal with.)

Source: Blake, Kohli and Rother: Markov random fields for vision and image processing. 2011.

A conditional random field or CRF (Lafferty et al. 2001), sometimes a discriminative random field (Kumar and Hebert 2003), is just a version of an MRF where all the clique potentials are conditioned on input features: [...]

The advantage of a CRF over an MRF is analogous to the advantage of a discriminative classifier over a generative classifier (see Section 8.6), namely, we don’t need to “waste resources” modeling things that we always observe. [...]

The disadvantage of CRFs over MRFs is that they require labeled training data, and they are slower to train[...]

Source: Kevin P. Murphy: Machine Learning: A Probabilistic Perspective

Answering my question:

If I fix the values of the observed nodes of an MRF, does it become a CRF?

Yes. Fixing the values is the same as conditioning on them. However, you should note that there are differences in training, too.

Watching many of the lectures about PGM (probabilistic graphical models) on coursera helped me a lot.