Markov Networks
Markov Networks A B C D Undirected graphical models Potential functions defined over cliques
Markov Networks A B C D Undirected graphical models Potential functions defined over cliques Weight of Feature i Feature i
Hammersley-Clifford Theorem If Distribution is strictly positive (P(x) > 0) And Graph encodes conditional independences Then Distribution is product of potentials over cliques of graph Inverse is also true. (“Markov network = Gibbs distribution”)
Markov Nets vs. Bayes Nets Property Markov Nets Bayes Nets Form Prod. potentials Potentials Arbitrary Cond. probabilities Cycles Allowed Forbidden Partition func. Z = ? Z = 1 Indep. check Graph separation D-separation Indep. props. Some Inference MCMC, BP, etc. Convert to Markov
Inference in Markov Networks Goal: compute marginals & conditionals of Exact inference is #P-complete Conditioning on Markov blanket is easy: Gibbs sampling exploits this
Markov Chain Monte Carlo Gibbs Sampler 1. Start with an initial assignment to nodes 2. One node at a time, sample node given others 3. Repeat 4. Use samples to compute P(X) Convergence: Burn-in + Mixing time Many modes Multiple chains
Other Inference Methods Belief propagation (sum-product) Mean field / Variational approximations
MAP Inference Iterated conditional modes Simulated annealing Graph cuts Belief propagation (max-product)
Learning Weights Maximize likelihood (or posterior) Convex optimization: gradient ascent, quasi-Newton methods, etc. Requires inference at each step (slow!) Feature count according to data Feature count according to model
Pseudo-Likelihood Likelihood of each variable given its Markov blanket in the data Does not require inference at each step Very fast numerical optimization Ignores non-local dependences
Learning Structure Feature search Evaluation 1. Start with atomic features 2. Form conjunctions of pairs of features 3. Select best and add to feature set 4. Repeat until no improvement Evaluation Likelihood, K-L divergence Approximation: Previous weights don’t change