1. Markov Networks

2. Markov Networks A B C D Undirected graphical models
Potential functions defined over cliques

3. Markov Networks A B C D Undirected graphical models
Potential functions defined over cliques
Weight of Feature i
Feature i

4. 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”)

5. 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

6. Inference in Markov Networks
Goal: compute marginals & conditionals of
Exact inference is #P-complete
Conditioning on Markov blanket is easy:
Gibbs sampling exploits this

7. 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

8. Other Inference Methods
Belief propagation (sum-product)
Mean field / Variational approximations

9. MAP Inference Iterated conditional modes Simulated annealing
Graph cuts
Belief propagation (max-product)

10. 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

11. 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

12. 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