1. General Gibbs Distribution
Representation
Probabilistic
Graphical
Models
Markov Networks
General Gibbs Distribution

2. P(A,B,C,D) A D B C

3. Consider a fully connected pairwise Markov network over X1,…,Xn where each Xi has d values. How many parameters does the network have?
O(dn)
O(nd)
O(n2d2)
O(nd)
Not every distribution can be represented
as a pairwise Markov network

4. Gibbs Distribution Parameters: General factors i (Di)  = {i (Di)}
0.25 c2
0.35 b2
0.08
0.16 a2
0.05
0.07 a3
0.15
0.21
0.09
0.18
Parameters:
General factors i (Di)
 = {i (Di)}

5. Gibbs Distribution

6. Induced Markov NetworkB C
Induced Markov network H has an edge Xi―Xj whenever

7. Factorization P factorizes over H if there exist such that
H is the induced graph for 

8. Which Gibbs distribution would induce the graph H?
All of the above
Graph structure doesn’t uniquely determine parameterization

9. Flow of Influence A D B C
Influence can flow along any trail, regardless of the form of the factors

10. Active Trails A trail X1 ─ … ─ Xn is active given Z if no Xi is in Z AD B C

11. Summary
Gibbs distribution represents distribution as a product of factors
Induced Markov network connects every pair of nodes that are in the same factor
Markov network structure doesn’t fully specify the factorization of P
But active trails depend only on graph structure

12. END END END