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PGM - Gibbs Distribution

PGM - Gibbs Distribution

Dec 25, 2023

Related: Boltzmann/Gibbs Distribution

Probabilistic Graphical Models & Gibbs Distribution

A probability distribution 𝐏𝐅(𝐗) is a Gibbs Distribution over a graphical model 𝒢 = ⟨𝐗, 𝐃, 𝐒, 𝐅, 𝐂⟩ if it can be written as

  • 𝐏𝐅(𝐗) = (1/𝘡) * 𝛱1≤𝑖≤𝑚 [ 𝐹𝑖(𝑆𝑖) ]

where:

  • the set of variables (𝑆𝑖) in each factor 𝐹𝑖 form a clique in the primal graph of 𝒢
  •  graphical model syntax

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Theorem 1: Factorization Implies Conditional Independencies

 Click here to expand...

If 𝐏𝐅(𝐗) is a Gibbs Distribution for 𝒢, then the primal graph of 𝒢 is an I-Map for probability distribution 𝐏(𝐗):

  • 𝐈(𝐆) ⊆ 𝐈(𝐏)

proof, suppose:

  • 𝐴, 𝐵, and 𝐶 are disjoint sets of variables
  • 𝐴 is connected to 𝐵
  • 𝐶 is connected to 𝐵
  • 𝐵 separates 𝐴 from 𝐶

then we can write:

  • 𝐏(𝐴, 𝐵, 𝐶) = (1/𝘡) * 𝐹1(𝐴,𝐵) * 𝐹2(𝐵,𝐶)

Theorem 2: Conditional Independencies Implies Factorization

 Click here to expand...

If 𝐏(𝐗) is a positive distribution and the primal graph of 𝒢 is an I-Map for 𝐏(𝐗), then 𝐏(𝐗) is a Gibbs Distribution that factorizes over graphical model 𝒢

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