／var／logmarcus chiu

PGM - Gibbs Distribution - Bayesian Network

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BN is Gibbs Distribution with 𝙕 = 1

parameters of BN can be viewed as parameters for a Gibbs Distribution

  • take each CPT 𝐏(𝑋𝑖|π‘π‘Žπ‘Ÿπ‘’π‘›π‘‘π‘ -π‘œπ‘“(𝑋𝑖)) and view it as a factor of scope {𝑋𝑖, π‘π‘Žπ‘Ÿπ‘’π‘›π‘‘π‘ -π‘œπ‘“(𝑋𝑖)}
  • its partition function 𝙕 is 1, since it is already normalized:
    𝙕 = βˆ‘π‘₯1βˆŠπ‘‹1… βˆ‘π‘₯π‘›βˆŠπ‘‹π‘›[ βˆπ‘‹π‘–βˆŠπ— [ 𝐏(𝑋𝑖|π‘π‘Žπ‘Ÿπ‘’π‘›π‘‘π‘ -π‘œπ‘“(𝑋𝑖)) ] ]
    𝙕 = 1
BN with evidence 𝐞 is Gibbs Distribution with 𝙕 = 𝐏(𝐄=𝐞)
  • take each CPT 𝐏(𝑋𝑖|π‘π‘Žπ‘Ÿπ‘’π‘›π‘‘π‘ -π‘œπ‘“(𝑋𝑖)) and view it as a factor of scope {𝑋𝑖, π‘π‘Žπ‘Ÿπ‘’π‘›π‘‘π‘ -π‘œπ‘“(𝑋𝑖)}
  • its partition function 𝙕 is 𝐏(𝐄=𝐞):
    𝐏(𝑋𝑖|𝐄=𝐞) = 𝐏(𝑋𝑖, 𝐄=𝐞) / 𝐏(𝐄=𝐞)

thus any BN conditioned on evidence 𝐞 can be represented as a Markov Network