A graphical model 𝒢 is a tuple 𝒢 = ⟨𝐗, 𝐃, 𝐒, 𝐅, 𝐂⟩ where:

• 𝐗 = {𝑋₁, ..., 𝑋_𝑛} set of ordered variables
• 𝐃 = {𝐷₁, ..., 𝐷_𝑛} set of corresponding domains of each variable 𝑋_𝑖 (e.g. if 𝑋₁ is a boolean variable then 𝐷₁ = {true, false}). The size of each 𝐷_𝑖 corresponds to the cardinality of variable 𝑋_𝑖
• 𝐒 = {𝑆₁, ..., 𝑆_𝑚} set of variable scopes, where each variable scope 𝑆_𝑖 is a subset of 𝐗 (i.e. 𝑆_𝑖 ⊆ 𝐗)
• 𝐅 = {𝐹₁, ..., 𝐹_𝑚} set of factors/functions, where each factor/function 𝐹_𝑖 is defined over its corresponding variable scope 𝑆_𝑖 and maps any assignment over its scope to a real value
  • in context of Bayesian Networks, 𝐅 = set of conditional probability tables
  • in context of Markov Networks, 𝐅 = set of potential functions / factors

  • global function - is a function whose scope includes all variables (i.e. 𝑆_𝑖 = 𝐗)

  • local functions - is a function whose scope is a proper subset of variables (i.e. 𝑆_𝑖 ⊂ 𝐗)

• 𝐂 is a set of combination operators which defines how functions are combined. common combination operators are:

  • summation operator (𝛴)

  • multiplication operator (𝛱)

  • AND operator (∧) - for Boolean functions

  • relational join operator (⨝) - when the functions are relation

  • marginalization operator - for reasoning queries

  • max operator - e.g. = argmax_𝑦 [ 𝐹_𝑖(𝑥,𝑦) ] = 𝐹_𝑗(𝑥) where 𝐹_𝑗 is a new function with scope over variable 𝑥

the set of local functions can be combined in a variety of ways (e.g. combination operators) to generate a new local function or even a global function

and let:

• 𝐐 = 𝑄₁, ..., 𝑄_𝑟 be the set of query variables
• 𝐄 = 𝐸₁, ..., 𝐸_𝑠 be the set of evidence variables
• 𝐇 = 𝐻₁, ..., 𝐻_𝑡 be the remainder of the variables (called the hidden variables (non-evidence, non-query variables))
• 𝐪 = 𝑞₁, ..., 𝑞_𝑟 be a grounded instantiation of 𝐐
• 𝐞 = 𝑒₁, ..., 𝑒_𝑠 be a grounded instantiation of 𝐄

• 𝐡 = 𝘩₁, ..., 𝘩_𝑡 be a grounded instantiation of 𝐇

𝑋, 𝐸, and 𝑌 are disjoint exhaustive sets of all variables 𝐗

therefore:

• the complete set of variables is 𝐗 = 𝐐 ∪ 𝐄 ∪ 𝐇 = {𝑋₁, ..., 𝑋_𝑛} = {𝑄₁, ..., 𝑄_𝑟, 𝐸₁, ..., 𝐸_𝑠, 𝐻₁, ..., 𝐻_𝑡}
• the complete set of instantiations is 𝒙 = 𝐪 ∪ 𝐞 ∪ 𝐡 = {𝑥₁, ..., 𝑥_𝑛} = {𝑞₁, ..., 𝑞_𝑟, 𝑒₁, ..., 𝑒_𝑠, ℎ₁, ..., ℎ_𝑡}

• product comes from a joint probability 𝐏(𝐐=𝐪, 𝐇=𝐡, 𝐄=𝐞) being factorized into a product of conditional probabilities
• when we take the log of products it becomes sum of factors
  

more details: Probabilistic Inference - Query/Task Types (Posterior Conditional - Prior Marginal / Probability of Evidence - MPE - MAP)

• Daphne Koller's Video Lectures