Sum-product and belief updates

When computing the distribution of one variable (or a subset of variables), the main operation is the marginalization, which consists of summing over a variable (or a subset of variables) to eliminate it from the main expression. If we call ϕ a factor in the factorization of the joint distribution, we can use the following properties to generalize and improve the variable elimination algorithm we saw in the previous section:

  • Commutativity: Sum-product and belief updates
  • Associativity: Sum-product and belief updates
  • And if

If again we apply it to the joint distribution P(ABCD)

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