# 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**:**Associativity**:- And if

If again we apply it to the joint distribution *P(ABCD)* from the previous section, we again obtain:

In the end, the main expression that comes again and again is the sum-product over a factor, which can be written as .

So in general, if we can find a good ordering of the factors or the variables in the case of a directed graphical model as we saw until now, we can, by applying the previous sum-product formula, eliminate step by step each variable until we obtain the desired subset.

The ordering must...