If we multiplied them together, then we could sum over E . Now we can sum over A to obtain the answer. So this is a big savings over creating the full joint distribution. The total number of additions and multiplications will be much less as well. The SPI algorithm works as follows.
Q be a list of all variables in N except the query variable Q .
These are the variables we will sum over. We then normalize the remaining cells so that the table again sums to 1. We can do something analogous with the belief network representation. When we are told that variable V has value val , we delete all cells involving all other values of of V from all of the ptables in the belief network. These tables now no longer involve the variable V .