Before going into the details of computing outer probabilities we describe their use in obtaining the expected rule counts needed for the E-step in grammar estimation.
Let 
denote the expected number of uses of
production 
in the derivation of string x.
Alternatively, is the expected
number of times that
is used for prediction in a complete Earley path generating x.
Let 
be the number of occurrences of predicted states
based on production along a path 
.
The last summation is over all predicted states based on production
.
The quantity 
is the sum of the probabilities of all paths
passing through 
.
Inner and outer probabilities have been defined such that this
quantity is obtained precisely as the
product of the corresponding of 
and 
.
Thus, the expected usage count for a rule can be computed as
The sum can be computed after completing both forward and backward passes (or during the backward pass itself) by scanning the chart for predicted states.