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## Absolute discounting and backing-off

The basic idea is to subtract a constant value from the counts of the observed events. Thus we define the model for absolute discounting and backing-off:

Using the same manipulations as for linear discounting , i.e. separating the singletons, ordering and carrying out the sums, we have for the leaving-one-out  log-likelihood function:

For the backing-off distribution , we obtain the same equation as for linear discounting.  For the discounting parameter , we obtain the following equation after separating the term with r=2:

For this equation, there is no closed-form solution. We will derive an upper and a lower bound for . Obviously, the right hand side is always greater than or equal to zero, and thus we get the upper bound:

Lower bounds are obtained by observing the inequalities

Using the additional inequality ; we have the lower bound:

Using the inequality

this can be simplified to the more convenient bound:

EAGLES SWLG SoftEdition, May 1997. Get the book...