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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 tex2html_wrap_inline45703, we obtain the same equation as for linear discounting.  For the discounting parameter tex2html_wrap_inline45761, 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 tex2html_wrap_inline45761. 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 tex2html_wrap_inline46395; we have the lower bound:
Using the inequality
this can be simplified to the more convenient bound:


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