next up previous contents index
Next: Physical characterisation and description Up: Final note: the mathematics Previous: Linear interpolation

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:


equation10509

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:


eqnarray10525

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:
eqnarray9178
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:
eqnarray9187
Lower bounds are obtained by observing the inequalities
eqnarray10548
Using the additional inequality tex2html_wrap_inline46395; we have the lower bound:
eqnarray10556
Using the inequality
eqnarray10564
this can be simplified to the more convenient bound:
eqnarray9191

       



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