Degrees of freedom

In an earlier post questioning whether we still need standard tables, we used the AIC to choose between models.  For the AIC we assign a "cost" to the inclusion of extra parameters, and I had counted each mortality rate actually used in the model-fitting as a parameter.  However, there is no set approach for this, and there are a number of arguable positions which could be taken:

1. Count each mortality rate in the table as a parameter.  This would arise because the selection of the table is supposed to be done before fitting any model.  However, it seems harsh to count rates which are not actually used.
2. Count each mortality rate actually used as a parameter.  This is my preferred approach, but the rates themselves are not parameters in the same way as the other parts of the model.
3. Count the number of parameters used in the graduation of the standard table.  However, this presupposes that this can be established, and will not be available for every table.
4. Do not count the standard table as having any parameters at all.  Here we are viewing the standard table as a fundamental part of the model by providing a structure for the progression with age.  In this scenario, the standard table is operating much as the link function in a GLM, or the choice of mortality law in a survival model.

This question is an example of  a wider problem in statistics: assuming a model is fixed, when in fact it is selected from a (possibly large) class of alternative models. Fortunately, in our example the choice does not make any difference.  Even taking the most favourable position of not counting the standard table as contributing any parameters at all yields an AIC of 3099.  The simpler, table-free model has a better AIC of 3097, meaning the standard table has worsened the model fit regardless of which of the four views above is taken.