Result · D-REPARAM-INVARIANCE
D partitions survive reparameterization
Derived from the literaturepublication: derived from published
Meaning
An invertible change of parameter coordinates preserves the D-optimal partition.
Scientific context
The determinant criterion is invariant under invertible reparameterization: changing the parameter basis changes the objective by the same constant for every rule, so the ranking of rules is untouched. This is a known property rather than a contribution, but it is a practical reason to prefer the determinant criterion over alternatives that lack it.
Statement
Under an invertible local parameter transformation, I_q transforms by congruence and logdet changes only by a quantizer-independent constant.
Relationships
Rests on
Enables
- Canonical score parameterization for linear mixturesOPEN-MIXTURE-PARAMETERIZATION
- Frozen-rule vector retention: plug-in CLT for the geometric-mean D-retention with a consistent influence-function varianceRETENTION-PLUGIN-CLT-FROZEN-VECTOR
- Reported versus true retention of a frozen label map under a score error: an exact two-term expansion in the Fisher-whitened score errorSCORE-ERROR-RETENTION-BUDGET
Raises
Open next door
Local graph
Proof
U6. Reparameterization invariance of D
Under an invertible local parameter transformation, transforms by congruence and changes only by a quantizer-independent constant. D-optimal partitions are therefore invariant under invertible reparameterization.