Mixture
The Bayesian information criterion with an additional penalty for having selected the component families from a catalog.
Searching a catalog of families for every component and then reporting a criterion computed on the winner is optimistic, in the direction that inflates the number of components. The penalty added here is inspired by the extended information criterion for large model spaces, inserting the size of the family search space into a general model-space term.
This formula is an analogue and not a result: the extended criterion is not established for mixtures of heterogeneous families. The tuning constant must be calibrated against held-out likelihood rather than assumed.
Parameters
the number of candidate families per component
the tuning constant, within the unit interval; zero recovers the ordinary criterion
Functions
Computes the criterion from a log-likelihood already known to be finite.
A fresh instance, for use where a criterion must not be shared.