  The Akaike information criterion (AIC) is a common tool for model selection.
It is frequently used in violation of regularity conditions at parameter space
singularities and boundaries. The expected AIC is generally not asymptotically
equivalent to its target at singularities and boundaries, and convergence to
the target at nearby parameter points may be slow. We develop a generalized AIC
for candidate models with or without singularities and boundaries. We show that
the expectation of this generalized form converges everywhere in the parameter
space, and its convergence can be faster than that of the AIC. We illustrate
the generalized AIC on example models from phylogenomics, showing that it can
outperform the AIC and gives rise to an interpolated effective number of model
parameters, which can differ substantially from the number of parameters near
singularities and boundaries. We outline methods for estimating the often
unknown generating parameter and bias correction term of the generalized AIC.
