  We leverage the Minimum Description Length (MDL) principle as a model
selection technique for Bernoulli distributions and compare several types of
MDL codes. We first present a simplistic crude two-part MDL code and a
Normalized Maximum Likelihood (NML) code. We then focus on the enumerative
two-part crude MDL code, suggest a Bayesian interpretation for finite size data
samples, and exhibit a strong connection with the NML approach. We obtain
surprising impacts on the estimation of the model complexity together with
superior compression performance. This is then generalized to the case of the
multinomial distributions. Both the theoretical analysis and the experimental
comparisons suggest that one might use the enumerative code rather than NML in
practice, for Bernoulli and multinomial distributions.
