  We provide two results concerning the optimality of the maximum mutual
information (MMI) decoder. First, we prove that the error exponents of the
typical random codes under the optimal maximum likelihood (ML) decoder and the
MMI decoder are equal. As a corollary to this result, we also show that the
error exponents of the expurgated codes under the ML and the MMI decoders are
equal. These results strengthen the well known result due to Csisz\'ar and
K\"orner, according to which, these decoders achieve equal random coding error
exponents, since the error exponents of the typical random code and the
expurgated code are strictly higher than the random coding error exponents, at
least at low coding rates. While the universal optimality of the MMI decoder,
in the random-coding error exponent sense, is easily proven by commuting the
expectation over the channel noise and the expectation over the ensemble, when
it comes to typical and expurgated exponents, this commutation can no longer be
carried out. Therefore, the proof of the universal optimality of the MMI
decoder must be completely different and it turns out to be highly non-trivial.
