  The likelihood decoder is a stochastic decoder that selects the decoded
message at random, using the posterior distribution of the true underlying
message given the channel output. In this work, we study a generalized version
of this decoder where the posterior is proportional to a general function that
depends only on the joint empirical distribution of the output vector and the
codeword. This framework allows both mismatched versions and universal (MMI)
versions of the likelihood decoder, as well as the corresponding ordinary
deterministic decoders, among many others. We provide a direct analysis method
that yields the exact random coding exponent (as opposed to separate upper
bounds and lower bounds that turn out to be compatible, which were derived
earlier by Scarlett et al. We also extend the result from pure channel coding
to combined source and channel coding (random binning followed by random
channel coding) with side information available to the decoder. Finally,
returning to pure channel coding, we derive also an expurgated exponent for the
stochastic likelihood decoder, which turns out to be at least as tight (and in
some cases, strictly so) as the classical expurgated exponent of the maximum
likelihood decoder, even though the stochastic likelihood decoder is
suboptimal.
