  We consider the question of sequential prediction under the log-loss in terms
of cumulative regret. Namely, given a hypothesis class of distributions,
learner sequentially predicts the (distribution of the) next letter in sequence
and its performance is compared to the baseline of the best constant predictor
from the hypothesis class. The well-specified case corresponds to an additional
assumption that the data-generating distribution belongs to the hypothesis
class as well. Here we present results in the more general misspecified case.
Due to special properties of the log-loss, the same problem arises in the
context of competitive-optimality in density estimation, and model selection.
For the $d$-dimensional Gaussian location hypothesis class, we show that
cumulative regrets in the well-specified and misspecified cases asymptotically
coincide. In other words, we provide an $o(1)$ characterization of the
distribution-free (or PAC) regret in this case -- the first such result as far
as we know. We recall that the worst-case (or individual-sequence) regret in
this case is larger by an additive constant ${d\over 2} + o(1)$. Surprisingly,
neither the traditional Bayesian estimators, nor the Shtarkov's normalized
maximum likelihood achieve the PAC regret and our estimator requires special
"robustification" against heavy-tailed data. In addition, we show two general
results for misspecified regret: the existence and uniqueness of the optimal
estimator, and the bound sandwiching the misspecified regret between
well-specified regrets with (asymptotically) close hypotheses classes.
