  We consider the problem of predicting the next observation given a sequence
of past observations, and consider the extent to which accurate prediction
requires complex algorithms that explicitly leverage long-range dependencies.
Perhaps surprisingly, our positive results show that for a broad class of
sequences, there is an algorithm that predicts well on average, and bases its
predictions only on the most recent few observation together with a set of
simple summary statistics of the past observations. Specifically, we show that
for any distribution over observations, if the mutual information between past
observations and future observations is upper bounded by $I$, then a simple
Markov model over the most recent $I/\epsilon$ observations obtains expected KL
error $\epsilon$---and hence $\ell_1$ error $\sqrt{\epsilon}$---with respect to
the optimal predictor that has access to the entire past and knows the data
generating distribution. For a Hidden Markov Model with $n$ hidden states, $I$
is bounded by $\log n$, a quantity that does not depend on the mixing time, and
we show that the trivial prediction algorithm based on the empirical
frequencies of length $O(\log n/\epsilon)$ windows of observations achieves
this error, provided the length of the sequence is $d^{\Omega(\log
n/\epsilon)}$, where $d$ is the size of the observation alphabet.
  We also establish that this result cannot be improved upon, even for the
class of HMMs, in the following two senses: First, for HMMs with $n$ hidden
states, a window length of $\log n/\epsilon$ is information-theoretically
necessary to achieve expected $\ell_1$ error $\sqrt{\epsilon}$. Second, the
$d^{\Theta(\log n/\epsilon)}$ samples required to estimate the Markov model for
an observation alphabet of size $d$ is necessary for any computationally
tractable learning algorithm, assuming the hardness of strongly refuting a
certain class of CSPs.
