Just Wing It: Optimal Estimation of Missing Mass in a Markovian Sequence
Ashwin Pananjady Vidya Muthukumar Andrew Thangaraj
We study the problem of estimating the stationary mass -- also called the
unigram mass -- that is missing from a single trajectory of a discrete-time,
ergodic Markov chain. This problem has several applications -- for example,
estimating the stationary missing mass is critical for accurately smoothing
probability estimates in sequence models. While the classical Good--Turing
estimator from the 1950s has appealing properties for i.i.d. data, it is known
to be biased in the Markov setting, and other heuristic estimators do not come
equipped with guarantees. Operating in the general setting in which the size of
the state space may be much larger than the length $n$ of the trajectory, we
develop a linear-runtime estimator called \emph{Windowed Good--Turing}
(\textsc{WingIt}) and show that its risk decays as
$\widetilde{\mathcal{O}}(\mathsf{T_{mix}}/n)$, where $\mathsf{T_{mix}}$ denotes
the mixing time of the chain in total variation distance. Notably, this rate is
independent of the size of the state space and minimax-optimal up to a
logarithmic factor in $n / \mathsf{T_{mix}}$. We also present a bound on the
variance of the missing mass random variable, which may be of independent
interest. We extend our estimator to approximate the stationary mass placed on
elements occurring with small frequency in $X^n$. Finally, we demonstrate the
efficacy of our estimators both in simulations on canonical chains and on
sequences constructed from a popular natural language corpus.