Entropy estimation is one of the prototypical problems in distribution property testing. To consistently estimate the Shannon entropy of a distribution on $S$ elements with independent samples, the optimal sample complexity scales sublinearly with $S$ as $\Theta(\frac{S}{\log S})$ as shown by Valiant and Valiant \cite{Valiant--Valiant2011}. Extending the theory and algorithms for entropy estimation to dependent data, this paper considers the problem of estimating the entropy rate of a stationary reversible Markov chain with $S$ states from a sample path of $n$ observations.

The classic Mallows model is a widely-used tool to realize distributions on permutations.

The classic Mallows model is a widely-used tool to realize distributions on permutations.

Entropy estimation is one of the prototypical problems in distribution property testing. To consistently estimate the Shannon entropy of a distribution on $S$ elements with independent samples, the optimal sample complexity scales sublinearly with $S$ as $\Theta(\frac{S}{\log S})$ as shown by Valiant and Valiant \cite{Valiant--Valiant2011}. Extending the theory and algorithms for entropy estimation to dependent data, this paper considers the problem of estimating the entropy rate of a stationary reversible Markov chain with $S$ states from a sample path of $n$ observations.

Entropy estimation is one of the prototypical problems in distribution property testing.

Sampling from a continuous distribution in high dimensions is a fundamental problem in algorithm design. As sampling serves as a key subroutine in a variety of tasks in machine learning [AdFDJ03], statistical methods [RC99], and scientific computing [Liu01], it is an important undertaking to understand the complexity of sampling from families of distributions arising in applications. The more restricted problem of sampling from a particular family of distributions, which we call "well-conditioned distributions," has garnered a substantial amount of recent research effort from the algorithmic learning and statistics communities. This specific family is interesting for a number of reasons. First of all, it is practically relevant: Bayesian methods have found increasing use in machine learning applications [Bar12], and many distributions arising from these methods are well-conditioned, such as multivariate Gaussians, mixture models with small separation, and densities arising from Bayesian logistic regression with a Gaussian prior [DCWY18].