Missing Mass of Rank-2 Markov Chains

Estimation of missing mass with the popular Good-Turing (GT) estimator is well-understood in the case where samples are independent and identically distributed (iid). In this article, we consider the same problem when the samples come from a stationary Markov chain with a rank-2 transition matrix, which is one of the simplest extensions of the iid case. We develop an upper bound on the absolute bias of the GT estimator in terms of the spectral gap of the chain and a tail bound on the occupancy of states. Borrowing tail bounds from known concentration results for Markov chains, we evaluate the bound using other parameters of the chain. The analysis, supported by simulations, suggests that, for rank-2 irreducible chains, the GT estimator has bias and mean-squared error falling with number of samples at a rate that depends loosely on the connectivity of the states in the chain

Optimal prediction of Markov chains with and without spectral gap

We study the following learning problem with dependent data: Observing a trajectory of length $n$ from a stationary Markov chain with $k$ states, the goal is to predict the next state. For $3 \leq k \leq O(\sqrt{n})$, using techniques from universal compression, the optimal prediction risk in Kullback-Leibler divergence is shown to be $\Theta(\frac{k^2}{n}\log \frac{n}{k^2})$, in contrast to the optimal rate of $\Theta(\frac{\log \log n}{n})$ for $k=2$ previously shown in Falahatgar et al., 2016. These rates, slower than the parametric rate of $O(\frac{k^2}{n})$, can be attributed to the memory in the data, as the spectral gap of the Markov chain can be arbitrarily small. To quantify the memory effect, we study irreducible reversible chains with a prescribed spectral gap. In addition to characterizing the optimal prediction risk for two states, we show that, as long as the spectral gap is not excessively small, the prediction risk in the Markov model is $O(\frac{k^2}{n})$, which coincides with that of an iid model with the same number of parameters.Comment: 52 page

Learning Non-Parametric and High-Dimensional Distributions via Information-Theoretic Methods

Learning distributions that govern generation of data and estimation of related functionals are the foundations of many classical statistical problems. In the following dissertation we intend to investigate such topics when either the hypothesized model is non-parametric or the number of free parameters in the model grows along with the sample size. Especially, we study the above scenarios for the following class of problems with the goal of obtaining minimax rate-optimal methods for learning the target distributions when the sample size is finite. Our techniques are based on information-theoretic divergences and related mutual-information based methods. (i) Estimation in compound decision and empirical Bayes settings: To estimate the data-generating distribution, one often takes the following two-step approach. In the first step the statistician estimates the distribution of the parameters, either the empirical distribution or the postulated prior, and then in the second step plugs in the estimate to approximate the target of interest. In the literature, the estimation of empirical distribution is known as the compound decision problem and the estimation of prior is known as the problem of empirical Bayes. In our work we use the method of minimum-distance estimation for approximating these distributions. Considering certain discrete data setups, we show that the minimum-distance based method provides theoretically and practically sound choices for estimation. The computational and algorithmic aspects of the estimators are also analyzed. (ii) Prediction with Markov chains: Given observations from an unknown Markov chain, we study the problem of predicting the next entry in the trajectory. Existing analysis for such a dependent setup usually centers around concentration inequalities that uses various extraneous conditions on the mixing properties. This makes it difficult to achieve results independent of such restrictions. We introduce information-theoretic techniques to bypass such issues and obtain fundamental limits for the related minimax problems. We also analyze conditions on the mixing properties that produce a parametric rate of prediction errors