arXiv subjects

Prafulla Chandra

Publications and source records attributed to Prafulla Chandra.

4 recordsLinked to original sources

arXiv2022

Missing Mass Estimation from Sticky Channels

Distribution estimation under error-prone or non-ideal sampling modelled as "sticky" channels have been studied recently motivated by applications such as DNA computing. Missing mass, the sum of probabilities of missing letters, is an important quantity that plays a crucial role in distribution estimation, particularly in the large alphabet regime. In this work, we consider the problem of estimation of missing mass, which has been well-studied under independent and identically distributed (i.i.d) sampling, in the case when sampling is "sticky". Precisely, we consider the scenario where each sample from an unknown distribution gets repeated a geometrically-distributed number of times. We characterise the minimax rate of Mean Squared Error (MSE) of estimating missing mass from such sticky sampling channels. An upper bound on the minimax rate is obtained by bounding the risk of a modified Good-Turing estimator. We derive a matching lower bound on the minimax rate by extending the Le Cam method.

math.ST↗

arXiv2021

Missing $g$-mass: Investigating the Missing Parts of Distributions

Estimating the underlying distribution from \textit{iid} samples is a classical and important problem in statistics. When the alphabet size is large compared to number of samples, a portion of the distribution is highly likely to be unobserved or sparsely observed. The missing mass, defined as the sum of probabilities $\text{Pr}(x)$ over the missing letters $x$, and the Good-Turing estimator for missing mass have been important tools in large-alphabet distribution estimation. In this article, given a positive function $g$ from $[0,1]$ to the reals, the missing $g$-mass, defined as the sum of $g(\text{Pr}(x))$ over the missing letters $x$, is introduced and studied. The missing $g$-mass can be used to investigate the structure of the missing part of the distribution. Specific applications for special cases such as order-$\alpha$ missing mass ($g(p)=p^{\alpha}$) and the missing Shannon entropy ($g(p)=-p\log p$) include estimating distance from uniformity of the missing distribution and its partial estimation. Minimax estimation is studied for order-$\alpha$ missing mass for integer values of $\alpha$ and exact minimax convergence rates are obtained. Concentration is studied for a class of functions $g$ and specific results are derived for order-$\alpha$ missing mass and missing Shannon entropy. Sub-Gaussian tail bounds with near-optimal worst-case variance factors are derived. Two new notions of concentration, named strongly sub-Gamma and filtered sub-Gaussian concentration, are introduced and shown to result in right tail bounds that are better than those obtained from sub-Gaussian concentration.

math.ST↗

arXiv2021

How good is Good-Turing for Markov samples?

The Good-Turing (GT) estimator for the missing mass (i.e., total probability of missing symbols) in $n$ samples is the number of symbols that appeared exactly once divided by $n$. For i.i.d. samples, the bias and squared-error risk of the GT estimator can be shown to fall as $1/n$ by bounding the expected error uniformly over all symbols. In this work, we study convergence of the GT estimator for missing stationary mass (i.e., total stationary probability of missing symbols) of Markov samples on an alphabet $\mathcal{X}$ with stationary distribution $[\pi_x:x \in \mathcal{X}]$ and transition probability matrix (t.p.m.) $P$. This is an important and interesting problem because GT is widely used in applications with temporal dependencies such as language models assigning probabilities to word sequences, which are modelled as Markov. We show that convergence of GT depends on convergence of $(P^{\sim x})^n$, where $P^{\sim x}$ is $P$ with the $x$-th column zeroed out. This, in turn, depends on the Perron eigenvalue $\lambda^{\sim x}$ of $P^{\sim x}$ and its relationship with $\pi_x$ uniformly over $x$. For randomly generated t.p.ms and t.p.ms derived from New York Times and Charles Dickens corpora, we numerically exhibit such uniform-over-$x$ relationships between $\lambda^{\sim x}$ and $\pi_x$. This supports the observed success of GT in language models and practical text data scenarios. For Markov chains with rank-2, diagonalizable t.p.ms having spectral gap $\beta$, we show minimax rate upper and lower bounds of $1/(n\beta^5)$ and $1/(n\beta)$, respectively, for the estimation of stationary missing mass. This theoretical result extends the $1/n$ minimax rate for i.i.d. or rank-1 t.p.ms to rank-2 Markov, and is a first such minimax rate result for missing mass of Markov samples.

cs.IT↗

arXiv2020

Convergence of Chao Unseen Species Estimator

Support size estimation and the related problem of unseen species estimation have wide applications in ecology and database analysis. Perhaps the most used support size estimator is the Chao estimator. Despite its wide spread use, little is known about its theoretical properties. We analyze the Chao estimator and show that its worst case mean squared error (MSE) is smaller than the MSE of the plug-in estimator by a factor of $\mathcal{O} ((k/n)^4)$, where $k$ is the maximum support size and $n$ is the number of samples. Our main technical contribution is a new method to analyze rational estimators for discrete distribution properties, which may be of independent interest.

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