相关论文: Bounds on the Entropy of a Function of a Random Va…

It is well known that the entropy $H(X)$ of a finite random variable is always greater or equal to the entropy $H(f(X))$ of a function $f$ of $X$, with equality if and only if $f$ is one-to-one. In this paper, we give tights bounds on…

A novel, non-trivial, probabilistic upper bound on the entropy of an unknown one-dimensional distribution, given the support of the distribution and a sample from that distribution, is presented. No knowledge beyond the support of the…

Various lower bounds are established for the entropy of sums, products and their combinations. First, we derive a prime-field analogue of a version of the entropy power inequality established by Tao over torsion-free groups. Next, we prove…

Let $\mathsf{N}_{\rm d}\left[X\right]=\frac{1}{2\pi {\rm e}}{\rm e}^{2\mathsf{H}\left[X\right]}$ denote the entropy power of the discrete random variable $X$ where $\mathsf{H}\left[X\right]$ denotes the discrete entropy of $X$. In this…

This paper provides tight bounds on the R\'enyi entropy of a function of a discrete random variable with a finite number of possible values, where the considered function is not one-to-one. To that end, a tight lower bound on the R\'enyi…

This paper derives new bounds on the difference of the entropies of two discrete random variables in terms of the local and total variation distances between their probability mass functions. The derivation of the bounds relies on maximal…

We study the relative entropy between the empirical estimate of a discrete distribution and the true underlying distribution. If the minimum value of the probability mass function exceeds an $\alpha > 0$ (i.e. when the true underlying…

The entropy power inequality (EPI) provides lower bounds on the differential entropy of the sum of two independent real-valued random variables in terms of the individual entropies. Versions of the EPI for discrete random variables have…

How low can the joint entropy of $n$ $d$-wise independent (for $d\ge2$) discrete random variables be, subject to given constraints on the individual distributions (say, no value may be taken by a variable with probability greater than $p$,…

We derive an asymptotic lower bound on the Shannon entropy $H$ of sums of $N$ arbitrary iid discrete random variables. The derived bound $H \geq \frac{r(X)}{2}\log(N) + {\it cst}$ is given in terms of the incommensurability rank $r(X)$ of…

A bound for functional $\Delta(F)=\sup_{x\in\mathbb R}|F(x)-\Phi(x)|$ is obtained, which is uniform for all distribution functions $F$ of random variables with zero mean-value and unity variance. Moreover, a two-point distribution is found,…

Entropy and differential entropy are important quantities in information theory. A tractable extension to singular random variables-which are neither discrete nor continuous-has not been available so far. Here, we present such an extension…

Shannon entropy is widely used to quantify the uncertainty of discrete random variables. But when normalized to the unit interval, as is often done in practice, it no longer conveys the alphabet sizes of the random variables being studied.…

A new upper bound on the relative entropy is derived as a function of the total variation distance for probability measures defined on a common finite alphabet. The bound improves a previously reported bound by Csisz\'ar and Talata. It is…

We give the proof of a tight lower bound on the probability that a binomial random variable exceeds its expected value. The inequality plays an important role in a variety of contexts, including the analysis of relative deviation bounds in…

Entropy rate is a real valued functional on the space of discrete random sources which lacks a closed formula even for subclasses of sources which have intuitive parameterizations. A good way to overcome this problem is to examine its…

We establish a lower bound on the entropy of weighted sums of (possibly dependent) random variables $(X_1, X_2, \dots, X_n)$ possessing a symmetric joint distribution. Our lower bound is in terms of the joint entropy of $(X_1, X_2, \dots,…

Given a bounded class of functions G and independent random variables X1, . . . , Xn, we provide an upper bound for the expectation of the supremum of the empirical process over elements of G having a small variance. Our bound applies in…

The weak law of large numbers implies that, under mild assumptions on the source, the Renyi entropy per produced symbol converges (in probability) towards the Shannon entropy rate. This paper quantifies the speed of this convergence for…

We consider families of finite quantum graphs of increasing size and we are interested in how eigenfunctions are distributed over the graph. As a measure for the distribution of an eigenfunction on a graph we introduce the entropy, it has…