A Proof of Entropy Minimization for Outputs in Deletion Channels via Hidden Word Statistics

07/30/2018
by   Arash Atashpendar, et al.
0

From the output produced by a memoryless deletion channel from a uniformly random input of known length n, one obtains a posterior distribution on the channel input. The difference between the Shannon entropy of this distribution and that of the uniform prior measures the amount of information about the channel input which is conveyed by the output of length m, and it is natural to ask for which outputs this is extremized. This question was posed in a previous work, where it was conjectured on the basis of experimental data that the entropy of the posterior is minimized and maximized by the constant strings 000... and 111... and the alternating strings 0101... and 1010... respectively. In the present work we confirm the minimization conjecture in the asymptotic limit using results from hidden word statistics. We show how the analytic-combinatorial methods of Flajolet, Szpankowski and Vallée for dealing with the hidden pattern matching problem can be applied to resolve the case of fixed output length and n→∞, by obtaining estimates for the entropy in terms of the moments of the posterior distribution and establishing its minimization via a measure of autocorrelation.

READ FULL TEXT

page 1

page 2

page 3

page 4

research
03/25/2020

From Information Theory Puzzles in Deletion Channels to Deniability in Quantum Cryptography

From the output produced by a memoryless deletion channel with a uniform...
research
02/02/2018

From Clustering Supersequences to Entropy Minimizing Subsequences for Single and Double Deletions

A binary string transmitted via a memoryless i.i.d. deletion channel is ...
research
02/07/2022

The Input and Output Entropies of the k-Deletion/Insertion Channel

The channel output entropy of a transmitted word is the entropy of the p...
research
11/08/2021

Sequence Reconstruction Problem for Deletion Channels: A Complete Asymptotic Solution

Transmit a codeword x, that belongs to an (ℓ-1)-deletion-correcting code...
research
03/21/2020

Hidden Words Statistics for Large Patterns

We study here the so called subsequence pattern matching also known as h...
research
12/28/2020

Objective Bayesian Analysis for the Differential Entropy of the Gamma Distribution

The use of entropy related concepts goes from physics, such as in statis...
research
04/23/2018

Statistical Estimation of Conditional Shannon Entropy

The new estimates of the conditional Shannon entropy are introduced in t...

Please sign up or login with your details

Forgot password? Click here to reset