Soft Guessing Under Log-Loss Distortion Allowing Errors

TL;DR

This paper extends the problem of soft guessing under log-loss to include errors, characterizing the minimal expected guessing cost using smooth Rényi entropy and analyzing asymptotic behavior for memoryless sources.

Contribution

It introduces a new framework for soft guessing with errors and links the minimal cost to smooth Rényi entropy, providing asymptotic analysis for stationary sources.

Findings

Minimal expected guessing cost characterized by smooth Rényi entropy.

Asymptotic analysis conducted for stationary, memoryless sources.

Extension of previous work to include error probabilities.

Abstract

This paper deals with the problem of soft guessing under log-loss distortion (logarithmic loss) that was recently investigated by [Wu and Joudeh, IEEE ISIT, pp. 466--471, 2023]. We extend this problem to soft guessing allowing errors, i.e., at each step, a guesser decides whether to stop the guess or not with some probability and if the guesser stops guessing, then the guesser declares an error. We show that the minimal expected value of the cost of guessing under the constraint of the error probability is characterized by the smooth R\'enyi entropy. Furthermore, we carry out an asymptotic analysis for a stationary and memoryless source.