On Two-Stage Guessing

Abstract

Stationary memoryless sources produce two correlated random sequences Xn and Yn. A guesser seeks to recover Xn in two stages, by first guessing Yn and then Xn. The contributions of this work are twofold: (1) We characterize the least achievable exponential growth rate (in n) of any positive ρ-th moment of the total number of guesses when Yn is obtained by applying a deterministic function f component-wise to Xn. We prove that, depending on f, the least exponential growth rate in the two-stage setup is lower than when guessing Xn directly. We further propose a simple Huffman code-based construction of a function f that is a viable candidate for the minimization of the least exponential growth rate in the two-stage guessing setup. (2) We characterize the least achievable exponential growth rate of the ρ-th moment of the total number of guesses required to recover Xn when Stage 1 need not end with a correct guess of Yn and without assumptions on the stationary memoryless sources producing Xn and Yn.