On two-stage guessing

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.