On Two-Stage Guessing

Abstract: Stationary memoryless sources produce two correlated random sequences $X<sup>n$ and $Y<sup>n$. A guesser seeks to recover $X<sup>n$ in two stages, by first guessing $Y<sup>n$ and then $X<sup>n$. The contributions of this work are twofold: (1) We characterize the least achievable exponential growth rate (in $n$) of any positive $\rho$-th moment of the total number of guesses when $Y<sup>n$ is obtained by applying a deterministic function $f$ component-wise to $X<sup>n$. We prove that, depending on $f$, the least exponential growth rate in the two-stage setup is lower than when guessing $X<sup>n$ 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 $\rho$-th moment of the total number of guesses required to recover $X<sup>n$ when Stage 1 need not end with a correct guess of $Y<sup>n$ and without assumptions on the stationary memoryless sources producing $X<sup>n$ and $Y<sup>n$.