Mismatched Guesswork

Abstract

We study the problem of mismatched guesswork, where we evaluate the number of symbols $y \in \mathcal{Y}$ which have higher likelihood than $X \sim μ$ according to a mismatched distribution $ν$. We discuss the role of the tilted/exponential families of the source distribution $μ$ and of the mismatched distribution $ν$. We show that the value of guesswork can be characterized using the tilted family of the mismatched distribution $ν$, while the probability of guessing is characterized by an exponential family which passes through $μ$. Using this characterization, we demonstrate that the mismatched guesswork follows a large deviation principle (LDP), where the rate function is described implicitly using information theoretic quantities. We apply these results to one-to-one source coding (without prefix free constraint) to obtain the cost of mismatch in terms of average codeword length. We show that the cost of mismatch in one-to-one codes is no larger than that of the prefix-free codes, i.e., $D(μ\| ν)$. Further, the cost of mismatch vanishes if and only if $ν$ lies on the tilted family of the true distribution $μ$, which is in stark contrast to the prefix-free codes. These results imply that one-to-one codes are inherently more robust to mismatch.