Soft Guessing Under Logarithmic Loss

We study a lossy variant of the Massy-Arikan guessing problem where instead of guessing the exact value of a discrete random variable, the goal is to guess a good soft reconstruction: a probability distribution under which the true realization has low uncertainty. The remaining uncertainty after guessing is measured through the logarithmic loss. We derive single-shot lower and upper bounds for the corresponding guessing moments. These bounds are exponentially tight in the asymptotic regime. Moreover, we establish a connection between our proposed soft guessing problem and the problem of variable-length lossy source coding under logarithmic loss.