Hierarchical guessing with a fidelity criterion

Arikan and Merhav (1998) studied the problem of guessing a random vector X within distortion D, and characterized the best attainable exponent E(D,/spl rho/) of the /spl rho/th moment of the number of required guesses G(X) until the guessing error falls below D. We extend these results to a multistage, hierarchical guessing model, which allows for a faster search for a codeword vector at the encoder of a rate-distortion codebook. In the two-stage case of this model, if the target distortion level is D/sub 2/, the guesser first makes guesses with respect to (a higher) distortion level D/sub 1/, and then, upon his/her first success, directs the subsequent guesses to distortion D/sub 2/. As in the above-mentioned earlier paper, we provide a single-letter characterization of the best attainable guessing exponent, which relies heavily on well-known results on the successive refinement problem. We also relate this guessing exponent function to the source-coding error exponent function of the two-step coding process.