On Guessing The Realization Of An Arbitrarily Varying Source

Abstract

We present a method to guess the realization of an arbitrarily varying source. Let TU be the type of the unknown state sequence. Our method results in a guessing moment that is within Kn (TU ) + O(log n/n) of the minimum attainable guessing moment with full knowledge of source statistics, i.e., with knowledge of the sequence of states s. The quantity Kn (TU ) + O(log n/n) can be interpreted as the penalty one pays for not knowing the sequence of states s of the source. Kn (TU ) by itself is the penalty one pays for guessing with the additional knowledge that the state sequence belongs to type TU . Conversely, given any guessing strategy, for every type TU , there is a state sequence belonging to this type whose corresponding source forces a guessing moment penalty of at least Kn (TU )−O(log n/n). I. GUESSING UNDER SOURCE MISMATCH Let X be a random variable on a finite setXwith probability mass function (PMF) given by (P (x) : x ∈ X). Consider the problem of guessing the realization of this random variable X by asking questions of the form “Is X equal to x?”, stepping through the elements of X, until the answer is “Yes” ([1], [2]). Massey [1] and Arikan [2] considered guessing strategies, i.e., sequences of guesses, and sought to lowerbound the minimum expected number of guesses. For a given guessing strategy G, let G(x) denote the number of guesses required when X = x. The strategy that minimizes the expected number of guesses, E [G(X)], proceeds in the decreasing order of source probabilities. Let us denote this optimum guessing order that depends on the source PMF P by GP . Arikan [2] showed that the exponent of the minimum value, i.e., log [minG E [G(X)]] = log [E [GP (X)]], satisfies H1/2(P )− log(1 + ln |X|) ≤ log [E [GP (X)]] ≤ H1/2(P ), where Hα(P ) is the Rényi entropy of order α > 0. For ρ > 0, Arikan [2] also considered minimization of (E[G(X)ρ]) over all guessing strategies G; GP minimizes this value, and the exponent of the minimum value satisfies [2] Hα(P )− log(1 + ln |X|) ≤ 1 ρ log [E [GP (X)]] ≤ Hα(P ), (1) where α = 1/(1 + ρ). Throughout this paper, ρ > 0, α = 1/(1 + ρ), and therefore α ∈ (0, 1). Suppose now that we do not know the true PMF P , but guessed assuming a PMF Q, where Q 6= P . Let this lead to a guessing strategy GQ. Thus GQ is not matched to the source. We might therefore anticipate that 1 ρ log [E [GQ(X) ]] is larger. Analogously, for any arbitrary guessing strategy G, we may think of an associated PMF QG for which G is the optimum guessing strategy. If QG 6= P , G may not be matched to the source. Setting α = 1/(1 + ρ), we claim (without going into details) that for any guessing strategy G, the redundancy defined by R(G) ∆ = 1 ρ log [E [G(X)ρ]]− 1 ρ log [E [GP (X)]]

Cite this paper

@inproceedings{SundaresanOnGT, title={On Guessing The Realization Of An Arbitrarily Varying Source}, author={Rajesh Sundaresan} }