Filtering: the case for "noisier" data

Suppose there is some discrete variable X of interest, which can only be observed after passing through 2 channels (Q and R). You are limited to n noisy observations of X and then must estimate the value of X. Your one control is a parameter k which determines the level of correlation in your observed data. Specifically, X is first transmitted through the channel Q n/k times to yield variables Y₁,..., Y_nk/ and then each Y_i is transmitted through channel R k times to yield variables Z_i,₁,..., Z_I,k. How should k be chosen to maximize the probability of successfully guessing the correct value of X? While k ≡ 1 yields data points Z_i,1 which are conditionally independent given the value of X, we find that this does not always mean that k ≡ 1 is the optimal choice. In fact, many simple situations yield cases where the optimal value of k is greater than 1. We explore this phenomenon and present both theoretical and empirical results.