The minimization is over all conditional distributions p(y∣y)
for which the joint distribution p(y,y) = p(y)p(y∣y) satisfies
the average distortion constraint (i.e. average distortion ≤
D).
The Rate Distortion Theorem states that R(D) is the max-
imum achievable rate of information transmission which does
not exceed the distortion D. Cover and Thomas (1991) or
Dembo and Zeitouni (1998) provide details.
More to the point, however, is the following: Pairs of se-
quences (yr,yr) can be defined as distortion typical; that is,
for a given average distortion D , defined in terms of a partic-
ular measure, pairs of sequences can be divided into two sets,
a high probability one containing a relatively small number
of (matched) pairs with d(yn,yrn) ≤ D, and a low probabil-
ity one containing most pairs. As n → ∞, the smaller set
approaches unit probability, and, for those pairs,
p(yr) ≥ p(yr∣yr)eχp[-nl(Y,Y)].
(28)
Thus, roughly speaking, I (Y, Y) embodies the splitting cri-
terion between high and low probability pairs of paths.
For the theory of interacting information sources, then,
I(Y, Y) can play the role of H in the dynamic treatment that
follows.
The rate distortion function can actually be calculated in
many cases by using a Lagrange multiplier method - see Sec-
tion 13.7 of Cover and Thomas (1991).
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16
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