1. Fano's inequality
Intuition: If Y is a function of X, Y = g(X), then H(Y|X) =0 . If we can only estimate X from Y with some probability of error P(e), we can image that H(X|Y) will be small if P(e) is small ,and H(X|Y)->0 , as P(e) -> 0.
Proof: we model the process as a Markov Chain
X------> Y-------->
we define E as follows:
E = 1, if
= 0, if
We expand in two ways,
for the second equation,
(1)
,which means
for the first equation, since E is a function of X and , then
now, (2)
From (1),(2), we obtain
(3)
Next , we will prove
From data-processing inequality, we have
since
,we obtain
(4)
From (3),(4)
viz
2. Converse for source coding theorem.
---->,------> form a markov chain.
(i.i.d)
( H(x) = H(x|y) + I(x;y) )
(data processing inequality in the above Markov chain)
( I(x;y)<= H(y))
( H(x)<= log|X| )
(Fano's inequality)
,which means