Why Von Neumann corrector only works when the input bits are independent of each other and same bias ?
The basic Von Neumann dibiaser works like this: you generate two bits X, Y, and then output a bit (or not output a bit) with this logic:
X=0 Y=0 -> No output
X=0 Y=1 -> Output a 0
X=1 Y=0 -> Output a 1
X=1 Y=1 -> No output
If X and Y are uncorrelated and share the same bias (that is, the probability of them being 0 is $p$ and the probability of 1 is $(1-p)$), then it immediately follows that the probability of the above procedure outputing a 0 is $p(1-p)$ and the probability of a 1 is $p(1-p)$, which is identical (and there's a probability $p^2 + (1-p)^2$ of not outputing anything - we'll just run it again with two more input bits). In addition, because we assume that the input bits are independent, running the debiaser multiple times on independent input bits will generate independent outputs.
However, you're question is "what if the bits aren't independent or don't share the same bias").
In that case, the above logic doesn't hold.
If the bits contain different biases, that is, if the first bit is 0 with probability $p$, and the second bit is 0 with probability $q \ne p$), then probability of a 0 is $p(1-q) = p - pq$ and a probability of a 1 is $q(1-p) = q - pq$, as we assumed that $p \ne q$, these are obviously different, and so the output of the dibiaser is biased.
If the bits share the same average bias, but are correlated, what may happen is that sequence bits from the debiaser may be correlated. For example, if a
01 sequence is followed by another
01 sequence 90% of the time (and similarly for a
10 sequence), then adjacent outputs from the debiaser will themselves be strongly correlated, that is, the dibiaser has failed to turn the raw input string into something 'random'.
Previous answer (which just gave counterexamples):
Consider a case where the even input bits are 0 90% of the time, and the odd input bits are 1 90% of the time.
In that case, the output of the Von Neumann corrector is even more strongly biased than the input.
Exercise for you: devise a similar counterexample where the bits are not independent of each other...
Paul complained that my example was "too extreme" (whatever that means - I had thought that any counterexample would apply). In any case, I'll respond with another, even more extreme, example.
Consider a raw source that generated pairs of bits with this probability distribution:
- 00 with probability 1/3
- 01 with probability 1/3
- 11 with probability 1/3
(with each pair being uncorrelated with any other pair).
Simple computation shows that each pair has Shannon entropy of $log_2(3) \approx 1.5850$ bits (or about $0.7925$ entropy bits per bit).
However, if we run that through a Von Neumann corrector, we get a stream that has, well, 0 bits of entropy...