Basic question about bits of evidence vs. bits of information:

I want to know the value of a random bit. I'm collecting evidence about the value of this bit.

First off, it seems weird to say "I have 33 bits of evidence that this bit is a 1." What is a bit of evidence, if it takes an infinite number of bits of evidence to get 1 bit of information?

Second, each bit of evidence gives you a likelihood multiplier of 2. E.g., a piece of evidence that says the likelihood is 4:1 that the bit is a 1 gives you 2 bits of evidence about the value of that bit. Independent evidence that says the likelihood is 2:1 gives you 1 bit of evidence.

But that means a one-bit evidence-giver is someone who is right 2/3 of the time. Why 2/3?

Finally, if you knew nothing about the bit, and had the probability distribution Q = (P(1)=.5, P(0)=.5), and a one-bit evidence giver gave you 1 bit saying it was a 1, you now have the distribution P = (2/3, 1/3). The KL divergence of Q from P (log base 2) is only 0.0817, so it looks like you've gained .08 bits of information from your 1 bit of evidence. ???