Why is the 20 multiplied by the million? Why not 0.95 multiplied by million?
Using the same argument I would accept infinite odds that my username is quickthrower2 so there is infinite information?
Using the same argument I would accept infinite odds that my username is quickthrower2 so there is infinite information?
Gaining a Shannon entropy bit means learning the answer to a yes-no question that had 1:1 odds.
Gaining a log-odds evidence bit means doubling your best-guess odds on a question you are uncertain about, from X:Y to (2X):Y.
One Shannon bit is worth arbitrarily many evidence bits, because a Shannon bit takes you from 1:1 odds to UNBOUNDEDLYHUGE:1 odds. So... yeah, actually, reading your username is worth infinite bits of log-odds evidence on what your username is! (Ignoring practical issues like the small chance of computer malfunctions, of course.)
And to answer your initial question: the 20 just came from the assertion they'd bet 20:1. That was arbitrary.
Going from 1:4 to 1:2 means that the event has become twice as likely. But going from 2:1 to 4:1 does not: it means that the complementary event has become half as likely.
Based on this, we can't do math with odds treating them identically to ratios.
If you do the math correctly, the two types of information measure are basically the same thing.
In the original comment, the evidence update was stated as going from 20:1 to 1:1000000 and it was claimed this was approximately 24 bits of evidence. The update is from 2^4.3:1 to 2^-19.9:1. Subtracting the exponents you get 4.3 - -19.9 = 24.2 which is approximately 24 as claimed. The "20" in 20:1 is correctly accounted for by the ~4 additional bits of evidence on top of updating from 1:1000000 to 1:1.
Clearly evidence bits behave very differently from entropy bits. Acquiring a single entropy bit is an update from 1:1 to 0:1 which is 2^0:1 to 2^-infinity:1. It's worth an unbounded number of evidence bits. It's important not to mix these two things up.
Or perhaps you can provide a reference to a justification of this type of calculation?
I think you're just wrong about needing everything to be in the form X:1 or 1:X. When I compute the ratio of 1000000:1 divided by 1:20 it gives 1000000:(1/20) then scaling both sides by the same factor gives 20000000:1.
I would be very surprised if you can find any reference at all to the number you describe as 'evidence bits', or anything equivalent, made by anyone who can show an understanding of basic probability, statistics, or information theory.
I understand how you get 20,000,000 as the answer to the calculation you carry out. My point is that that number is not meaningful in any way.
For example, suppose you are trying to estimate how much rounding errors in a pseudo random number generator betray that it is not a true exact representation of the random process. One way to quantify this is to compute the expected bits of evidence revealed per call to the RNG.