Board Thread:General Discussion/@comment-26006155-20200107155350/@comment-32182236-20200205144018

The rigged coin was simply an example. In practice, the best approach is to approximate the value, rather than coming up with a random guess. But in this particular case, I simply needed a value-Any value, to demonstrate how the formula works.

Of course, the question to ask is-How to we approximate the value of $ P(A) $?

Perhaps 50% per assumption made?

The .001 value I gave in the example was simply because the vast majority of coins aren't rigged. If we were actually trying to use it in practice, $ P(A) $ would be approximated by looking at the actual ratio of rigged coins to non-rigged coins. (It actually wouldn't work at all, since it assumes that coins are either rigged to ALWAYS show up heads, or it's fair. It's possible that the coin is rigged to show up heads, say, 80% of the time. I'm trying to work on a way to generalize the idea to allow for more than just a binary set of hypotheses.

The conclusion? P(A|B) is our new probability that hypothesis A is true. And once again, if we find more evidence, we put that new probability back into the equation. The old evidence still matters, after all.

Once all the evidence has been taken into account this way, we have a probability that the hypothesis we're testing is true. If the value we get is .95, then that means there's a 95% chance that the hypothesis is true. (I had replaced A and B with H and E, respectively, to help make the idea clearer. It's easier to remember that H means hypothesis, rather than A being hypothesis.)

Now to explain where the complex denominator I gave comes from.

Jacky's formula is correct. However, one cannot simply approximate $ P(B) $ willy-nilly. In order to calculate it correctly, we have to separate it into two separate parts-The probability that it would appear, and that Event A happened, and the probability that it would appear, and that Event A didn't happen.

In other words..

$ P(B)=P(B|A)P(A)+P(B|¬A)P(¬A) $

The $ ¬ $ symbol is a negator-So $ ¬A $ means refers to event $ A $ not happening.

If we substitute $ P(B) $ for the thing we need to calculate it, we get my formula.

$ P(A|B)=\frac{P(B|A)P(A)}{P(B|A)P(A)+P(B|¬A)P(¬A)} $

As you can see, the first term of the denominator is the same as the numerator. This fact can reduce the number of steps we need to take-Once we calculate the numerator, we can substitute the first term with the value of the numerator, leaving only the second to calculate.

Another thing that will only help out is this:

$ P(A)+P(¬A)=1 $

So if you know what $ P(A) $ is, you can calculate $ P(¬A) $ easily-Just subtract P(A) from 1, and there you go!

And when we're using it to test hypothesis, then it helps to replace "A" with H, for hypothesis, and "B" with E, for the evidence we're putting in the equation.

$ P(H|E)=\frac{P(E|H)P(H)}{P(E|H)P(E)+P(E|¬H)P(¬H)} $

..The math should render in an external link.. Which I'll link to once I find it.