I’m hoping, when I get round to it, to give a full explanation of Bayes’ Theorem, it’s use and different forms of it. For now, since I’ve just been formalising the derivations in preparation for a paper I’m writing, I thought I might as well type it up, and no reason not to share in case people want to have a look.
So we begin with a basic axiom of probability theory:
This is to say that the probability of both A and B being the case is the probability of B being the case multiplied by the probability of A given B. One can also put this the other way round:
Since both these latter halves are equal to , it follows that:
Now, if we divide both sides by , we get:
This, in short, is Bayes’ Theorem, which says that the probability of A given B is equal to the probability of A, multiplied by the probability of B given A, divided by the probability of B.
Now, to get to the odds form, we need to do a few more things: firstly, we note that:
And so we can deduce that:
The odds form allows us to compare and directly. To get further towards this, we can go through the whole process again, this time using in place of . This will eventually give us:
From this, we find we can divide by , which gives us the following:
This may look confusing, but we can note that the denominators of both the top half and of the lower half are the same – if we multiply top and bottom by that denominator, we get the much simpler equation:
Separate out some of the terms on the right hand side, and you get:
And you now have the odds form of Bayes’ Theorem! Perfect. As I said, I won’t go into its use or anything here: this is purely to provide the formal derivation for future reference. I hope you won’t be too disappointed, therefore, if you find that there is nothing at all interesting to you in this post.