I'm looking at Dr. Anderson's explanation of Bayesian statistics for a simple haystack model, from http://www.ar-tiste.com/pwa-on-bayes.html and reproduced below:
There are two sources of hay, one with no needles at all and one with up to 9 needles per stack. Let's assign precisely probability 1/2, for the sake of argument, to the case where I'm buying from the needle-free source. (This represents the "null hypothesis" in this example.) If I'm dealing with the potentially needly hay, let's assume that p= (1/2)(1/10) for 0,1, . . . ,9 needles in anyone stack.
I search for needles in one stack, and find none. What do I now know? I know that this outcome had p = 1/2 for needle-free hay, p = 1/20 for needly hay; hence the probability of this outcome is 10 times as great if the hay is needle free. The new "a posteriori" probability of the null hypothesis is therefore 10/11 = (1/2)(1/2+1/20 ) rather than 1/2. Clearly I should buy this hay if that is a good enough bet.
Now suppose I was an ordinary statistician: I would simply say my expected number of needles per stack is now down to 0 +/- 2.5, and to get to 90% certainty I must search at least ten more haystacks, which is ten times as boring.
I have difficulty following this logic:
- (1/2)(1/2+1/20) = 11/40 by my calculation. I think this is the probability of buying two haystacks from our source with no needles. I think the null hypothesis is that our source is the needle-free seller. I'm not sure where 10/11 comes from.
- how is 0 +/- 2.5 derived? What does 0-2.5 expected # of needles actually mean?
He seems to be jumping a few, doubtless trivial, steps that I've missed. I'm not an ordinary or bayesian statistician but would like a basic understanding of the argument. If someone can help it would be appreciated.
thanks