"The Analysis of Experimental Data", Prof Lindley describes a blind trial where the subject is asked to identify the process of making a cup of tea. The probability of them giving a correct answer is P.
The author doesn't really believe they have this ability but doesn't rule it out either so gives a prior of 1.6(1-P) for P>0.5.
He initially guessed that they'd get it right 50% of the time, therefore P=0.5, therefore the prior is 0.8. He then says "the prior value of this probability was .8, which drops to .59 when 1 error is made in 6".
My question is: where does he get this number .59 from?
I thought he found the probability at which the posterior was greatest and fed it into the formula for the prior (that is, he found that P=0.63 yielded the highest posterior probability). This appears to be what he did for in the example where there were 6 successses and no failures.
But this value of 0.63 does not appear to agree with his graph in Figure 2. Nor does it agree with the value that yields the maximum of the posterior for 5 successes and 1 failure, that is
posterior(P) ~ (likelihood of 5 successes and 1 failure in that order).(prior) = (P5(1-P)).(1-P)
The maximum of this function does not equal 0.63 (at least in my attempt to solve it).
As ever, any guidance would be much appreciated...