I’m not asking you to translate it across model problems; I’m asking you to explain it in more detail than you have yet done above, if you are willing.

]]>Never mind Gott, the argument at 8.01pm on 20th was yours; can you fill it out please?

]]>Phillip, I confess that I don’t fully understand your brief argument at 8.01pm on 20th, and as a Bayesian who finds that it fails in special cases I take that to mean it contains a subtle error. Unless you fill it out, the only thing I can do is to give the Bayesian version, as follows:

Define the proposition

A = “my stop is in the middle 95% of all stops”

Then prior(A|J) = 0.95. Define

B = “my stop is the 18th”

To calculate posterior(A|B,J) we must use Bayes’ theorem:

posterior(A|B,J) = (1/K) prior(A|J)likelihood(B|J,A)

where

K = prior(A|J)likelihood(B|J,A) + prior(~A|J)likelihood(B|J,~A)

The priors are known, the likelihoods can be found, but even then you have to play intuitive tricks to get the probability for the number of stations – when Bayes’ theorem allows you to consider this proposition directly.

]]>“100 per cent sure is strong prior information. The whole point is whether one can say anything at all if one has essentially no information.”

The whole point is to make a general statement that doesn’t fail in special cases.

]]>Phillip: to make my critique more specific, with 95% probability you are in the first 95% of the track, or the middle 95%, or the end 95%, or – crucially – the subset of track comprising the first 47.5% plus the last 47.5%.

]]>Phillip,

My station problem is already a simple model problem and I don’t think your new model problem is any simpler. I’m reluctant to analyse it because there can be subtle differences between a thing and an analogy to it. We are in disagreement about the railway problem; let’s try to settle it there.

Let proposition A = “the middle 95% of stations are in the urn” and B = “my station is in the middle 95%”.

Then you are reasoning (I think) that

p(B|A) = 0.95 = p(A|B).

This is the usual error appearing in intuitive non-Bayesian reversion; not all people who are female are pregnant, but all people who are pregnant are female.

]]>Phillip, Your statement “Assuming that all are likely, then with 95 per cent confidence I am in the middle 95 per cent of the stops” is not contingent on the prior information you have, and I am showing you by means of extreme prior information that it can therefore fail. It is therefore untrustworthy – which, in problems involving inference, is another way of saying wrong.

]]>OK Phillip, let me sharpen the question. Suppose your prior info is such that you are 100% certain that the railway has 997 stops. This is still a special case of the probabilistic situation, and can therefore be used to test assertions based on intuition rather than Bayes’ theorem. In that case, are you still 95% confident that you are in the middle 95% of all the stops?

]]>Busy weekend Philip, as you can judge from my warning of delayed responses on Friday. Then and in the continuum case I set the scene by giving the Bayesian (ie, correct) answer. Before I analyse your argument, here’s a question: What if, before the ‘experiment’, you were 99.999% certain that there were 997 stations?

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