IJ Good's paper (1967) gives a solution to Ayer's problem on why we should make new observations.

I'm trying to follow the steps in his solution.

His assumptions are that:

  1. There are r mutually exclusive and exhaustive hypothesis H1, H2,..., Hr.

  2. On some evidence E the prior probabilities are pi = P(H|E).

  3. There's an observation that has possible outcomes E1, E2,..., Et, where P(Ek| Hi) = pik (i = 1, 2,..., r; k = 1, 2, ..., t).

Then he lets qik = P(Hi|E^Ek) = pipik/ Sigmai(pipik) be the posterior of Hi if Ek occurs.

This is not the posterior I get if I use Bayes Theorem:

P(Hi|E^Ek) = P(Ek|Hi^E) P(Hi|E) / Sigmai(P(Ek|Hi^E) P(Hi|E))

is not equal to:

P(Ek|Hi) P(Hi|E) / Sigmai(P(Ek|Hi) P(Hi|E)

Which is what's implied by Good's formula.

Where's my mistake? Does P(Ek|Hi^E) = P(Ek|Hi) in this formula? Why?

Thanks for your time.

  • I don't know this paper, so I can't say what the author had in mind. But one possibility is that the observations are independent conditional on H. Then P(E_k | H_j, E) = P(E_k | H_j). – Dan Hicks Jun 15 '18 at 14:58

I think your observation is correct. The Bayesian formula requires us to conditionalise throughout on E, where E (without subscript) is the evidence we already have, i.e. it is the background information, and is part of the prior probability of the hypotheses. Good is assuming that P(Ek | Hi ⋀ E) = P(Ek | Hi) which means in effect that the probabilities of the various possible results of the experiment that yield the Ek values are independent of the background E. I'm not convinced this is a safe assumption.

This may not matter in the context of Good's argument. Good's main point in his paper is that the principle of total evidence can be justified on the basis that it serves to maximise the expected utility of a decision based on the evidence, provided the cost of acquiring the evidence is negligible. Or, to put it another way, the expected utility of using free information is never negative. This would hold for any individual reasoner, whatever the E that represents the background information for that reasoner.

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