Thursday, May 9, 2013

Bayes theorem and the nature of confirmation

To: Chris M.
From: Geoffrey Klempner
Subject: Bayes theorem and the nature of confirmation
Date: 14th January 2010 15:51

Dear Christian,

Thank you for your email of 5 January, with your one hour timed essay for the University of London BA Logic module, in response to the question, 'What role, if any, should Bayes theorem play in the explanation of confirmation?'

This is a knowledgeable answer which succeeds in covering a fair bit of ground. As such, it would easily score mid-60's. However, I think that a picky examiner would consider that you have not properly answered the question. This is because the question is, What role should Bayes theorem play in the EXPLANATION of confirmation? You do say something about this. However, you have interpreted the question as being, What role should Bayes theorem play in CONFIRMATION? (See your paragraph 4.)

These are clearly different issues. What role should Bayes theorem play in confirmation? To a considerable extent, this is about experimental and scientific practice. There are arguments to be made on both sides regarding the role of Bayes theorem -- that is to say, how heavy a reliance should be placed on it. But the question didn't ask for that.

The question is about 'explanation' in a philosophical sense, that is to say, accounting for our grasp or use of a particular concept, in this case the concept of 'confirmation'. To put the question in different words, what light does Bayes theorem shed on the concept of confirmation and the philosophical problems of confirmation? How successfully does the theorem account for our prior intuitions regarding confirmation? How successfully does it deal with the paradoxes of confirmation?

The first point to make about Bayes theorem (in effect, you say this but in a roundabout way) is that the theorem is directly derivable from the definition of conditional probability. The proof only takes a couple of lines. If you accept the definition of conditional probability you cannot, consistently, deny the truth of Bayes theorem. How this formula is to be applied is of course a different matter.

I think you also need to say something about probability and confirmation, as such. Intuitively, if a theory is considered more probable after the discovery of a piece of evidence than it was prior to the discovery of that piece of evidence, then we would regard that evidence as giving some degree of confirmation to that theory. Why? No theory has, or will ever be 'confirmed' in an absolute sense, i.e. proved beyond any possibility of doubt. So all 'confirmation' is a matter of 'increasing probability'. Or is it?

Suppose we have two theories A and B. Does it go without saying that if theory A has probability x and theory B has probability x+n, that this constitutes 'justification' for rejecting theory A in favour of theory B, or at least preferring theory B over theory A? I don't think this goes without saying. Is it always the case that rejecting one theory in favour of another, or preferring one theory to another is a matter of comparing probabilities? That seems even less likely to me. There are surely many cases where we just wouldn't know where to begin in assigning a probability, but where one theory clearly is better, according to various criteria, than another.

One thing we can say is that calculating probabilities is very useful in confirmation, but there are many times when we can't make meaningful probability judgements. There are many cases where one would speak of 'confirming' a hypothesis where we are not in a position to calculate probabilities. Surely, this limits to some extent the role of Bayes theorem in EXPLAINING confirmation.

As it happens, I am somewhat sceptical about probability, as a concept, outside clearly delineated cases like calculations of relative frequency, or calculating probability a priori from mathematical models. To retreat into calling probability 'subjective' muddies the waters even further, because we then have to reckon with the complicating issue of risk aversion.

You mention that there are a number of paradoxes of probability. Have you come across the 'Trading Paradox?' I give you and your friend each an envelope. I tell you both that there is a certain amount of money in each envelope (a cheque), and that the amount in one envelope is double the amount in the other envelope. It can be any amount, I haven't given any indication.

You reason as follows. The amount I have is A. Either my friend has 2A or he has A/2. The chances are the same (0.5). However, if we exchange envelopes, and he has 2A, then I gain A. If, on the other hand, he has A/2, I lose A/2. A 50-50 bet where I stand to gain twice as much as I stand to lose is a good bet!

Or maybe not. (Because your friend reasons in exactly the same way.) But if this reasoning is invalid, where is the fallacy? Of course, there is another way of describing the situation. Either you have A and your friend has 2A, or you have 2A and your friend has A. Now, the paradox disappears. But WHY is this the correct description, while the previous description was not?

This was in fact given as a competition in the journal Analysis a number of years ago, and none of the entrants came up with a fully satisfactory solution.

To get back to your essay, a lot of the things you say probably (!) could be recast as an answer to the question about the explanation of our notion of confirmation. E.g. the point about Hempel's paradox. But, clearly, as you also point out, there are issues with confirmation (such as Goodman's paradox) which Bayes theorem sheds no light on.

The moral is, in an exam, take extra care to read the question!

All the best,

Geoffrey