Re: A simple but confusing question

From: Ian Jermyn (Ian.Jermyn_at_sophia.inria.fr)
Date: 10/06/04 

Date: Wed, 6 Oct 2004 11:11:30 +0200

Hello George.

"George Kahrimanis" <anakreon@hol.gr> a écrit dans le message de
news:415EE5B7.755A1E83@hol.gr...
> Good Day, Ian.
>
> I just do not concede that a prior always exists!

You are allowing your use of language to confuse you, what Wittgenstein
would call a 'grammatical mistake'. Priors are not things, and thus the
meaning of the word 'exists' here is not its conventional one; it remains to
be defined in this context. I do not know whether you are aware of the
theorem that probability theory is the unique extension of classical logic
to situations of uncertain knowledge under a few incontestable criteria of
rationality. Saying that you 'do not concede that a prior always exists' is
precisely analogous to saying of classical logic 'I just do not concede that
premises always exist'. Of course one may not have any premises (one can
call that 'existence' if one wants, but then one is in danger of making a
grammatical mistake by assuming that the word 'existence' here obeys the
same rules as it does in more conventional uses), but this is not a
statement about the way one should reason, but merely a statement that
reasoning is impossible because we cannot even get started. Equally, there
may be circumstances in which our prior knowledge is so vaguely expressed
that it is impossible to convert it into prior probability distributions,
but then we cannot proceed at all.

> (My caveat is, in principle even if you had "the right" model,
> there still would be a chance that the outcome will look like
> a "deviation". But this is beside the point now.)

Yes of course. This is a question of model selection.

> The underlying (or "true") pdf for x2 is ~ N(w,1), but it is
> useless to us inasmuch w is unknown. Morever, assuming "no
> prior knowledge", one refrains from admitting any pdf for w.
> However, x2-x1 ~ N(0,2), so that "x2, given x1" ~ N(x1,2).
> This is the predictive pdf.

I do not understand the utility of your definition of 'underlying' (I have
no idea of what you might mean by 'true') and 'predictive' probabilities.
The definition is not general enough to characterize two categories of
probability distribution. In fact, it is hard to see how to move beyond the
specific case you describe: you seem to be defining them as 'probability of
one of two pieces of data unconditioned on the other' and 'probability of
one of two pieces of data conditioned on the other' . As far as I can see,
the terminology can only introduce confusion, without introducing any
benefit, by implying that there are two such categories, when in fact there
are only probabilities of propositions, conditioned on other propositions.

> Back to the balls in the bucket, or particle decays. If we
> have no clue for the ratio of white balls in the bucket (or
> for the rate of A->B decays) then we can still look for a
> solution to *this* mathematical problem: find the probability
> of the next outcome being white. (I gave an answer without proof,
> in my previous piece in this thread.) This case is *very* tricky,
> because we are used to think in terms of the "true"/underlying
> probability, so that the predictive probability in this case is
> regarded as the same thing as the true probability. I admit that
> I feel somewhat like Alice in Wonderland, when I say "in default
> of any knowledge about the way the bucket has been filled, or in
> default of a dependable theory regarding the decay of particle A,
> I refrain from even defining the "true" probability; I define
> only predictive probability" which turns out as interval-valued.
> I would be surprised if you rushed to adopt this view.

I do not understand what you are saying in this paragraph because I do not
understand your use of the word 'true'.

Best,

Ian. 

-- 
--------------------------------------------------
Ian Jermyn
ianjermyn@wanadoo.fr

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