Ronald Parr writes:
> However you interpret Bayes rule, you make the assumption that the
> evidence upon which you are conditioning is germane to the proposition
> in question.
This is a demonstrably false statement, as I demonstrated in a previous post.
Bayes' Rule handles cases where the conditioning information is irrelevant
with just as much ease as it handles cases where the conditioning information
is relevant. Any assumptions of relevance or germane-ness are contained in
the prior information X of P(H | D, X) (H being the hypothesis, D the data).
So rather than attacking Bayes' Rule, you should be looking with a critical
eye at the prior information X used in any proposed induction scenario.
> [Kevin's formal argument for induction]
>
> The problem is that you have described a hypothesis about the machine
> that *has genenerated* the symbols. [...]
> To condition (in a non-vacuous way) on what you have previously seen,
> you need the assumption that the machine that generates the symbol at
> the next time step is related to the one you have been observing - and
> this is where the circularity comes in.
First of all, I did not give an argument *for* induction; instead I
showed, for one particularly simple universe, how one could compare the
hypothesis that induction is impossible (because there is no relation
between the state of the universe at different time steps) with a
hypothesis that allows induction.
Secondly, there is no circularity because I have no *general* assumption
that the machine that generates the symbol at the next time step is
related to the one I have been observing. *One* of the two hypotheses
makes this assumption; the other does not. I allow for both hypotheses
and then try to use the data to compare them.
A better objection one might make -- and maybe this is the real
objection -- is that I should have included a third hypothesis IR. This
hypothesis says that
P(x_1,...,x_T | IR, X) = P(x_1,...,x_T | I, X)
P(x_i | IR, X) = 1/2 for i > T
The x_i, for i > T, are independent of each other and x_1,...,x_T,
if IR is assumed.
If we detect some strong regularity in the past (up through time T),
this would provide strong evidence against hypothesis R (random
universe), but would not distinguish between hypotheses IR and I.
The answer at this point, of course, is to continue collecting data, and
then compare IR and I based on the evidence x_1,...,x_(T+N) for some
sufficiently large N.
