Hello all,
These days I am working on my dissertation, and I would like to
include some comments on the history of the concept of probability.
I have come across David Hume's treatment of induction. Briefly, he
argues that we have no reason (by which I believe he intended deduction
as the only kind of ``reason'') to suppose the future will be similar
to the past, so we cannot argue from similar circumstances in the
past to similar outcomes in the present or future. He argues in
particular that the past does not entail or necessitate the future,
thus throwing doubt on the then-accepted notion of causality.
All this seems to put the empirical sciences in a rather weak position.
It has been a favorite pastime of philosophers in the centuries since
Hume to argue against him -- perhaps the best-known attempt is Popper's
theory of falsifiability, which seeks to support the sciences by
non-inductive arguments. Frankly, I can't tell if Popper succeeded.
Probability, in its scientific aspects, seems to succumb to Humean
arguments just the same as other sciences. There is a purely formal
part of probability which is immune to arguments about the past, future,
observations, reality, etc., yet when we try to connect probability
with the ``real world,'' we run into just the sort of difficulties
Hume pointed out.
This is especially disconcerting because a formal theory of induction
is easily stated in terms of probability: induction is the computation
of the posterior for an unobserved phenomenon conditioned on
observed phenomena. From this definition, we can create pleasant
belief network models of induction (such as all varieties of hidden Markov
models) and derive their formal properties.
But if there is no such thing as induction, how can we make use of
probabilistic models? Doubtless I am missing something: on the one
hand, Hume's anti-inductive arguments seem unanswerable, but on the
other, we all seem to reason inductively every day with great success.
I can think of one resolution that seems worth pursuing: one can derive
laws of probability by deductive arguments, as did R.T. Cox. Then statements
of degrees of belief about unobserved phenomena are logically entailed
by statements concerning observed phenomena. Yet the particular statements
which result depend crucially on the assumed relation (namely the
conditional probability) of the unobserved given the observed, and that
relation, if not a priori, must be determined empirically, so we are
back where we started.
So I wonder if anyone would care to comment on the connection between
induction and probability -- can probability work without induction?
What is the relation between the two -- shall we found probability on
induction or vice versa? Was Hume wrong? Was Popper right? Your comments,
and especially pointers to the literature, are very much appreciated.
Skeptically yours,
Robert Dodier
