>On Wed, 24 Jun 1998 R.G.Cowell@city.ac.uk wrote:
>
> mainly because you have to give your prior probability distribution
> over the set of possible initial conditions in order to perform the
> marginalisation you ask for, and that prior is something that
> depends upon your state of knowledge of the world.
>On Wed, 24 Jun 1998, Kathryn Blackmond Laskey wrote:
>
> That begs the question of what marginal distribution you're going to use
> to marginalize out all possible initial conditions.
>
> Kathy
Let me restate the initial "challenge" that I was responding to:
> On Fri, 19 Jun 1998, Kevin S. Van Horn wrote:
>
> Another example: When you say that a flipped coin has a 1/2 probability of
> landing heads up, this is a statement about *you*, not a statement about the
> coin. There is no physical property of the coin that gives it a probability
> 1/2 (or 3/4, or 1/3, etc.) of landing heads up. In fact, if you were to
> measure the initial location, orientation, angular momentum, linear momentum,
> and mass distribution of the coin sufficiently accurately, you could predict
> exactly which way it would fall. We say there is a 1/2 probability of heads
> simply because the way the coin falls usually has a very sensitive dependence
> on the above initial conditions, and we have insufficient information to
> make reliable predictions about how it will fall. Note, however, that it
> is possible to flip a coin in a manner that appears to tumble about just
> like a "normal" coin flip, but with highly predictable and repeatable
> results.
> [...]
> Probability is not a physical
> property or quantity that can be measured. I challenge you to show me a
> probability that is an actual physical property, and not just a statement
> of one's state of knowledge.
Of course, if you take the Bayesian definition of probability as
degrees of belief, then _by definition_ probability is not a physical
quantity. In that case there is no challenge; it can trivially be shown
that probabilities are not physical.
However, I interpreted the above paragraph as an argument _supporting_
the Bayesian definition, by attempting to discredit--even given the
frequentist definition of probability--the frequentist notion that
probalities represent objective physical quantities. The argument,
as I understand it, says that the probability of the coin landing heads
is just a manifestation of our uncertainty of the initial conditions
of the coin toss. As our certainty of the initial conditions increases,
the probability (i.e. the ML calculation) of the coin coming up heads
ranges over different values, and indeed will be 1 (or 0) if we know the
initial conditions perfectly.
My argument is that the ML calculation averaged over all conditions which
are not intrinsic properties of the coin (i.e. the initial conditions)
yields a well-defined value that is different for different coins. Why
isn't this then a "physical property" of the coin?
Regards,
Denver.