From: Tom McCabe (firstname.lastname@example.org)
Date: Tue Aug 28 2007 - 10:15:22 MDT
--- Norman Noman <email@example.com> wrote:
> > "Rare" covers a very large spectrum of
> > from 10^-1 to 10^-100000000000000 (and so on). In
> > order for something to qualify as a "miracle", its
> > probability would have to be so low that the event
> > provides enough Bayesian evidence to overturn our
> > current model. Eg, if we saw Mount Everest
> > spontaneously levitate several hundred meters in
> > air, this would require a new model of physics. If
> > saw a large chunk of rock spontaneously break off
> > Mount Everest, it would be rare, but it would not
> > qualify as a miracle because it wouldn't require
> > laws of physics to explain.
> I can randomly mash keys on the keyboard for a few
> minutes and produce a
> text string with a 1 in
> 1000000000000000000000000000000000000000000 chance
> of appearing. Rareness is not the sole prerequisite
> for miracles.
"Suppose I flip a coin twenty times. If I believe the
coin is fair, the best prediction I can make is to
predict an even chance of heads or tails on each flip.
If I believe the coin is fair, I assign the same
probability to every possible sequence of twenty
coinflips. There are roughly a million (1,048,576)
possible sequences of twenty coinflips, and I have
only 1.0 of probability mass to play with. So I assign
to each individual possible sequence a probability of
(1/2)^20 - odds of about a million to one; -20 bits or
I made an experimental prediction and got a score of
-60 decibels! Doesn't this falsify the hypothesis?
Intuitively, no. We do not flip a coin twenty times
and see a random-looking result, then reel back and
say, why, the odds of that are a million to one. But
the odds are a million to one against seeing that
exact sequence, as I would discover if I naively
predicted the exact same outcome for the next sequence
of twenty coinflips. It's okay to have theories that
assign tiny probabilities to outcomes, so long as no
other theory does better. But if someone used an
alternate hypothesis to write down the exact sequence
in a sealed envelope in advance, and she assigned a
probability of 99%, I would suspect the fairness of
the coin. Provided that she only sealed one envelope,
and not a million.
That tells us what we ought common-sensically to
answer, but it doesn't say how the common-sense answer
arises from the math. To say why the common sense is
correct, we need to integrate all that has been said
so far into the framework of Bayesian revision of
belief. When we're done, we'll have a technical
understanding of the difference between a verbal
understanding and a technical understanding."
> > Conservation of mass (adjusted)
> > > The mass of a closed system of substances will
> > > remain constant, regardless
> > > of the processes acting inside the system,
> > > when people are being
> > > turned into frogs.
> > >
> > > Yes, I can see how after a few years, that
> > > wrinkle would be all
> > > ironed out.
> > The laws of physics as we now understand them
> > work like that. If we discovered a violation of
> > it would probably be a violation that occurs
> > everywhere, but is simply too small to notice
> > normal circumstances.
> Yes, that was kind of my point.
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