Re: Bayesian boxes and Independence of Scales

From: <GSLevy.domain.name.hidden>
Date: Tue, 11 May 1999 20:01:13 EDT

Hi Jacques,
I was refering to a problem posed by ids.domain.name.hidden (Iain Stewart):

>
> Let box one contain m pounds and box two 2m pounds, where m is
> any real positive number. You may choose a box look at the
> amount of money inside and decide either to keep this amount or
> the other box's amount. (For this exercise you may think of
> cheques with real numbers on it and a special bank account where
> you can actually bank any real number of pounds.)

This is a well know problem in probabilities. I am not sure if the solution
which I propose here is also known: The expected value for the money in the
box can be calculated as the "average" between (1/2)m and 2m. If the average
is taken as an arithmetic average we get 1.25m which implies that we should
switch our choice. This obviously does not make sense. However, if the
"average" is taken as a geometrical average (equivalently, an arithmetic
average along a logarithmic scale) then we get sqrt(2m x 0.5m) = m which
implies that switching results in exactly the same expected value as not
switching, which makes sense.
Another way to state the same thing is in terms of logarithms:

>Expected Value = (1/2) (Log(m/2) + Log(2m)) =
>(1/2) (Log(m) - Log2 + Log(m) + Log2) = Log(m)

Another way to state the same thing is that the distribution is such that
there are as many number greater than m as smaller than m. (m is smack in the
middle of the scale, and a sufficient reason for this to occur is that the
scale must be logarithmic). I'll leave it to the mathematicians to explain
how there can be "more" or "less" real numbers in a given interval depending
on the scale used. This certainly goes against the concept of transfinite
numbers the way they are understood now.

As to the relationship with MWI I would like to recall the proposal I made a
few months ago about extending the Cosmological Principle to the MW. I'll
provide some background first by recalling that the conventional Cosmological
Principle, assumes that in the large scale, the properties of the Universe
are identical everywhere, and that no matter "where" we are, the universe
looks approximately the same in every direction. (This principle has been
extended by Bondi and Gold in the Steady State Theory: The Universe looks the
same no matter "when" we look at it. Recent advances in cosmology have shown
that the Universe actually does not look the same in every direction and at
any time. A reason for these variations is that these variations are
"fact-like" and the size of the sample or field of view that we are
observing is not large enough.
So I propose to make the size of the field of view as large as the MW. The
Cosmological Principle thus extended, then state that the MW looks identical
from any point in the MW. Rephrasing this in terms of probabilities of life
and death: From the PERSPECTIVE OF THE OBSERVER The probability of continued
life for a 25 year old OBSERVER is identical to the probability of continued
life for of a 100 year old very sick OBSERVER.

Going back to the independence of scale concept discussed in terms of picking
the right scale for the Bayesian probability problem. It appears that the
expected values for continued life for the young and old man are identical
just like the expected value for the money in the boxes. I don't know how far
this analogy can be carried. However, the point is that the frame of
reference must be selected from the point of view of the OBSERVER. The MW
will then appear to be identical on the ""large scale."" (This is an
understatement? :-))

Cheers! Don't worry about the path not taken. You took it.

George
Received on Tue May 11 1999 - 17:04:17 PDT