[extropy-chat] Indifference (was Coin Flip Paradox)

[The Avantguardian]

--- gts <gts_2000 at yahoo.com> wrote:
> Think about it (and you too, Jef)...
> 
> On what logical grounds can proposition A imply or
> entail proposition B?
> 
> A: "No information is available about the true
> probabilities of the two  
> possible outcomes X and Y."
> 
> B: "Outcomes X and Y are therefore equiprobable."
> 
> Does A imply B, logically? I think not! If you or
> Jef can show me  
> otherwise then please do.

Because an empty scale will balance too, Gordon. Let
me try to explain it to you from a frequentist point
of view. I will assume that you are a very detailed
oriented person because if you weren't, I don't think
we would be having this conversation. So let me get a
few key assumptions out of the way:

1. P(X) is the unknown *true probability* of X. (The
Bayesian in me shudders at this, but I will try to
argue this one from a frequentist stand point.)

2. X and Y are mutually exclusive. i.e.
P(X|Y)=P(Y|X)=0

3. If X doesn't happen, then Y will happen or put
another way, Y is the "non-occurance" of X or Y = X'.
i.e. P(Y)=P(X')=1-P(X)

4. There exists a set S of all possible events and
their "non-occurances" that may or may not happen i.e.
{E1, E2, E3 ... En} & {E1',E2',E3' ... En'}. This
gigantic, possibly infinite, set would include very
rare events like you winning the lotto or getting hit
by lightning and very likely events like the sun
rising tomorrow or you taking a breath in the next
hour as well as everything in between like whether a
traffic light will turn red before you get to an
intersection or not.

5. Every element Ei of S has a characteristic *true
probability* P(Ei) that governs how likely it is to
occur as well as a converse probability P(Ei')=1-P(Ei)
which governs how likely it is to NOT occur.   

6. Event X and its converse Y are elements of the set
S.

Now that we have all our assumptions explicitly
stated, let's get to the logic.

Now let us say that we want to try to estimate the
*true probability* of X but we have no data regarding
what that probability might be. So let us assign the
probability of X a random variable P(X). Now we have
no idea what the shape of the distribution of P(X)
might be but we do know that it is a real number that
lies somewhere on the interval [0,1].

So now what a frequentist would do is start drawing
pairs of elements as samples from the set S and
measure their *true probabilities*. Lets say on the
first draw you get E1 and E2 with P1 and P2 of .0001
and .75 respectively. Then on the next draw you get E3
and E1' with P3 and P1' of .985 and .9999. Then on the
next you draw you get E2' and E3' with probability .25
and .015 repectively.

Next you calculate the average of the probabilities
that you get in each draw:

Average of draw 1: (.0001+.75)/2=.37505
Average of draw 2: (.985+.9999)/2=.99245
Average of draw 3: (.25+.015)/2=.1325

Now you calculate the mean average you get with your
draws:
(.37505+.99245+.1325)/3= .5

Now admittedly I rigged this particular example to
come out at exactly .5 but do you not see that if the
set of everything that can happen or not includes
every event and its converse then for every
probability P(E) there exists P'(E')=1-P(E). So if you
take enough samples of specific events and measure
their average probability (relative frequency at
infinity) you cannot help but get .5 as the average
probability of SOMETHING happening at all. This is
guaranteed by the Law of Large Numbers and the Central
Limit Theorem just like Von Mises' results were.

So if you don't have any information regarding what
the *true probability* of a specific event is, then it
only makes sense that you assign it the average
probability of *all possible events* or .5 because you
will be within one standard deviation of the *true
probability* 2/3rds of the time.

That concludes my sloppy logical proof of the
Principle of Indifference using frequentist rationale.
I like Jef, am tired of discussing the Priciple of
Indifference with you. If this doesn't convince you,
you will just have to go through life not believeing
in the Principle of Indifference. :) Other topics in
the category of probability/randomness however are
fine.



Stuart LaForge
alt email: stuart"AT"ucla.edu

"If we all did the things we are capable of doing, we would literally astound ourselves." - Thomas Edison


 
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