Pei, gts and others:
I will now try to rephrase my ideas about indefinite probabilities
and betting, since my prior
exposition was not well-understood.
What I am suggesting is pretty different from Walley's ideas about
betting and imprecise probabilities, and so
far as I can tell is also different from other theorists' betting
scenarios related to imprecise
probabilities (though I have not read every single paper in that
literature, just a fairly
large quasi-random sampling). But of course there are close
relations among all these
different approaches.
So, here goes...
Suppose we have a category C of discrete events, e.g. a set of tosses
of a certain coin
which has heads on one side and tails on the other.
Next, suppose we have a predicate S, which is either True or False
(boolean values)
for each event within the above event-category. C For example, if C
is a set of tosses
of a certain coin, then S could be the event "Heads". S is a
function from events into
Boolean values.
If we have an agent A, and the agent A has observed the evaluation of
S on n different
events, then we will say that n is the amount of evidence that A has
observed
regarding S; or we will say that A has made n observations
regarding S.
Now consider a situation with three agents: the House, the Gambler,
and the Meta-gambler.
As the names indicate, the House is going to run a gambling
operation, involving
generating repeated events in category C, and proposing bets
regarding the outcome
of future events in C.
More interestingly, House is also going to propose bets to the Meta-
gambler, regarding
the behavior of the Gambler.
Specifically, suppose the House behaves as follows.
After the Gambler makes n observations regarding S, House offers
Gambler the opportunity to
make a "dutch book" type bet regarding the outcome of the next
observation of S.
That is, House offers Gambler the opportunity:
"
You must set the price of a promise to pay $1 if the next observation
of S comes out
True, and $0 if there it does not. You must commit that I will be
able to choose either to buy
such a promise from you at the price you have set, or require you to
buy such a promise from
me.
In other words: you set the odds, but I decide which side of the bet
will be yours.
"
Assuming the Gambler does not want to lose money, the price Gambler
sets in such a bet, is the
"operational subjective probability" that Gambler assigns that the
next observation of S will come
out True.
As an aside, House might also offer Gambler the opportunity to bet on
sequences of observations, e.g.
it might offer similar "dutch book" price-setting opportunities
regarding predicates like "The next 5
observations of S made will be in the ordered pattern (True, True,
True, False, True)"
Next, suppose Gambler thinks that: For each sequence Z of {True,
False} values emerging
from repeated observation of S, any permutation of Z has the same
(operational subjective)
probability as Z.
Then, Gambler thinks that the series of observations of S is
"exchangeable", which means
intuitively that S's subjective probability estimates are really
estimates of the "underlying
probability of S being true on a random occasion."
Various mathematical conclusions follow from the assumption that
Gambler does not want to lose
money, or the assumption that Gambler believes in exchangeability.
This is all stuff de Finetti
did more than half a century ago. I am repeating it slowly just to
set the stage for the next part,
which is more original.
Next, let's bring Meta-gambler into the picture.
Suppose that House, Gambler and Meta-gambler have all together been
watching n
observations of S.
Now, House is going to offer Meta-gambler a special opportunity.
Namely, he is going to bring
Meta-gambler into the back room for a period of time (which happens
to be where the cocaine
and whores are kept -- trust me, I lived in Vega$ for 4 years -- but
let's keep the digressions to a
minimum.... ;-). During this period of time, House and Gambler will
be partaking in a gambling
process involving the predicate S.
Specifically, while Meta-gambler is in the back room, House is going
to show Gambler k new
observations of S. Then, after the k'th observation, House is going
to come drag Meta-gambler
out of the back room, away from the pleasures of the flesh and back
to the place where gambling
on S occurs.
House then offers Gambler the opportunity to set the price of yet
another dutch-book bet on yet another
observation of S.
Before Gambler gets to set his price, though, Meta-gambler is going
to be given the opportunity
of placing a bet regarding what price Gambler is going to set.
Specifically, House is going to allow Meta-gambler to set the price
of a dutch-book bet on a proposition
of Meta-gambler's choice, of the form:
Q = "Gambler is going to bet an amount p that lies in the interval
[L,U]"
For instance Meta-gambler might propose
"Let Q be the proposition that Gambler is going to bet an amount
lying in [.4, .6] on this next observation of S.
I'll set at 30 cents the price of a promise defined as follows: To
pay $1 if Q comes out True, and $0 if it does
not. I will commit that you will be able to choose either to buy
such a promise from me at this price, or
require me to buy such a promise from you."
I.e., Meta-Gambler sets the price corresponding to Q, but House gets
to determine which side of the bet
to take.
Let us denote the price set by Meta-gambler as b; and let us assume
that Meta-gambler does not want to
lose money.
Then, b is Meta-gambler's subjective probability assigned to the
statement that:
"Gambler's subjective probability for the next observation of S being
True lies in [L,U]."
OK ... the sordid little tale is now done....
This is a betting-game-based foundation for what we call "indefinite
probabilities" in the Novamente
system. Specifically, the indefinite probability
<L,U,b,k>
attached to S means that
"The estimated odds are b that after k more observations of S, the
estimated probability of S will lie in [L,U]"
or in other words
"[L,U] is a b-level credible interval for the estimated probability
of S after k more observations."
In a Novamente context, there is no explicit separation between the
Gambler and the Meta-gambler; the same
AI system makes both levels of estimate. But this is of course not
problematic, so long as the two components
(p-estimation and b-estimation) are carried out separately.
This doesn't really add anything practical to the indefinite
probabilities framework as already formulated, it
just makes clearer the interpretation of the indefinite probabilities
in terms of de Finetti style betting games.
-- Ben
-----
This list is sponsored by AGIRI: http://www.agiri.org/email
To unsubscribe or change your options, please go to:
http://v2.listbox.com/member/?list_id=303
