As I understand it, his idea was that if you set your operational
subjective probability
(as defined e.g. in the betting game I suggested) equal to the
correct conditional
probability, then you won't be subject to losing $$ in Dutch Book
arrangements...
My terminological error was in using the term "dutch book" too
broadly, whereas
I see now that the term should just to mean a bet that will lead you
to a net loss
regardless of the outcome...
The actual betting arrangement I suggested in my post was the same
one used
by de Finetti to define operational subjective probability (except
that I use it in a
more complicated way, to get at indefinite probability)
Ben
On Feb 7, 2007, at 6:41 PM, Pei Wang wrote:
> I don't really care about what label you use, but wonder if you
get de
> Finetti's idea right, which is largely motivated by the worry about
> Dutch Book.
>
> Pei
>
> On 2/7/07, Ben Goertzel <[EMAIL PROTECTED]> wrote:
>>
>> Ok, sorry if I used the term wrong. The actual game is clearly
>> defined though even if I
>> attached the wrong label to it. I will resubmit the post with
>> corrected terminology...
>>
>> ben
>>
>> On Feb 7, 2007, at 6:21 PM, Pei Wang wrote:
>>
>> > Ben,
>> >
>> > Before going into the details of your description, I feel
that your
>> > usage of "Dutch book" is different from what it usually means
for
>> > subjectivist (http://en.wikipedia.org/wiki/Dutch_book) --- it is
>> not a
>> > special type of betting procedure, but a sure win (or loss)
>> setting.
>> > Therefore, "you set the odds, but I decide which side of the bet
>> will
>> > be yours" is not a Dutch Book.
>> >
>> > Pei
>> >
>> > On 2/7/07, Ben Goertzel <[EMAIL PROTECTED]> wrote:
>> >>
>> >> 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
>> >>
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