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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