H Peii,
As a mathematician (although I've forgotten nearly all the math I used
to know in the many years since I actually practiced serious
mathematics)
I take a slightly different attitude...
The mathematics of indefinite probabilities is what it is ... and is
actually
all we need to guide the integration of the indefinite probability
formalism
with Novamente (an integration that has already occurred, codewise).
Explaining the meaning underlying the mathematics, conceptually, is
interesting to me, and I find it worthwhile in terms of expanding and
enriching the way I think about the mathematics...
But I don't think there is one correct conceptual explanation of
ordinary
probability, nor of indefinite probability. I think it's fine to go
in the direction
> "evidence of belief" -->
> "amount of evidence" --> "degree of belief" --> "betting preference".
or in some other order. In fact the account I gave in my prior email
was
more like
> "evidence of belief" -->
> "amount of evidence" --> "betting preference" --> "degree of belief"
where indefinite probabilities were concerned. That is, the notion of
evidence and amount of evidence were introduced in the set up of
the bet posed to the Meta-gambler.
The main reason I decided to formulate an explanation of indefinite
probabilities in terms of betting, was just to make clear the
relationship
between indefinite probabilities and traditional subjective
probabilities.
One could also justify indefinite probabilities using a Cox-style
approach, via simply positing multiple agents, and assuming that
each agent obeys Cox's axioms when reasoning about the future
plausibility assessments of the other agents.
I don't consider either of these justifications particularly superior to
the other, nor particularly necessary to the application to AGI. But I
do find them interesting.
-- Ben
On Feb 7, 2007, at 7:20 PM, Pei Wang wrote:
> Ben,
>
> What you just wrote makes sense.
>
> However, I always feel that defining probability by betting preference
> is to "put the cart before the horse".
>
> To me, the best part of the subjective approach is to strongly argue
> that "probability" is nothing but "degree of belief", and that for the
> same belief, different systems can legally attach different degrees to
> it. Since betting preference depends on the related degrees of belief,
> the former can be used to measure the latter. However, to explain the
> latter by the former is to get the causal relation backwards. In this
> aspect, I don't agree with the subjective approach.
>
> To me, the right order of definition is "evidence of belief" -->
> "amount of evidence" --> "degree of belief" --> "betting preference".
> Both one-component and two-component measurements should and can be
> defined in this way.
>
> Though I'll need more time to comment on the details, I don't feel
> good about the overall picture.
>
> Pei
>
> On 2/7/07, Ben Goertzel <[EMAIL PROTECTED]> wrote:
>> 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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>> >> >
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