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Philosophy of Science, 69 (September 2002) pp. S98-S103. 0031-8248/2002/69supp-0009
Copyright 2002 by The Philosophy of Science Association. All rights reserved.
It All Adds Up: The Dynamic Coherence of Radical
Probabilism
S. L. Zabell
Northwestern University
http://www.journals.uchicago.edu/PHILSCI/journal/issues/v69nS3/693009/693009.
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Send requests for reprints to the author, Departments of Mathematics and Statistics, 2033 Sheridan
Rd., Northwestern University, Evanston, IL 60208; [email protected].
Acknowledgements. I would like to thank Brian Skyrms for suggesting this topic and providing
several references. Thanks also to Persi Diaconis and Teddy Seidenfeld for helpful conversations.
Brian Skyrms (1987, 1990, 1993, 1997) has discussed the role of dynamic
coherence arguments in the theory of personal or subjective probability. In
particular, Skyrms (1997) both reviews and discusses the utility of martingale
arguments in establishing the convergence of beliefs within the context of radical
probabilism. The classical martingale converence theorem, however, assumes the
countable additivity of the underlying probability measure; an assumption rejected
by some subjectivists such as Bruno de Finetti (see, e.g., de Finetti 1930 and
1972). This brief note has a very modest goal: to briefly consider the extent to which
Skyrms's argument can be extended to the finitely additive case.
1. Dynamic Coherence.
Let us begin by briefly reviewing Skyrms's argument. Let ( ,
, P) be a
probability space; that is, is a set (the sample space), is a -algebr> of subsets
of , and P is a countably additive probability measure on
. By definition, a
random variable X on ( , , P) is an -measurable function>X:
Let
n
1
2
3
R.
... be an increasing sequence of sub- -algebras of> ; intuitively
represents the information available to us at time n. Let X1, X2, X3, ... be a
sequence of random variables on the probability space. One says that X1, X2, X3, ...
is a martingale with respect to {
n
: n 1} provided:
1. Xn is n measurable;
2. E [ | Xn| ] < ;
3. E [ Xn+1>| n] = Xn almost surely.
The celebrated Doob martingale convergence theorem (see, for example, either
Billingsley, 1995 or Durrett, 1996) assures us that if the expectations E[| Xn|] are
bounded, then such a sequence of random variables converges almost surely to an
-measurable random variable>X:
Skyrms's clever observation is the following. Suppose that Xn represents your
subjective probability regarding an uncertain outcome on the basis of the
information available to you at time n; and let
n=
(X1, X2, ... , Xn), the -field
generated b> the random variables X1, X2, ... . , Xn. (No particular model such as
Bayesian updating is being assumed here, the revision has taken place in a "black
box".) Then the resulting sequence
1,
2,
3,
... is an increasing sequence of sub-
-algebras o> and each of the three conditions above in the definition of a
martingale must be satisfied: the first because a random variable is (by definition)
measurable with respect to the -algebr> it generates; the second because Xn is a
probability (and therefore 0 Xn 1); and the third is an instance of the "principal
principle",
and can be justified on the grounds of a Dutch book coherence argument (see
Skyrms, 1997, Section 5).
Thus the sequence X1, X2, X3, ... is a martingale. Because of the very special
nature of the random variables in question (they are nonnegative and bounded), the
martingale convergence theorem applies to the sequence, and therefore
(subjective) probability one is necessarily assigned to the event that the sequence
of probability assessments will converge to a limiting value.
What are the potential vulnerabilities in such an argument? There are two
obvious points of attack. First, there is the key martingale assumption that E[ Xn+1 ||
X1, X2, ... , Xn] = Xn almost surely. It is certainly possible, given the black box nature
of the process, that this might not hold but, as Skyrms discusses, this would violate
a natural coherence condition.
That leaves the assumption of countable additivity, implicitly invoked in the use
of the MCT (martingale convergence theorem). Is it really necessary? That question
is the focus of this note.
2. The Finitely Additive Version of the MCT.
In an important paper, Purves and Sudderth (1976) have discussed the
extension of many of the classical limit theorems of probability to the finitely additive
setting. Their approach is based on the concept of a strategy. Here are the basic
definitions.
Let X and Y be nonempty sets. From now on, a probability on a set means a
finitely additive probability measure defined on all the subsets of the set; and a
conditional probability on Y given X is a function that assigns to each element x X
a probability (in this sense) on Y. A strategy on X × Y is then a pair ( 0,
consisting of a probability
0
on X and a conditional probability
1 on
1)
Y given X.
Every such strategy in turn determines a probability on X × Y: given a set S X × Y,
if Sx = { y Y: (x, y) S}, then
(The existence and uniqueness of this integral is a standard topic in finitely additive
integration theory.)
Probability measures on X × Y arising in this way are said to be strategic; and it
is an unexpected feature of finitely additive probability measures on a product
space X × Y that not all such probabilities are strategic.
In the martingale convergence theorem, probability statements are made about
infinite sequences of observations. Such probabilities are built up from probabilities
defined on more basic objects, sets of finite sequences. Thus, given a set X, let X*
denote the set of all finite sequences of elements in X (including the null sequence).
A strategy in this extended context is then defined to be a conditional probability
on X given X*. (The earlier definition of strategy only assigned conditional
probabilities given sequences of length zero and one.)
Let
0
denote the conditional probability on X that assigns to the empty
sequence (that is, the unconditional probability on X). It is apparent that one can, in
the manner discussed earlier, use (in conjunction with
0) to
obtain a strategic
measure on Xn, the Cartesian product of X with itself n times, for all n 1. It is less
obvious, but a fundamental result of Dubins and Savage (1965, 7 21) that such a
strategy determines a positive linear functional on the bounded real-valued
functions on X , the space of infinite sequences of elements of X, endowed with
the discrete topology. This in turn means that the strategy determines a probability
on certain special subsets of X , the sets that are at once both open and closed in
the discrete topology. Purves and Sudderth (1976) prove that the probability can in
fact be extended to a much larger class of sets
( ); this class is an algebra (but
not a -algebra) and is quit> rich: it contains the Borel sets.
This extension of in turn permits the establishment of a martingale
convergence theorem in the finitely additive setting (Purves and Sudderth 1976,
Theorem 7.3). The set of sequences for which convergence does not occur is
contained in an
( )-set that has (extended) probability zero. (Purves and
Sudderth (1983) discuss some other examples of zero-one laws in the finitely
additive setting.)
3. Strategic Measures and Coherence.
One remaining question is the justification for considering strategies and
strategic measures. Lane and Sudderth (1985) have advanced an attractive notion
of coherence that singles out precisely the strategic measures. This goes briefly as
follows. Suppose P is a probability on X × Y, and q is a conditional probability on Y
given X. If A X × Y is a set, let IA denote the indicator function of the set (that is,
IA(x) = 1, x A, IA (x) = 0, x A), and let Ax = { y Y: (x, y) A}. Interpreting P and q as
prices for gambles on (x, y) and y given x, respectively, a bet of c on A yields a net
return of
and, letting S X and B = {(x, y): y Bx}, a bet of d on Bx conditional on x S yields a
net return of
The Lane and Sudderth notion of coherence is then very simple: viewing the
posted odds as those of a bookie, if a gambler is limited to a finite number of
transactions of each of these two types, then the bookie is coherent provided the
gambler cannot ensure a positive return. That is, one cannot find functions
m
and
1,
... ,
n
such that
1,
... ,
Lane and Sudderth then prove that a pair (P,q) is coherent in this sense if and
only if P is a finitely additive probability and (letting P0 denote the marginal of P on
X)
for every subset A X × Y.
4. Concluding Comments.
Strategic probability measures are also sometimes termed disintegrable. Dubins
(1975, Theorem 1) demonstrated that this property is equivalent to another,
apparently different one, the earlier property of conglomerability, discovered by de
Finetti (1930 and 1972, 98). The Lane-Sudderth notion of coherence is closely
related to an earlier notion of coherence introduced by Freedman and Purves
(1969) and later studied by Heath and Sudderth (1978) and Lane and Sudderth
(1985). This earlier notion (and implicitly the later Lane-Sudderth notion) has been
criticized by Kadane, Schervish, and Seidenfeld (1996), appealing to a result of
Schervish, Seidenfeld, and Kadane (1984). This important result states that every
finitely additive probability that is not countably additive fails to be disintegrable (or,
as discussed above, conglomerable) with respect to some countable partition.
The comments of Lane and Sudderth, however, appear to adequately address
this point. They point out that this argument, which appears to argue that a
probability must be countably additive in order to be coherent, runs afoul of a simple
objection: if countable partitions are permitted, then it seems arbitrary to exclude
uncountable partitions. But then even countably additive probabilities can fail to
satisfy this restriction.
In a later survey paper, Sudderth (1994) returned to this issue, and his brief
comment seems very much to the point:
Several authors (see, for example, Heath and Sudderth (1972) or Skyrms
(1984)) have remarked that if a gambler is allowed to make countably many
bets, then P must be countably additive to avoid a sure loss. However, de
Finetti (1972, p. 91) considered such arguments to be circular because they
rely on the usual conventions about infinite sums which are tantamount to an
assumption of countable additivity.
References
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Billingsley, Patrick (1995), Probability and Measure, 3rd ed. New York: John
Wiley and Sons. First citation in article
de Finetti, Bruno (1930), "Sulla proprietà coglomerativa delle probabilità
subordinate," Rend. R. Inst. Lombardo Milano 63: 414 418. First citation in article
de Finetti, Bruno (1972), Probability, Induction, and Statistics. New York: John
Wiley and Sons. First citation in article
Dubins, Lester E. (1975), "Finitely Additive Conditional Probablities,
Conglomerability and Disintegrations," The Annals of Probability 3: 89 99. First
citation in article
Dubins, Lester E., and Leonard J. Savage (1965), How to Gamble If You Must:
Inequalities for Stochastic Processes. New York: McGraw-Hill. (Second edition,
Dover Press, New York, 1976.) First citation in article
Durrett, Richard (1996), Probability: Theory and Examples, 2nd ed. Belmont,
Calif.: Duxbury Press. First citation in article
Freedman, David A., and Roger A. Purves (1969), "Bayes' Method for Bookies,"
The Annals of Mathematical Statistics 40: 1177 1186. First citation in article
Heath, David C., and William D. Sudderth (1972), "On a Theorem of de Finetti,
Oddsmaking, and Game Theory," The Annals of Mathematical Statistics 43: 2072
2077. First citation in article
Heath, David C., and William D. Sudderth (1978), "On Finitely Additive Priors,
Coherence, and Extended Admissibility," The Annals of Statistics 6: 333 345. First
citation in article
Kadane, Joseph B., Mark J. Schervish, and Teddy Seidenfeld (1996), "Reasoning to
a Foregone Conclusion," Journal of the American Statistical Association 91: 1228
1236. (Paper reprinted in Kadane, Schervish, and Seidenfeld 1999, Section 3.7.)
First citation in article
Kadane, Joseph B., Mark J. Schervish, and Teddy Seidenfeld (1999), Rethinking the
Foundations of Statistics. Cambridge University Press.
Lane, David A., and William D. Sudderth (1984), "Coherent Predictive Inference,"
Sankhy 46, Series A: 166 185.
Lane, David A., and William D. Sudderth (1985), "Coherent Predictions Are
Strategic," The Annals of Statistics 13: 1244 1248. First citation in article
Purves, Roger A., and William D. Sudderth (1976), "Some Finitely Additive
Probability," The Annals of Probability 4: 259 276. First citation in article
Purves, Roger A., and William D. Sudderth (1983), "Finitely Additive Zero-one
Laws," Sankhy 45, Series A: 32 37. First citation in article
Schervish, Mark J., Teddy, Seidenfeld, and Joseph B. Kadane (1984), "The Extent
of Nonconglomerability of Finitely Additive Measures," Zeitschrift fur
Wahrscheinlichkeitstheorie und verwandte Gebiete 66: 205 226. First citation in
article
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




Skyrms, Brian (1984), Pragmatics and Empiricism. Yale University Press. First
citation in article
Skyrms, Brian (1987), "Dynamic Coherence and Probability Kinematics,"
Philosophy of Science 54: 1 20. First citation in article
Skyrms, Brian (1990), The Dynamics of Rational Deliberation. Harvard University
Press. First citation in article
Skyrms, Brian (1993), "Discussion: A Mistake in Dynamic Coherence Arguments,"
Philosophy of Science 60: 320 328. First citation in article
Skyrms, Brian (1997), "The Structure of Radical Probabilism," Erkenntnis 45: 285
297. First citation in article
Sudderth, William D. (1994), "Coherent Inference and Prediction in Statistics," in
D. Prawitz, B. Skryms, and D. Westerst;arahl (eds.), Logic, Methodology and
Philosophy of Science IX Elsevier. The Annals of Probability 4: 259 276.