Abstract
Philosophers cannot agree on whether the rule of Countable Additivity should be an axiom of probability. Edwin T. Jaynes attacks the problem in a way which is original to him and passed over in the current debate about the principle: he says the debate only arises because of an erroneous use of mathematical infinity. I argue that this solution fails, but I construct a different argument which, I argue, salvages the spirit of the more general point Jaynes makes. I argue that in Jaynes’s objective Bayesianism we might have good reasons to adopt Countable Additivity, and some of the major problems this adoption is known to entail need not worry us. In particular, I propose to adopt this new angle on Countable Additivity in Jon Williamson’s version of objective Bayesianism.
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Notes
If a measure is countably additive, then it is also finitely additive. Let P be a countably additive measure. We want to show that \( P\left( \bigcup \nolimits _{n=1}^{N} A_n \right) = \sum \nolimits _{n=1}^{N} P(A_n) \), for \(A_1, \dots , A_n\) pairwise disjoint. We can extend this sequence by a countably infinite sequence of empty sets: \(A_{n+1}, A_{n+2}, \dots = \emptyset \). We now can write \( P\left( \bigcup \nolimits _{n=1}^{N} A_n \right) = P\left( \bigcup \nolimits _{n=1}^{\infty } A_n \right) = \sum \nolimits _{n=1}^{\infty } P(A_n) = P(A_1) + \cdots + P(A_N) + P(\emptyset ) + P(\emptyset )+ \cdots = \sum \nolimits _{n=1}^{N} P(A_n)\). For an example of a probability function which is FA but not CA, see function Q below.
n here is the number of propositions treated; while Jaynes speaks of propositions, I will use the terms proposition and event interchangeably to mean elements of the domain of the probability functions treated.
I use this terminology to reflect Jaynes’s, although ‘unbounded’ might be more natural.
Note, however, that while this function exists, it is not trivial to define, if we require it to be defined for all subsets of \(\mathbb {N}\): see Kadane and O’Hagan (1995) and my discussion below.
For example, the Vitali sets are members of the power set of \([0,1]\subset \mathbb {R} \) for which the Lebesgue measure is not defined.
A real interval can be partitioned into at most countably many non-empty intervals, as each non-empty sub-interval must contain a rational number because rational numbers are dense in \(\mathbb {R}\), and there are countably many rational numbers.
The property of operators in deductive logic Jaynes refers to is usually called functional completeness: through conjunction and negation all other logical operators can expressed; it is not obvious that Jaynes’s “adequacy” property in probability is a natural parallel, but this is unimportant for the present argument.
Kadane and O’Hagan treat probability distributions over the natural numbers: events are represented by the sets \(\{1\},\{2\}, \dots \), instead of propositions \(\{A_1, A_2,\dots \}\), but the argument is identical
We can see this easily as follows: if \( P(1)=P(2)=\cdots =P(n)=0 \), \( P\left( \bigcup \nolimits _{i=1}^n i \right) = P(1) + \cdots + P(n) = 0 \). For the second remark, see the following: \(P\left( \bigcup \nolimits _{i=1}^n i \right) = 0 \), but \( P\left( \bigcup \nolimits _{i=1}^n i \right) + P\left( \bigcup \nolimits _{i=n+1}^{\infty } i \right) = 1 \) by FA, so \( P\left( \bigcup \nolimits _{i=n+1}^{\infty } i \right) = 1 \).
Some paraphrase is needed to give context present in their paper.
I am grateful to an anonymous referee for highlighting this point.
De Finetti’s argument, re-translated from the Italian, is as follows (this is a comment on a proof such as the above, where it is claimed that if our betting quotients do not abide by CA, then we are open to a certain loss):
But this is a bit of a vicious circle, because only if I knew complete additivity to be valid could I think of extending the notion of ‘fair combination of bets’ to combinations of infinite bets, and of basing them on the series of the betting odds (de Finetti 1949, 12) [emphasis as in the original, which follows: Un motivo che tenderebbe ad avvalorare l’additività completa: se le probabilità \(p_n\) hanno somma \(p<1\), stipulando tutte le infinite scommesse posso ricevere in ogni caso 1 pagando p, e quindi avrei un’incongruenza. Ma è un po’ un circolo vizioso, perchè solo se sapessi valida l’additività completa potrei pensare di estendere la nozione di ‘combinazione di scommesse equa’ a combinazioni di infinite scommesse, e di basarle sulla serie delle quote di scommessa.]
Howson (2008) studies this argument but is puzzled by it, as, he writes, are many authors before him. In the English version quoted by Howson and others, the word serie (series) is wrongly translated as ‘sequence’
Note also that \(I_{CAJ}\) does not imply \(I_{CAW}\), as the the latter principle is about bets, on which \(I_{CAJ}\) is completely silent. That is, supposing it true that the probabilities of elementary events should determine the probabilities of the compound events they form, we still have no indication on whether bets should be countably additive or not. It is a tenable position to uphold \(I_{CAJ}\) but deny the interpretation of degrees of belief as bets, thus, for example, believing \(I_{CAJ}\), the countable additivity of degrees of belief, but not the countable additivity of bets.
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Elliot, C. E.T. Jaynes’s Solution to the Problem of Countable Additivity. Erkenn 87, 287–308 (2022). https://doi.org/10.1007/s10670-019-00195-2
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DOI: https://doi.org/10.1007/s10670-019-00195-2