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Iterative Probability Kinematics Horacio Arl o-Costa and Richmond H. Thomason

Summary: Iterative Probability Kinematics
Horacio Arlo-Costa and Richmond H. Thomason
ABSTRACT. Following the pioneer work of Bruno De Finetti [12], conditional prob-
ability spaces (allowing for conditioning with events of measure zero) have been studied
since (at least) the 1950's. Perhaps the most salient axiomatizations are Karl Popper's
in [30], and Alfred Renyi's in [32]. Non-standard probability spaces [33] are a well
know alternative to this approach. Vann McGee proposed in [29] a result relating both
approaches by showing that the standard values of in nitesimal probability functions
are representable as Popper functions, and that every Popper function is representable
in terms of the standard real values of some in nitesimal measure.
Our main goal in this article is to study the constraints on (qualitative and prob-
abilistic) change imposed by an extended version of McGee's result. We focus on an
extension capable of allowing for iterated changes of view. Such extension, we argue,
seems to be needed in almost all considered applications. Since most of the available
axiomatizations stipulate (de nitionally) important constraints on iterated change, we
propose a non-question-begging framework, Iterative Probability Systems (IPS) and
we show that every Popper function can be regarded as a Bayesian IPS. A generalized
version of McGee's result is then proved and several of its consequences considered.
In particular we note that our proof requires the imposition of Cumulativity, i.e. the


Source: Andrews, Peter B. - Department of Mathematical Sciences, Carnegie Mellon University


Collections: Mathematics