Probability Theory and Probability Semantics

Book Information Probability Theory and Probability Semantics. By P. Roeper and H. Leblanc. University of Toronto Press. Toronto. 1999. Pp. xii + 240. Hardback, US$65.00.     

Probability kinematics

Probability kinematics is studied in detail within the framework of elementary probability theory. The merits and demerits of Jeffrey's and Field's models are discussed. In particular, the principle of maximum relative entropy and other principles are used in an epistemic justification of generalized conditionals. A representation of conditionals in terms of Bayesian conditionals is worked out in the framework of external kinematics.     

Probability matching

The class of symmetric path-independent models with experimenter-controlled events is considered in conjunction with two-choice probability learning experiments. Various refinements of the notion of probability matching are defined, and the incidence of these properties within this class is studied. It is shown that the linear models are the only models of this class that predict a certain phenomenon that we call stationary probability matching. It is also shown that models within this class that possess an additional property called marginal constancy predict approximate probability matching.This research grew out of questions posed by William K. Estes. We are also indebted to Professor Estes for his encouragement and assistance at all stages of this research. During the course of this research J. I. Y. received support from the U. S. Public Health Service (N. I. M. H.). M. F. N.'s present address is the University of Pennsylvania. J. I. Y.'s present address is the University of Minnesota.     

Causal Probability

Examples growing out of the Newcomb problem have convinced many people that decision theory should proceed in terms of some kind of causal probability. I endorse this view and define and investigate a variety of causal probability. My definition is related to Skyrms' definition, but proceeds in terms of objective probabilities rather than subjective probabilities and avoids taking causal dependence as a primitive concept.     

Default Probability

A probability may be called “default” if it is neither derived from preestablished probabilities nor based on considerations of frequency or symmetry. Default probabilities presumably arise through reasoning based on causality and similarity. This article advances a model of default probability based on a featural approach to similarity. The accuracy of the model is assessed by comparing its predictions to the probabilities provided by undergraduates asked to reason about mammals.     

Foundations of Probability

The foundations of probability are viewed through the lens of the subjectivist interpretation. This article surveys conditional probability, arguments for probabilism, probability dynamics, and the evidential and subjective interpretations of probability.     

Probability gate statistics

A probability gate presents a fixed but controllable transmission probability p to pulses arriving at its input. This paper describes some statistical properties of the transformation imposed by a gate on the input distribution of interpulse intervals. The output distribution of interpulse intervals is derived in terms of the input density and p. A simple relation is shown to hold between the moment generating functions, and the first four output moments are presented in terms of the input moments and p. Certain restrictions on the kinds of obtainable output distributions are discussed. In particular, a condition is established that specifies when the output density is a “replica” of the input distribution.     

Iterative Probability Kinematics

Following the pioneer work of Bruno De Finetti [12], conditional probability 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 [31], and Alfred Renyi's in [33]. Nonstandard probability spaces [34] are a well know alternative to this approach. Vann McGee proposed in [30] a result relating both approaches by showing that the standard values of infinitesimal probability functions are representable as Popper functions, and that every Popper function is representable in terms of the standard real values of some infinitesimal measure.Our main goal in this article is to study the constraints on (qualitative and probabilistic) 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 (definitionally) 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 principle that a proposition that is accepted at any stage of an iterative process of acceptance will continue to be accepted at any later stage. The plausibility and range of applicability of Cumulativity is then studied. In particular we appeal to a method for defining belief from conditional probability (first proposed in [42] and then slightly modified in [6] and [3]) in order to characterize the notion of qualitative change induced by Cumulative models of probability kinematics. The resulting cumulative notion is then compared with existing axiomatizations of belief change and probabilistic supposition. We also consider applications in the probabilistic accounts of conditionals [1] and [30].     

Resurrecting Logical Probability

The logical interpretation of probability, or ``objective Bayesianism'– the theory that (some) probabilitiesare strictly logical degrees of partial implication – is defended.The main argument against it is that it requires the assignment ofprior probabilities, and that any attempt to determine them by symmetryvia a ``principle of insufficient reason' inevitably leads to paradox.Three replies are advanced: that priors are imprecise or of little weight, sothat disagreement about them does not matter, within limits; thatit is possible to distinguish reasonable from unreasonable priorson logical grounds; and that in real cases disagreement about priorscan usually be explained by differences in the background information.It is argued also that proponents of alternative conceptions ofprobability, such as frequentists, Bayesians and Popperians, areunable to avoid committing themselves to the basic principles oflogical probability.     

概率与命题

本文的主题是揭示信念对象。关于信念对象最流行的观点是指称主义的观点,指称主义者或把个体与性质组成的命题视为信念对象,或把可能世界集视为信念对象,但若从贝叶斯的确证理论出发来考察信念对象的话,就会发现指称主义与贝叶斯主义间的冲突,由于贝叶斯主义是一种相当成功的理论,因而指称主义是错误的。贝叶斯主义需要的是一种非指称的信念对象,对贝叶斯理论而言,首内涵可以扮演置信对象这一角色。不过首内涵缺乏结构,缺乏指称信息,前者可以用借助于结构化首内涵来解决,后者可以在前者的基础上借助于丰富化内涵进而借助于丰富化的命题来解决。     

Probability and logic

The paper is an attempt to show that the formalism of subjective probability has a logical interpretation of the sort proposed by Frank Ramsey: as a complete set of constraints for consistent distributions of partial belief. Though Ramsey proposed this view, he did not actually establish it in a way that showed an authentically logical character for the probability axioms (he started the current fashion for generating probabilities from suitably constrained preferences over uncertain options). Other people have also sought to provide the probability calculus with a logical character, though also unsuccessfully. The present paper gives a completeness and soundness theorem supporting a logical interpretation: the syntax is the probability axioms, and the semantics is that of fairness (for bets).     

Subjunctive Conditional Probability

There seem to be two ways of supposing a proposition: supposing “indicatively” that Shakespeare didn’t write Hamlet, it is likely that someone else did; supposing “subjunctively” that Shakespeare hadn’t written Hamlet, it is likely that nobody would have written the play. Let P(B//A) be the probability of B on the subjunctive supposition that A. Is P(B//A) equal to the probability of the corresponding counterfactual, A □→B? I review recent triviality arguments against this hypothesis and argue that they do not succeed. On the other hand, I argue that even if we can equate P(B//A) with P(A □→B), we still need an account of how subjunctive conditional probabilities are related to unconditional probabilities. The triviality arguments reveal that the connection is not as straightforward as one might have hoped.