STATISTICAL THEORIES

Briefly, a statistical theory [Appendix n, §2] contains a set of experimental propositions, P ; a set of states, S ; and a mapping, p , from PxS to the

interval [0,1] ; such that the quantity p(A,X) is predicted by the theory as the probability that proposition A is verified if tested on state X . A deterministic theory is a special case of a statistical theory, with p(A,A.) 6 {0,1} for all A e P , X e S . Elements of this characterisation are examined below.

Propositions

Experimental propositions essentially correspond to yes/no experiments which may be performed on members of the class of systems described by the theory. The two possible results of testing proposition A e P may be labelled by 1 and 0 , corresponding respectively to verification (a "yes" result) and falsification (a "no" result). It is not necessary that P contains all possible propositions (i.e., yes/no experiments) for the class of systems under consideration (this would amount to a "completeness" condition for the theory).

equivalent with respect to the theory if the predictions of the theory do not distinguish between them, i.e., if p(A,X) = p(B,A.) for all states \ . Equivalence is denoted by A = B . It follows that propositions may be represented up to equivalence by mappings, from the set of states to the interval [0,1] , where for A e P the mapping A is defined by

A(k) := p(A,A.) (2.1)

for all X e S . These mappings will be called the abstract propositions of the theory, and the set of such propositions denoted by 7 . Note that this improves on the notation of Hall [1988], where the distinction between P and 7 is made by context. The relations A = B , A = B are equivalent. In general a statistical theory will be denoted by (S,P,p) , although the incomplete representation (S,7) may also be used in what follows.

States

States are sometimes regarded as descriptions of ensembles (e.g., in the "statistical interpretation" of quantum mechanics [Ballentine, 1970]). However, in what follows I wish to regard states as descriptions of individual systems. The connection between the two types of state is the notion of a preparation procedure. An ensemble state may be identified with the preparation procedure which generates a "random" member of that ensemble, while a system state may be identified with the preparation procedure which generates a system in that state. The equivalence of ensemble and system state descriptions lies in the fact that only so-called "random" members of the former are used in physical predictions. The main conceptual advantage of single system state descriptions lies in the fact that experiments are not in general performed on ensembles, but on individual systems. A second advantage is that discussion of the word "random" may be avoided.

Probability

The definition of statistical theories above inherits a weakness from standard probability theory, in that the meaning of the word "probability" in connection with the quantity p(A,X) is not defined empirically, but taken as a primitive concept. This should somehow be resolved in a physical context to make predictions of the theory testable. In §2 of Appendix II I consider a sequence of results r(A .A^) , r(A2,^2),...a’(AN,A^f) (where r(A,A) ) is 1 or 0 if A is respectively verified or falsified on X ), and propose that a statistical theory predicts the behaviour

^theoretical ^ N as N -.<*>, (2.2a) where N yes N := X r(A i ) i=l (2.2b)

is the total number of verifications, and

^theoretical yes N := X p ( A A . ) . i= l (2.2c)

In the case p(A.,A.) = p for each value of i , equations (2.2) imply N = pN

i i yes

for sufficiently large N , justifying the interpretation of p(A,A) as a probability. However, the question "how large is sufficiently large?" remains (this question also applies in particular to standard quantum mechanics — e.g., how many photons determine an interference pattem?).

Graham [1973], in the context of the many-worlds interpretation of quantum mechanics, and more generally Finkelstein [1972] note that statistical predictions for single-system measurements may be transformed into deterministic predictions for ensemble measurements. In particular, if a proposition is tested on each member of an ensemble, then the probability that the relative frequency of verifications has a particular value is either 0 or 1 . Thus statistical predictions involving individual systems become deterministic predictions about ensembles.

However, this is unsatisfactory for assigning an empirical meaning to probabilities, as in practice the "ensemble limit" N — can not be realised. At best only finite subensembles are available for experimentation, the behaviour of which is not deterministic in general.

Before leaving (unresolved) the question of how probabilities are to be physically interpreted, I shall mention briefly the interpretation of de Finetti [1974; see especially §§3, 5, 7] based on the idea that probability is subjective. Probabilities are taken to reflect a degree of belief of the occurrence of an event, and vary according to the individual and available information. This information may include observed frequencies, and the connection of subjective probabilities with relative frequencies is provided by a suitable inteipretation of the law of large numbers [de Finetti, 1974; §7.5.6]. I shall not attempt to provide here an adequate discussion of de Finetti’s viewpoint. However, I feel that the notion of purely subjective probability cannot provide a compelling explanation for the existence of (successful) statistical theories such as quantum mechanics, which organise a wide range of physical phenomena into an objective structure. To interpret statistical theories as merely generating "probabilities", which may or may not agree with an individual's own subjective calculations, appears to ignore this organisational structure (unless perhaps this structure is interpreted as reflecting the common thought structure of "subjective" individuals!).

One further point must still be made about the characterisation of statistical theories given at the beginning of this section. In particular, a probability is assigned to an individual event (measurement of A on X ) rather than to "many" events. This is justified by equations (2.2): while the probabilistic nature of the event is evident only within the context of a large number of events, the contribution p(A.,A.) appears as an independent term in the expression for Nyes°retlCal ’ re^ ect^n§ individual nature of the contribution.