Mario Zanotti

When are probabilistic explanations possible

The primary criterion of adequacy of a probabilistic causal analysis is that the causal variable should render the simultaneous phenomenological data conditionally independent. The intuition back of this idea is that the common cause of the phenomena should factor out the observed correlations. So we label the principle the common cause criterion. If we find that the barometric pressure and temperature are both dropping at the same time, we do not think of one as the cause of the other but look for a common dynamical cause within the physical theory of meteorology. If we find fever and headaches positively correlated, we look for a common disease as the source and do not consider one the cause of the other. But we do not want to suggest that satisfaction of this criterion is the end of the search for causes or probabilistic explanations. It does represent a significant and important milestone in any particular investigation.

On using random relations to generate upper and lower probabilities

For a variety of reasons there has been considerable interest in upper and lower probabilities as a generalization of ordinary probability. Perhaps the most evident way to motivate this generalization is to think of the upper and lower probabilities of an event as expressing bounds on the probability of the event. The most interesting case conceptually is the assignment of a lower probability of ~ r o and an upper probability of one to express maximum ignorance. Simplification of standard probability spaces is given by random variables that map one space into another and usually simpler space. For example, if we flip a coin a hundred times, the sample space describing the possible outcome of each flip consists of 21°° points, but by using the random variable that simply counts the number of heads in each sequence of a hundred flips we can construct a new space that contains only 101 points. Moreover, the random variable generates in a direct fashion the appropriate probability measure on the new space. What we set forth in this paper is a similar method for generating upper and lower probabilities by means of random relations. The generalization is a natural one; we simply pass from functions to relations, and the multivalued character of the relations leads in an obvious way to upper and lower probabilities. The generalization from random variables to random relations also provides a method for introducing a distinction between indeterminacy and uncertainty that we believe is new in the literature. Both of these concepts are defined in a purely set-theoretical way and thus do not depend, as they often do in informal discussions, on explicit probability considerations. Random variables, it should be noted, possess uncertainty but not indeterminacy. In this sense, the concept of indeterminacy is a generalization that goes strictly beyond ordinary probability theory, and thus provides a means of expressing the intuitions of those philosophers who are not satisfied with a purely probabilistic notion of indeterminacy. Section I is devoted to set-theoretical concepts. The initial developments

On the Determinism of Hidden Variable Theories with Strict Correlation and Conditional Statistical Independence of Observables

The main purpose of this note is to prove a lemma about random variables, and then to apply this lemma to the characterization of local theories of hidden variables by Bell (1964, 1966) and Wigner (1970), which are focused around Bell’s inequality. We use the results of the lemma in two different ways. The first is to show that the assumptions of Bell and Wigner can be weakened to conditional statistical independence rather than conditional determinism because determinism follows from conditional independence and the other assumptions that are made about systems of two spin−1/2 particles.

Performance models of American indian students on computer-assisted instruction in elementary mathematics

This investigation applied experimentally the use of predictive-control models integrated into computer-assisted instruction (CAI) as discussed earlier by Suppes, Fletcher, and Zanotti (1973). Many of those who are engaged in curriculum reform efforts have been dissatisfied with classical evaluations that simply compare the pre- and post-treatment achievement of experimental and control groups. It is natural to seek a more predictivecontrol approach that can be used as an integral part of the curriculum in order to ensure greater benefits, especially for students who are educationally disadvantaged or handicapped and for whom global performance models derived from standard populations are inappropriate. Such an approach was discussed by Suppes, Fletcher, and Zanotti (1973), who developed a theory by which the amount of time a student spends on a curriculum is a function of his progress, and his achievements in given course objectives, which are individually set for each student, are expressed as post-treatment grade placement (GP). Using the approach described by Suppes et al. (1973), we were able to achieve precise individualization of instruction both in the amount of instruction for each student and in the goal set for him. Further, the approach separates global features of the curriculum described by a simple differential equation from parameters that are characteristic of individual students.

Existence of hidden variables having only upper probabilities

We prove the existence of hidden variables, or, what we call generalized common causes, for finite sequences of pairwise correlated random variables that do not have a joint probability distribution. The hidden variables constructed have upper probability distributions that are nonmonotonic. The theorem applies directly to quantum mechanical correlations that do not satisfy the Bell inequalities.

Necessary and sufficient qualitative axioms for conditional probability

In a previous paper (Suppes and Zanotti, 1976) we gave simple necessary and sufficient qualitative axioms for the existence of a unique expectation function for the set of extended indicator functions. As we defined this set of functions earlier, it is the closure of the set of indicator functions of events under function addition. In the present paper we extend the same approach to conditional probability. One of the more troublesome aspects of the qualitative theory of conditional probability is that A [ B is not an object in particular it is not a new event composed somehow from events A and B. Thus the qualitative theory rests on a quaternary relation A [ B 2 CI D, which is read: event A given event B is at least as probable as event C given event D. There have been a number of attempts to axiomatize this quaternary relation (Koopman, 1940a, 1940b; Aczel, 1961, 1966, p. 319; Luce, 1968; Domotor, 1969; Krantz et al., 1971; and Suppes, 1973). The only one of these axiomatizations to address the problem of giving necessary and sufficient conditions is the work of Domotor, which approaches the subject in the finite case in a style similar to that of Scott (1964). By using indicator functions or, more generally, extended indicator functions, the difficulty of A I B not being an object is eliminated, for AZ IB is just the indicator function of the set A restricted to the set B, that is, AZ IB is a partial function whose domain is B. In similar fashion if X is an extended indicator function, X I A is that function restricted to the set A. The use of such partial functions requires care in formulating the algebra of functions in which we are interested, for functional addition X IA + Y [ B will not be well defined when A +B but A n B =t= 8. Thus, to be completely explicit we begin with a nonempty set Q, the probability space, and an algebra F of events, that is, subsets of Q, with it understood that 9 is closed under union and complementation. Next we extend this algebra to the algebra F* of extended indicator functions, that is, the smallest semigroup (under function addition) containing the indicator functions of all events in F. This latter algebra is now

New Bell-type inequalities forN > 4 necessary for existence of a hidden variable

The purpose of this paper is to extend Bells inequalities to obtain some general necessary conditions for the existence of a joint probability distribution for any finite collection of Bell-type random variables. Our results show that forN > 4 many new elementary inequalities beyond those of Bell must be satisfied by any hidden variable theory.

Causality and Symmetry

This paper is concerned with inferences from phenomenological variables to hidden causes or hidden variables. A number of theorems of a general sort are stated. The paper concludes with a treatment of Bell’s inequalities and their generalization to more than four observables.