Probability models

Fintan Costello, Paul Watts

We test contrasting predictions of two recent models of proba-bility judgment: the quantum probability model (Busemeyeret al., 2011) and the probability theory plus noise model(Costello and Watts, 2014). Both models assume that peo-ple estimate probability using formal processes that follow orsubsume standard probability theory. The quantum probabil-ity model predicts people’s estimates should agree with oneset of probability theory identities, while the probability the-ory plus noise model predicts a specific pattern of violation ofthose identities. Experimental results show just the form of vi-olation predicted by the probability theory plus noise model.These results suggest that people’s probability judgments donot follow quantum probability: instead, they follow the rulesof standard probability theory, with the systematic biases seenin those judgments due to the effects of random noise.

Andrei Yu. Khrennikov

Our aim is to emphasize the role of mathematical models in physics, especially models of geometry and probability. We briefly compare developments of geometry and probability by pointing to similarities and differences: from Euclid to Lobachevsky and from Kolmogorov to Bell. In probability, Bell could play the same role as Lobachevsky in geometry. In fact, violation of Bell’s inequality can be treated as implying the impossibility to apply the classical probability model of Kolmogorov (1933) to quantum phenomena. Thus the quantum probabilistic model (based on Born’s rule) can be considered as the concrete example of the non-Kolmogorovian model of probability, similarly to the Lobachevskian model — the first example of the non-Euclidean model of geometry. This is the “probability model” interpretation of the violation of Bell’s inequality. We also criticize the standard interpretation—an attempt to add to rigorous mathematical probability models additional elements such as (non)locality and (un)realism. Finally, we compare embeddings of non-Euclidean geometries into the Euclidean space with embeddings of the non-Kolmogorovian probabilities (in particular, quantum probability) into the Kolmogorov probability space. As an example, we consider the CHSH-test.

G. E. H. Reuter

Wolfgang Muckenheim, Günther Ludwig, Christopher Dewdney +6 more

Peter D. Finch

Problems associated with setting up a probability model which generates the quantum theoretical probabilities for the two spin 1/2 particle system are examined. Arguments which claim to show that such a model cannot be constructed within classical probability theory under the assumption of local singlet states are also considered. It is shown that the model then in question is not a probability model in the sense that term is used elsewhere in science. An alternative model is proposed and its bearing on the Einstein-Bohr debate is briefly discussed.