Pearle's Hidden-Variable Model Revisited

Pearle (1970) gave an example of a local hidden variables model which exactly reproduced the singlet correlations of quantum theory, through the device of data-rejection: particles can fail to be detected in a way which depends on the hidden variables carried by the particles and on the measurement settings. If the experimenter computes correlations between measurement outcomes of particle pairs for which both particles are detected, he is actually looking at a subsample of particle pairs, determined by interaction involving both measurement settings and the hidden variables carried in the particles. We correct a mistake in Pearle's formulas (a normalization error) and more importantly show that the model is more simple than first appears. We illustrate with visualisations of the model and with a small simulation experiment, with code in the statistical programming language R included in the paper. Open problems are discussed.Comment: 19pp. This is now arXiv version 4 = final revision for journa

Statistics, ethics, and probiotica

A randomized clinical trial comparing an experimental new treatment to a standard therapy for a life-threatening medical condition should be stopped early on ethical grounds, in either of the following scenarios: (1) it has become overwhelmingly clear that the new treatment is better than the standard; (2) it has become overwhelmingly clear that the trial is not going to show that the new treatment is any better than the standard. The trial is continued in the third scenario: (3) there is a reasonable chance that the new treatment will finally turn out to be better than the standard, but we aren't sure yet. However, the (blinded) data monitoring committee in the "PROPATRIA" trial of an experimental probiotica treatment for patients with acute pancreatitis allowed the trial to continue at the half way interim analysis, in effect because there was still a good chance of proving that the probiotica treatment was very harmful to their patients. The committee did not know whether treatment A was the probiotica treatment or the placebo. In itself this should not have caused a problem, since it could easily have determined the appropriate decision under both scenarios. Were the decisions in the two scenarios different, then the data would have to be de-blinded, in order to determine the appropriate decision. The committee mis-read the output of SPSS, which reports the smaller of two one-sided p-values, without informing the user what it is doing. It seems that about 5 lives were sacrificed to the chance of getting a significant result that the probiotica treatment was bad for the patients in the trial

A nonparametric Bayesian approach to the rare type match problem

The "rare type match problem" is the situation in which the suspect's DNA profile, matching the DNA profile of the crime stain, is not in the database of reference. The evaluation of this match in the light of the two competing hypotheses (the crime stain has been left by the suspect or by another person) is based on the calculation of the likelihood ratio and depends on the population proportions of the DNA profiles, that are unknown. We propose a Bayesian nonparametric method that uses a two-parameter Poisson Dirichlet distribution as a prior over the ranked population proportions, and discards the information about the names of the different DNA profiles. This fits very well the data coming from European Y-STR DNA profiles, and the calculation of the likelihood ratio becomes quite simple thanks to a justified Empirical Bayes approach.Comment: arXiv admin note: text overlap with arXiv:1506.0844

Product-limit estimators of the gap time distribution of a renewal process under different sampling patterns

Nonparametric estimation of the gap time distribution in a simple renewal process may be considered a problem in survival analysis under particular sampling frames corresponding to how the renewal process is observed. This note describes several such situations where simple product limit estimators, though inefficient, may still be useful

On an Argument of David Deutsch

We analyse an argument of Deutsch, which purports to show that the deterministic part of classical quantum theory together with deterministic axioms of classical decision theory, together imply that a rational decision maker behaves as if the probabilistic part of quantum theory (Born's law) is true. We uncover two missing assumptions in the argument, and show that the argument also works for an instrumentalist who is prepared to accept that the outcome of a quantum measurement is random in the frequentist sense: Born's law is a consequence of functional and unitary invariance principles belonging to the deterministic part of quantum mechanics. Unfortunately, it turns out that after the necessary corrections we have done no more than give an easier proof of Gleason's theorem under stronger assumptions. However, for some special cases the proof method gives positive results while using different assumptions to Gleason. This leads to the conjecture that the proof could be improved to give the same conclusion as Gleason under unitary invariance together with a much weaker functional invariance condition.Comment: Revision 28-7-03: added reference Final revision 28-05-04. To appear in proceedings of "Quantum Probability and Infinite Dimensional Analysis", Greifswald, 2003; World Scientifi

The triangle wave versus the cosine: How classical systems can optimally approximate EPR-B correlations

The famous singlet correlations of a composite quantum system consisting of two spatially separated components exhibit notable features of two kinds. The first kind consists of striking certainty relations: perfect correlation and perfect anti-correlation in certain settings. The second kind consists of a number of symmetries, in particular, invariance under rotation, as well as invariance under exchange of components, parity, or chirality. In this note, I investigate the class of correlation functions that can be generated by classical composite physical systems when we restrict attention to systems which reproduce the certainty relations exactly, and for which the rotational invariance of the correlation function is the manifestation of rotational invariance of the underlying classical physics. I call such correlation functions classical EPR-B correlations. It turns out that the other three (binary) symmetries can then be obtained "for free": they are exhibited by the correlation function, and can be imposed on the underlying physics by adding an underlying randomisation level. We end up with a simple probabilistic description of all possible classical EPR-B correlations in terms of a "spinning coloured disk" model, and a research programme: describe these functions in a concise analytic way. We survey open problems, and we show that the widespread idea that "quantum correlations are more extreme than classical physics allows" is at best highly inaccurate, through giving a concrete example of a classical correlation which satisfies all the symmetries and all the certainty relations and which exceeds the quantum correlations over a whole range of settingsComment: This version, arXiv:1312.6403v.6, as accepted by "Entropy" 27 February 202