811,119 research outputs found
On classical models of spin
The reason for recalling this old paper is the ongoing discussion on the
attempts of circumventing certain assumptions leading to the Bell theorem
(Hess-Philipp, Accardi). If I correctly understand the intentions of these
Authors, the idea is to make use of the following logical loophole inherent in
the proof of the Bell theorem: Probabilities of counterfactual events A and A'
do not have to coincide with actually measured probabilities if measurements of
A and A' disturb each other, or for any other fundamental reason cannot be
performed simulaneously. It is generally believed that in the context of
classical probability theory (i.e. realistic hidden variables) probabilities of
counterfactual events can be identified with those of actually measured events.
In the paper I give an explicit counterexample to this belief. The "first
variation" on the Aerts model shows that counterfactual and actual problems
formulated for the same classical system may be unrelated. In the model the
first probability does not violate any classical inequality whereas the second
does. Pecularity of the Bell inequality is that on the basis of an in principle
unobservable probability one derives probabilities of jointly measurable random
variables, the fact additionally obscuring the logical meaning of the
construction. The existence of the loophole does not change the fact that I was
not able to construct a local model violating the inequality with all the other
loopholes eliminated.Comment: published as Found. Phys. Lett. 3 (1992) 24
On the relation between quantum mechanical probabilities and event frequencies
The probability `measure' for measurements at two consecutive moments of time
is non-additive. These probabilities, on the other hand, may be determined by
the limit of relative frequency of measured events, which are by nature
additive. We demonstrate that there are only two ways to resolve this problem.
The first solution places emphasis on the precise use of the concept of
conditional probability for successive measurements. The physically correct
conditional probabilities define additive probabilities for two-time
measurements. These probabilities depend explicitly on the resolution of the
physical device and do not, therefore, correspond to a function of the
associated projection operators. It follows that quantum theory distinguishes
between physical events and propositions about events, the latter are not
represented by projection operators and that the outcomes of two-time
experiments cannot be described by quantum logic.
The alternative explanation is rather radical: it is conceivable that the
relative frequencies for two-time measurements do not converge, unless a
particular consistency condition is satisfied. If this is true, a strong
revision of the quantum mechanical formalism may prove necessary. We stress
that it is possible to perform experiments that will distinguish the two
alternatives.Comment: 16 pages LATEX. Minor corrections, final version to appear in Ann.
Phy
Bayes and health care research.
Bayes’ rule shows how one might rationally change one’s beliefs in the light of evidence. It is the foundation of a statistical method called Bayesianism. In health care research, Bayesianism has its advocates but the dominant statistical method is frequentism.
There are at least two important philosophical differences between these methods. First, Bayesianism takes a subjectivist view of probability (i.e. that probability scores are statements of subjective belief, not objective fact) whilst frequentism takes an objectivist view. Second, Bayesianism is explicitly inductive (i.e. it shows how we may induce views about the world based on partial data from it) whereas frequentism is at least compatible with non-inductive views of scientific method, particularly the critical realism of Popper.
Popper and others detail significant problems with induction. Frequentism’s apparent ability to avoid these, plus its ability to give a seemingly more scientific and objective take on probability, lies behind its philosophical appeal to health care researchers.
However, there are also significant problems with frequentism, particularly its inability to assign probability scores to single events. Popper thus proposed an alternative objectivist view of probability, called propensity theory, which he allies to a theory of corroboration; but this too has significant problems, in particular, it may not successfully avoid induction. If this is so then Bayesianism might be philosophically the strongest of the statistical approaches. The article sets out a number of its philosophical and methodological attractions. Finally, it outlines a way in which critical realism and Bayesianism might work together.
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Existence of the signal in the signal plus background model
Searching for evidence of neutrino oscillations is an important problem in
particle physics. Suppose that evidence for neutrino oscillations from an LSND
experiment reports a significant positive oscillation probability, but that the
LSND result is not confirmed by other experiments. In statistics, such a
problem can be proposed as the detection of signal events in the Poisson signal
plus background model. Suppose that an observed count is of the form
, where the background and the signal are independent Poisson
random variables with parameters and respectively, is known
but is not. Some recent articles have suggested conditioning on the
observed bound for ; that is, if is observed, the suggestion is to
base the inference on the conditional distribution of given . This
suggestion is used here to derive an estimator of the probability of the
existence of the signal event. The estimator is examined from the view of
decision theory and is shown to be admissible.Comment: Published at http://dx.doi.org/10.1214/074921706000000653 in the IMS
Lecture Notes--Monograph Series
(http://www.imstat.org/publications/lecnotes.htm) by the Institute of
Mathematical Statistics (http://www.imstat.org
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