Prior Probabilities

Jaynes, E. T.

doi:10.1109/tssc.1968.300117

Cited by 1,426 publications

(853 citation statements)

References 9 publications

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“…The former, which underlies the definition of a state as a collection of expectation values, allows us to speak of probabilities before constructing the full theory of quantum measurement, while the frequency interpretation will support the solution of the measurement problem (see section 11.3.1). A similar bridge between the two interpretations is found in the purely classical set-up of selecting the non-informative prior probability distribution, the most controversial aspect of the Bayesian statistics [335,336,337] 106 .…”

Section: Paul Valérymentioning

confidence: 61%

Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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The quantum measurement problem, to wit, understanding why a unique outcome is obtained in each individual experiment, is currently tackled by solving models. After an introduction we review the many dynamical models proposed over the years for elucidating quantum measurements. The approaches range from standard quantum theory, relying for instance on quantum statistical mechanics or on decoherence, to quantum-classical methods, to consistent histories and to modifications of the theory. Next, a flexible and rather realistic quantum model is introduced, describing the measurement of the z-component of a spin through interaction with a magnetic memory simulated by a Curie-Weiss magnet, including N 1 spins weakly coupled to a phonon bath. Initially prepared in a metastable paramagnetic state, it may transit to its up or down ferromagnetic state, triggered by its coupling with the tested spin, so that its magnetization acts as a pointer. A detailed solution of the dynamical equations is worked out, exhibiting several time scales. Conditions on the parameters of the model are found, which ensure that the process satisfies all the features of ideal measurements. Various imperfections of the measurement are discussed, as well as attempts of incompatible measurements. The first steps consist in the solution of the Hamiltonian dynamics for the spin-apparatus density matrixD(t). Its off-diagonal blocks in a basis selected by the spin-pointer coupling, rapidly decay owing to the many degrees of freedom of the pointer. Recurrences are ruled out either by some randomness of that coupling, or by the interaction with the bath. On a longer time scale, the trend towards equilibrium of the magnet produces a final stateD(t f ) that involves correlations between the system and the indications of the pointer, thus ensuring registration. AlthoughD(t f ) has the form expected for ideal measurements, it only describes a large set of runs. Individual runs are approached by analyzing the final states associated with all possible subensembles of runs, within a specified version of the statistical interpretation. There the difficulty lies in a quantum ambiguity: There exist many incompatible decompositions of the density matrixD(t f ) into a sum of sub-matrices, so that one cannot infer from its sole determination the states that would describe small subsets of runs. This difficulty is overcome by dynamics due to suitable interactions within the apparatus, which produce a special combination of relaxation and decoherence associated with the broken invariance of the pointer. Any subset of runs thus reaches over a brief delay a stable state which satisfies the same hierarchic property as in classical probability theory; the reduction of the state for each individual run follows. Standard quantum statistical mechanics alone appears sufficient to explain the occurrence of a unique answer in each run and the emergence of classicality in a measurement process. Finally, pedagogical exercises are proposed and lessons for future works on m...

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Section: Paul Valérymentioning

confidence: 61%

Understanding quantum measurement from the solution of dynamical models

Allahverdyan¹,

Balian²,

2013

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“…Monitoring one's own prior experiences provides important information about the probability that certain acts and behaviors do not lead to successful outcomes and increases the probability that children will incorporate alternative means through observation of the acts of others. This is compatible with a causal Bayes net framework (Jaynes, 1983;Pearl, 1988;Russell & Norvig, 2002;Tenenbaum, Griffiths, & Kemp, 2006), which provides a mathematical formalism for combining information from one's own interventions (the prior experience in the current experiments) with information from observation (the adult's demonstration in the current experiments). The findings reported here provide clear evidence that young children seamlessly integrate interventions they perform with interventions they see performed by others to infer the most causally efficacious solution to a novel task, thus contributing developmental findings to the causal Bayes net literature (Gopnik & Schulz, 2007;Shon, Storz, Meltzoff, & Rao, 2007).…”

Section: Discussionmentioning

confidence: 75%

Prior experiences and perceived efficacy influence 3-year-olds' imitation.

Williamson¹,

Meltzoff²,

Markman³

2008

Developmental Psychology

115

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Children are selective and flexible imitators. They combine their own prior experiences and the perceived causal efficacy of the model to determine whether and what to imitate. In Experiment 1, children were randomly assigned to have either a difficult or an easy experience achieving a goal. They then saw an adult use novel means to achieve the goal. Children with a difficult prior experience were more likely to imitate the adult's precise means. Experiment 2 showed further selectivitychildren preferentially imitated causally efficacious versus nonefficacious acts. In Experiment 3, even after an easy prior experience led children to think their own means would be effective, they still encoded the novel means performed by the model. When a subsequent manipulation rendered the children's means ineffective, children recalled and imitated the model's means. The research shows that children integrate information from their own prior interventions and their observations of others to guide their imitation.

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“…Thus the scenario is still "impoverished" and leads to a similar pattern of choices as observed in the original version. Indeed, equal probability distributions are considered to be one of the least informative distributions in probability theory (Jaynes, 1968).…”

Section: Discussionmentioning

confidence: 99%

The Ellsberg 'Problem' and Implicit Assumptions under Ambiguity

Güney

Newell

2011

PsycEXTRA Dataset

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Empirical research has revealed that people try to avoid ambiguity in the Ellsberg problem and make choices inconsistent with the predictions of Expected Utility Theory. We hypothesized that people might be forming implicit assumptions to deal with the ambiguity resulting from the incomplete information in the problem, and that some assumptions might lead them to deviate from normative predictions. We embedded the Ellsberg problem in various scenarios that made one source of ambiguity (i.e., the implied distribution of the unknown number of the colored balls) explicit. Results of an experiment showed that more people chose consistently (and hence rationally) when the scenario encouraged them to think that the probability distribution of the number of balls was normal. The results give insight into the implicit assumptions that might lead to choices congruent with normative models.

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Prior Probabilities

Cited by 1,426 publications

References 9 publications

Understanding quantum measurement from the solution of dynamical models

Understanding quantum measurement from the solution of dynamical models

Prior experiences and perceived efficacy influence 3-year-olds' imitation.

The Ellsberg 'Problem' and Implicit Assumptions under Ambiguity

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