A loose and separate certainty: Caves, Fuchs and Schack on quantum probability one

Carlton Caves, Fuchs, and Schack (2002) have recently appealed to an argument of mine (Stairs, 1983) to address a problem for their subjective Bayesian account of quantum probability. The difficulty is that on the face of it, quantum mechanical probabilities of one appear to be objective, but in that case, the Born Rule would yield a continuum of probabilities between zero and one. If so, we end up with objective probabilities strictly between zero and one. The authors claim that objective probabilities of one leads to a dilemma: give up locality or fall into contradiction. I argue that this conclusion depends on an overly strong interpretation of objectivism about quantum probabilities.     

Measurement outcomes and probability in Everettian quantum mechanics

The decision-theoretic account of probability in the Everett or many-worlds interpretation, advanced by David Deutsch and David Wallace, is shown to be circular. Talk of probability in Everett presumes the existence of a preferred basis to identify measurement outcomes for the probabilities to range over. But the existence of a preferred basis can only be established by the process of decoherence, which is itself probabilistic.     

Non-monotonic probability theory for n-state quantum systems

In previous work, a non-standard theory of probability was formulated and used to systematize interference effects involving the simplest type of quantum systems. The main result here is a self-contained, non-trivial generalization of that theory to capture interference effects involving a much broader range of quantum systems. The discussion also focuses on interpretive matters having to do with the actual/virtual distinction, non-locality, and conditional probabilities.     

Probability without certainty: foundationalism and the Lewis–Reichenbach debate

Like many discussions on the pros and cons of epistemic foundationalism, the debate between C. I. Lewis and H. Reichenbach dealt with three concerns: the existence of basic beliefs, their nature, and the way in which beliefs are related. In this paper we concentrate on the third matter, especially on Lewis’s assertion that a probability relation must depend on something that is certain, and Reichenbach’s claim that certainty is never needed. We note that Lewis’s assertion is prima facie ambiguous, but argue that this ambiguity is only apparent if probability theory is viewed within a modal logic. Although there are empirical situations where Reichenbach is right, and others where Lewis’s reasoning seems to be more appropriate, it will become clear that Reichenbach’s stance is the generic one. We conclude that this constitutes a threat to epistemic foundationalism.     

Quantum probability and many worlds

We discuss the meaning of probabilities in the many worlds interpretation of quantum mechanics. We start by presenting very briefly the many worlds theory, how the problem of probability arises, and some unsuccessful attempts to solve it in the past. Then we criticize a recent attempt by Deutsch to derive the quantum mechanical probabilities from the non-probabilistic parts of quantum mechanics and classical decision theory. We further argue that the Born probability does not make sense even as an additional probability rule in the many worlds theory. Our conclusion is that the many worlds theory fails to account for the probabilistic statements of standard (collapse) quantum mechanics.     

Subjective confidence in forecasts

Forecasts have little value to decision makers unless it is known how much confidence to place in them. Those expressions of confidence have, in turn, little value unless forecasters are able to assess the limits of their own knowledge accurately Previous research has shown very robust patterns in the judgements of individuals who have not received special training in confidence assessment: Knowledge generally increases as confidence increases. However, it increases too swiftly, with a doubling of confidence being associated with perhaps a 50 per cent increase in knowledge. With all but the easiest of tasks, people tend to be overconfident regarding how much they know These results have typically been derived from studies of judgements of general knowledge. The present study found that they also pertained to confidence in forecasts. Indeed, the confidence-knowledge curves observed here were strikingly similar to those observed previously. The only deviation was the discovery that a substantial minority of judges never expressed complete confidence in any of their forecasts. These individuals also proved to be better assessors of the extent of their own knowledge Apparently confidence in forecasts is determined by processes similar to those that determine confidence in general knowledge. Decision makers can use forecasters assessments in a relative sense, in order to predict when they are more and less likely to be correct. However, they should be hesitant to take confidence assessments literally. Someone is more likely to be right when he or she is ‘certain’than when he or she is ‘fairly confident’; but there is no guarantee that the certain forecast will come true.     

Uncertainty and probability for branching selves

Everettian accounts of quantum mechanics entail that people branch; every possible result of a measurement actually occurs, and I have one successor for each result. Is there room for probability in such an account? The prima facie answer is no; there are no ontic chances here, and no ignorance about what will happen. But since any adequate quantum mechanical theory must make probabilistic predictions, much recent philosophical labor has gone into trying to construct an account of probability for branching selves. One popular strategy involves arguing that branching selves introduce a new kind of subjective uncertainty. I argue here that the variants of this strategy in the literature all fail, either because the uncertainty is spurious, or because it is in the wrong place to yield probabilistic predictions. I conclude that uncertainty cannot be the ground for probability in Everettian quantum mechanics.     

Objective probability and the mind-body relation

Objectiveprobability in quantum mechanics is often thought to involve a stochastic process whereby an actual future is selected from a range of possibilities. Everett's seminal idea is that all possible definite futures on the pointer basis exist as components of a macroscopic linear superposition. I demonstrate that these two conceptions of what is involved in quantum processes are linked via two alternative interpretations of the mind-body relation. This leads to a fission, rather than divergence, interpretation of Everettian theory and to a novel explanation of why a principle of indifference does not apply to self-location uncertainty for a post-measurement, pre-observation subject, just as Sebens and Carroll claim. Their Epistemic Separability Principle is shown to arise out of this explanation and the derivation of the Born rule for Everettian theory is thereby put on a firmer footing.     

Acceptance,values, and probability

This essay makes a case for regarding personal probabilities used in Bayesian analyses of confirmation as objects of acceptance and rejection. That in turn entails that personal probabilities are subject to the argument from inductive risk, which aims to show non-epistemic values can legitimately influence scientific decisions about which hypotheses to accept. In a Bayesian context, the argument from inductive risk suggests that value judgments can influence decisions about which probability models to accept for likelihoods and priors. As a consequence, if the argument from inductive risk is sound, then non-epistemic values can affect not only the level of evidence deemed necessary to accept a hypothesis but also degrees of confirmation themselves.     

Subjective intervention in formal models

Success in forecasting using mathematical/statistical models requires that the models be open to intervention by the user. In practice, a model is only one component of a forecasting system, which also includes the users/forecasters as integral components. Interaction between the user and the model is necessary to adequately cater for events and changes that go beyond the existing form of the model. In this paper we consider Bayesian forecasting models open to interventions, of essentially any form, to incorporate subjective information made available to the user. We discuss principles of intervention and derive theoretical results that provide the means to formally incorporate feedforward interventions into Bayesian models. Two example time series are considered to illustrate why and when such interventions may be necessary to sustain predictive performance.     

Newton and the classical theory of probability

Summary Probabilistic ideas and methods from Newton's writings are discussed in § 1: Newton's ideas pertaining to the definition of probability, his probabilistic method in chronology, his probabilistic ideas and method in the theory of errors and his probabilistic reasonings on the system of the world. Newton's predecessors and his influence upon subsequent scholars are dealt with in §2: beginning with his predecessors the discussion continues with his contemporaries Arbuthnot and De Moiver, then Bentley. The section ends with Laplace, whose determinism is seen as a development of the Newtonian determinism.An addendum is devoted to Lambert's reasoning on randomness and to the influence of Darwin on statistics. A synopsis is attached at the end of the article.Abbreviations PT abridged Philosophical Transactions of the Royal Society 1665–1800 abridged. London, 1809 - Todhunter I. Todhunter, History of the mathematical theory of probability, Cambridge, 1865 To the memory of my mother, Sophia Sheynin (1900–1970)     

An empirical approach to symmetry and probability

We often rely on symmetries to infer outcomes’ probabilities, as when we infer that each side of a fair coin is equally likely to come up on a given toss. Why are these inferences successful? I argue against answering this question with an a priori indifference principle. Reasons to reject such a principle are familiar, yet instructive. They point to a new, empirical explanation for the success of our probabilistic predictions. This has implications for indifference reasoning generally. I argue that a priori symmetries need never constrain our probability attributions, even for initial credences.     

Quantum probability from subjective likelihood: Improving on Deutsch's proof of the probability rule

I present a proof of the quantum probability rule from decision-theoretic assumptions, in the context of the Everett interpretation. The basic ideas behind the proof are those presented in Deutsch's recent proof of the probability rule, but the proof is simpler and proceeds from weaker decision-theoretic assumptions. This makes it easier to discuss the conceptual ideas involved in the proof, and to show that they are defensible.     

Boltzmann's probability distribution of 1877

Communicated by R. Stuewer     

Non-integrability and mixing in quantum systems: On the way to quantum chaos

In spite of the increasing attention that quantum chaos has received from physicists in recent times, when the subject is considered from a conceptual viewpoint the usual opinion is that there is some kind of conflict between quantum mechanics and chaos. In this paper we follow the program of Belot and Earman, who propose to analyze the problem of quantum chaos as a particular case of the classical limit of quantum mechanics. In particular, we address the problem on the basis of our account of the classical limit, which in turn is grounded on the self-induced approach to decoherence. This strategy allows us to identify the conditions that a quantum system must satisfy to lead to non-integrability and to mixing in the classical limit.