Probability in the Physical Sciences

Recently advocates of the propensity interpretation of fitness have turned critics. To accommodate examples from the population genetics literature they conclude that fitness is better defined broadly as a family of propensities rather than the propensity to contribute descendants to some future generation. We argue that the propensity theorists have misunderstood the deeper ramifications of the examples they cite. These examples demonstrate why there are factors outside of propensities that determine fitness. We go on to argue for the more general (...) thesis that no account of fitness can satisfy the desiderata that have motivated the propensity account. (shrink)

There exist several phenomena breaking the classical probability laws. The systems related to such phenomena are context-dependent, so that they are adaptive to other systems. In this paper, we present a new mathematical formalism to compute the joint probability distribution for two event-systems by using concepts of the adaptive dynamics and quantum information theory, e.g., quantum channels and liftings. In physics the basic example of the context-dependent phenomena is the famous double-slit experiment. Recently similar examples have been found in biological (...) and psychological sciences. Our approach is an extension of traditional quantum probability theory, and it is general enough to describe aforementioned contextual phenomena outside of quantum physics. (shrink)

In the absence of a fundamental theory that precisely predicts values for observable parameters, anthropic reasoning attempts to constrain probability distributions over those parameters in order to facilitate the extraction of testable predictions. The utility of this approach has been vigorously debated of late, particularly in light of theories that claim we live in a multiverse, where parameters may take differing values in regions lying outside our observable horizon. Within this cosmological framework, we investigate the efficacy of top-down anthropic reasoning (...) based on the weak anthropic principle. We argue contrary to recent claims that it is not clear one can either dispense with notions of typicality altogether or presume typicality, in comparing resulting probability distributions with observations. We show in a concrete, top-down setting related to dark matter, that assumptions about typicality can dramatically affect predictions, thereby providing a guide to how errors in reasoning regarding typicality translate to errors in the assessment of predictive power. We conjecture that this dependence on typicality is an integral feature of anthropic reasoning in broader cosmological contexts, and argue in favour of the explicit inclusion of measures of typicality in schemes invoking anthropic reasoning, with a view to extracting predictions from multiverse scenarios. (shrink)

This paper relates both to the metaphysics of probability and to the physics of time asymmetry. Using the formalism of decoherent histories, it investigates whether intuitions about intrinsic time directedness that are often associated with probability can be justified in the context of no-collapse approaches to quantum mechanics. The standard approach to time symmetry in the decoherent histories literature is criticised, and an alternative approach is proposed, based on two decoherence conditions within the one-vector formalism. In turn, considerations of forwards (...) and backwards decoherence and of decoherence and recoherence suggest that a time-directed interpretation of probabilities, if adopted, should be both contingent and perspectival. (shrink)

In any probabilistic theory, we say that a bipartite state ω on a composite system AB steers its marginal state ω B if, for any decomposition of ω B as a mixture ω B =∑ i p i β i of states β i on B, there exists an observable {a i } on A such that the conditional states $\omega_{B|a_{i}}$ are exactly the states β i . This is always so for pure bipartite states in quantum mechanics, a fact (...) first observed by Schrödinger in 1935. Here, we show that, for weakly self-dual state spaces (those isomorphic, but perhaps not canonically isomorphic, to their dual spaces), the assumption that every state of a system A is steered by some bipartite state of a composite AA consisting of two copies of A, amounts to the homogeneity of the state cone. If the state space is actually self-dual, and not just weakly so, this implies (via the Koecher-Vinberg Theorem) that it is the self-adjoint part of a formally real Jordan algebra, and hence, quite close to being quantum mechanical. (shrink)

We are inundated by scientific and statistical information, but what should we believe? How much should we trust the polls on the latest electoral campaign? When a physician tells us that a diagnosis of cancer is 90% certain or a scientist informs us that recent studies support global warming, what should we conclude? How can we acquire reliable statistical information? Once we have it, how do we evaluate it? Despite the importance of these questions to our lives, many of us (...) have only a vague idea of how to answer them. In this admirably clear and engaging book, Mark Battersby provides a practical guide to thinking critically about scientific and statistical information. The goal of the book is not only to explain how to identify misleading statistical information, but also to give readers the understanding necessary to evaluate and use statistical and statistically based scientific information in their own decision making. (shrink)

We are inundated by scientific and statistical information, but what should we believe? How much should we trust the polls on the latest electoral campaign? When a physician tells us that a diagnosis of cancer is 90% certain or a scientist informs us that recent studies support global warming, what should we conclude? How can we acquire reliable statistical information? Once we have it, how do we evaluate it? Despite the importance of these questions to our lives, many of us (...) have only a vague idea of how to answer them. In this admirably clear and engaging book, Mark Battersby provides a practical guide to thinking critically about scientific and statistical information. The goal of the book is not only to explain how to identify misleading statistical information, but also to give readers the understanding necessary to evaluate and use statistical and statistically based scientific information in their own decision making. (shrink)

We discuss a generalization of the standard notion of probability space and show that the emerging framework, to be called operational probability theory, can be considered as underlying quantal theories. The proposed framework makes special reference to the convex structure of states and to a family of observables which is wider than the familiar set of random variables: it appears as an alternative to the known algebraic approach to quantum probability.

Emch, G.G., Liu, C.: The Logic of Thermostatistical Physics. Springer, Berlin/ Heidelberg (2002) 11. Frigg, R., Werndl, C.: Entropy – a guide for the perplexed. Forthcoming in: Beisbart, C., Hartmann, S. (eds.) Probabilities in Physics. Oxford  ...

Recent discussions of theorigins of the thermodynamical temporal asymmetry (thearrow of time) by Huw Price and others arecritically assessed. This serves as amotivation for consideration of relationshipbetween thermodynamical and cosmologicalcauses. Although the project of clarificationof the thermodynamical explanandum is certainlywelcome, Price excludes another interestingoption, at least as viable as the sort ofAcausal-Particular approach he favors, andarguably more in the spirit of Boltzmannhimself. Thus, the competition of explanatoryprojects includes three horses, not two. Inaddition, it is the Acausal-Particular approachthat could benefit enormously (...) from dissociationfrom fanciful ideas of low-entropy futureboundary conditions entertained by Price. Novelrevolutionary developments in observationalcosmology, as well as in the nascentastrophysical discipline of physicaleschatology, have obliterated such hypotheses.Also, the Acausal-Anthropic approach wepropose, offers another clear instance ofdisteleological nature of the anthropicprinciple. (shrink)

The history of the mathematical probability includes two phases: 1) From Pascal and Fermat to Laplace, the theory gained in application fields; 2) In the first half of the 20th Century, two competing axiomatic systems were respectively proposed by von Mises in 1919 and Kolmogorov in 1933. This paper places this historical sketch in the context of the philosophical complexity of the probability concept and explains the resounding success of Kolmogorov’s theory through its ability to avoid direct interpretation. Indeed, unlike (...) experimental sciences, and despite its numerous applications, probability theory cannot be tested per se. Rather it relates to practical matters by means of transition hypotheses or bridging principles that match the structure of practical problems with abstract theory. In this respect probability theory has a very special status among scientific disciplines. (shrink)

It is an experimental fact that \ -decays produce in a cloud chamber at most one track and that this track points in a random direction. This seems to contradict the description of decay in Quantum Mechanics: according to Gamow a spherical wave is produced and moves radially according to Schrödinger’s equation. It is as if the interaction with the supersaturated vapor turned the wave into a particle. The aim of this note is to place this effect in the context (...) of Schrödinger’s Quantum Mechanics. We shall see that the properties of the initial wave function suggest the introduction of a semiclassical formalism in which the \ -wave can be described as a collection of semiclassical wavelets; each of them interacts with an atom and forms an entangled state. The interaction can be regarded as a semiclassical inelastic scattering event. The measurement selects the wave function of one of the ions, with a probability given by Born’s rule. This ion interacts with the atoms nearby leading to the formation of a droplet. One can reasonably assume that also the wavelet entangled with the selected ion has probability one to remain as part of the description of the system. The measurement process is therefore represented by a unitary operator. The wavelet remains sharply localized on a classical path \ It is still a probability wave: it determines the probability that another atom be ionized. This probability is essentially zero unless the atom lies on \ . This gives the visible track. (shrink)

The paper argues that the formulation of quantum mechanics proposed by Ghirardi, Rimini and Weber is a serious candidate for being a fundamental physical theory and explores its ontological commitments from this perspective. In particular, we propose to conceive of spatial superpositions of non-massless microsystems as dispositions or powers, more precisely propensities, to generate spontaneous localizations. We set out five reasons for this view, namely that it provides for a clear sense in which quantum systems in entangled states possess properties (...) even in the absence of definite values; it vindicates objective, single-case probabilities; it yields a clear transition from quantum to classical properties; it enables to draw a clear distinction between purely mathematical and physical structures, and it grounds the arrow of time in the time-irreversible manifestation of the propensities to localize. (shrink)

The concept of probability seems to have been inexplicable since its invention in the seventeenth century. In its use in science, probability is closely related with relative frequency. So the task seems to be interpreting that relation. In this paper, we start with predicted relative frequency and show that its structure is the same as that of probability. I propose to call that the ‘prediction interpretation’ of probability. The consequences of that definition are discussed. The “ladder”-structure of the probability calculus (...) is analyzed. The expectation of the relative frequency is shown to be equal to the predicted relative frequency. Probability is shown to be the most general empirically testable prediction. (shrink)

Bertand’s paradox is a fundamental problem in probability that casts doubt on the applicability of the indifference principle by showing that it may yield contradictory results, depending on the meaning assigned to “randomness”. Jaynes claimed that symmetry requirements solve the paradox by selecting a unique solution to the problem. I show that this is not the case and that every variant obtained from the principle of indifference can also be obtained from Jaynes’ principle of transformation groups. This is because the (...) same symmetries can be mathematically implemented in different ways, depending on the procedure of random selection that one uses. I describe a simple experiment that supports a result from symmetry arguments, but the solution is different from Jaynes’. Jaynes’ method is thus best seen as a tool to obtain probability distributions when the principle of indifference is inconvenient, but it cannot resolve ambiguities inherent in the use of that principle and still depends on explicitly defining the selection procedure. (shrink)

Rather than entailing that a particular outcome will occur, many scientific theories only entail that an outcome will occur with a certain probability. Because scientific evidence inevitably falls short of conclusive proof, when choosing between different theories it is standard to make reference to how probable the various options are in light of the evidence. A full understanding of probability in science needs to address both the role of probabilities in theories, or chances, as well as the role of probabilistic (...) judgment in theory choice. In this chapter, the author introduces and distinguishes the two sorts of probability from one another and attempt to offer a satisfactory characterization of how the different uses for probability in science are to be understood. A closing section turns to the question of how views about the chance of some outcome should guide our confidence in that outcome. (shrink)

To clarify and illuminate the place of probability in science Ellery Eells and James H. Fetzer have brought together some of the most distinguished philosophers ...

At a purely instrumental level, quantum theory is all about multiplication, addition and taking mod squares of complex numbers called probability amplitudes. The rules for combining amplitudes are deceptively simple. When two or more events are independent you multiply their respective probability amplitudes and when they are mutually exclusive you add them. Whenever you want to calculate probabilities you take mod squares of respective amplitudes. That’s it. If you are prepared to ignore the explanatory power of the theory (which you (...) should not) the rest is just a set of convenient mathematical tools developed for the purpose of book-keeping of amplitudes. Thus we tabulate amplitudes into state vectors and unitary matrices, and place them in Hilbert spaces. (shrink)

In this review article we present different formal frameworks for the description of generalized probabilities in statistical theories. We discuss the particular cases of probabilities appearing in classical and quantum mechanics, possible generalizations of the approaches of A. N. Kolmogorov and R. T. Cox to non-commutative models, and the approach to generalized probabilities based on convex sets.

One implication of Bell’s theorem is that there cannot in general be hidden variable models for quantum mechanics that both are noncontextual and retain the structure of a classical probability space. Thus, some hidden variable programs aim to retain noncontextuality at the cost of using a generalization of the Kolmogorov probability axioms. We generalize a theorem of Feintzeig to show that such programs are committed to the existence of a finite null cover for some quantum mechanical experiments, i.e., a finite (...) collection of probability zero events whose disjunction exhausts the space of experimental possibilities. (shrink)

The aim of this paper is to present and discuss a probabilistic framework that is adequate for the formulation of quantum theory and faithful to its applications. Contrary to claims, which are examined and rebutted, that quantum theory employs a nonclassical probability theory based on a nonclassical "logic," the probabilistic framework set out here is entirely classical and the "logic" used is Boolean. The framework consists of a set of states and a set of quantities that are interrelated in a (...) specified manner. Each state induces a classical probability space on the values of each quantity. The quantities, so considered, become statistical variables (not random variables). Such variables need not have a "joint distribution." For the quantum theoretic application, there is a uniform procedure that defines and determines the existence of such "joint distributions" for statistical variables. A general rule is provided and it is shown to lead to the usual compatibility-commutivity requirements of quantum theory. The paper concludes with a brief discussion of interference and the misunderstandings that are involved in the false move from interference to nonclassical probability. (shrink)

I describe several applications of the theory of n-universes through several different probabilistic situations. I describe fi rst how n-universes can be used as an extension of the probability spaces used in probability theory. The extended probability spaces thus defined allow for a finer modeling of complex probabilistic situations and fi ts more intuitively with our intuitions related to our physical universe. I illustrate then the use of n-universes as a methodological tool, with two thought experiments described by John Leslie. (...) Lastly, I model Goodman’s paradox in the framework of n-universes while also showing how these latter appear finally very close to goodmanian worlds. (shrink)

We clarify the significance of quasiprobability in quantum mechanics that is relevant in describing physical quantities associated with a transition process. Our basic quantity is Aharonov’s weak value, from which the QP can be defined up to a certain ambiguity parameterized by a complex number. Unlike the conventional probability, the QP allows us to treat two noncommuting observables consistently, and this is utilized to embed the QP in Bohmian mechanics such that its equivalence to quantum mechanics becomes more transparent. We (...) also show that, with the help of the QP, Bohmian mechanics can be recognized as an ontological model with a certain type of contextuality. (shrink)

In recent years quantum probability models have been used to explain many aspects of human decision making, and as such quantum models have been considered a viable alternative to Bayesian models based on classical probability. One criticism that is often leveled at both kinds of models is that they lack a clear interpretation in terms of psychological mechanisms. In this paper we discuss the mechanistic underpinnings of a quantum walk model of human decision making and response time. The quantum walk (...) model is compared to standard sequential sampling models, and the architectural assumptions of both are considered. In particular, we show that the quantum model has a natural interpretation in terms of a cognitive architecture that is both massively parallel and involves both co-operative and competitive interactions between units. Additionally, we introduce a family of models that includes aspects of the classical and quantum walk models. (shrink)

We show that the motion of particles may be essentially discontinuous and random.

Regardless or independent of any actuality or actualization and exempt from spatiotemporal and causal conditions, each individual possibility is pure. Actualism excludes the existence of individual pure possibilities, altogether or at least as existing independently of actual reality. In this paper, I demonstrate, on the grounds of my possibilist metaphysics—panenmentalism—how some of the most fascinating scientific discoveries in chemistry could not have been accomplished without relying on pure possibilities and the ways in which they relate to each other . The (...) discoveries are the following: Dan Shechtman’s discovery of quasicrystals; Linus Pauling’s alpha helix; the discovery of F. Sherwood Rowland and Mario J. Molina concerning the destruction of the atmospheric ozone layer; and Neil Bartlett’s noble gas compounds. On the grounds of the analysis of these cases, actualism must fail, whereas panenmentalism gains support. (shrink)

The revolutionary discovery of actual quasicrystals, thanks to Dan Shechtman’s stamina, is a golden opportunity to analyze once again the role that pure (“theoretical”) possibilities and saving them plays in scientific progress. Some theoreticians, primarily Alan Mackay, contributed to saving pure possibilities of quasicrystalline structures and to opening materials science for them. My analysis rests upon an original modal metaphysics—panenmentalism—which I introduced and have been developing since 1999, quite independently of any familiarity with modern crystallography, and which deals with saving (...) pure possibilities as indispensably contributing, inter alia, to our knowledge and sciences. (shrink)

Through the mental experiment that I suggest, it is possiblc to demonstrate that Hugh Everett’s quantum interpretation, known as of the “many universes”, is incongruent with the special theory of relativity.

The claims of science and the claims of probability combine in two ways. In one, probability is part of the content of science, as in statistical mechanics and quantum theory and an enormous range of "models" developed in applied statistics. In the other, probability is the tool used to explain and to justify methods of inference from records of observations, as in every science from psychiatry to physics. These intimacies between science and probability are logical sports, for while we think (...) science aims to say what happens, what has happened, what will happen, what would happen if other things were to happen, what could and could not happen, what will nearly happen, or what will approximate what will happen, probability claims say none of these things, or at least none of them about the phenomena with which science is concerned. On that, at least, almost all philosophical interpreters of probability since DeMoivre agree, with whatever reluctance. Consider some examples. (shrink)