Probability Theory

High-dimensional model representation for structural reliability analysis : Authors’ reply to comments by S. Rahman and H. Xu

by Rajib Chowdhury

International Journal for Numerical Methods in Biomedical Engineering
Volume 27, Issue 10, pages 1660–1664, October 2011

On equilibrium properties of evolutionary multiplayer games with random payoff matrices

by The Anh Han

co-authored with A. Traulsen and C. S. Gokhale.
Journal of Theoretical Population Biology (in press)

The analysis of equilibrium points in biological dynamical systems has been of great interest in a variety of... more The analysis of equilibrium points in biological dynamical systems has been of great interest in a variety of mathematical approaches to biology, such as population genetics, theoretical ecology or evolutionary game theory. The maximal number of equilibria and their classification based on stability have been the primary subjects of these studies, for example in the context of two-player games with multiple strategies. Herein, we address a different question using evolutionary game theory as a tool. If the payoff matrices are drawn randomly from an arbitrary distribution, what are the probabilities of observing a certain number of (stable) equilibria? We extend the domain of previous results for the two-player framework, which corresponds to a single diploid locus in population genetics, by addressing the full complexity of multi-player games with multiple strategies. In closing, we discuss an application and illustrate how previous results on the number of equilibria, such as the famous Feldman–Karlin conjecture on the maximal number of isolated fixed points in a viability selection model, can be obtained as special cases of our results based on multi-player evolutionary games. We also show how the probability of realizing a certain number of equilibria changes as we increase the number of players and number of strategies.

Objective Probability in Everettian Quantum Mechanics

by Alastair Wilson

Forthcoming in British Journal for Philosophy of Science.

David Wallace has given a decision-theoretic argument for the Born Rule in the context of Everettian quantum mechanics... more David Wallace has given a decision-theoretic argument for the Born Rule in the context of Everettian quantum mechanics (EQM). This approach promises to resolve some long-standing problems with probability in EQM, but it has faced plenty of resistance. One kind of objection (the ‘Incoherence problem’) charges that the requisite notion of decision-theoretic uncertainty is unavailable in the Everettian picture, so that the argument cannot gain any traction; another kind of objection grants the proof’s applicability and targets the premises. In this paper I propose some novel principles connecting the physics of EQM with the metaphysics of modality, and argue that in the resulting framework the Incoherence problem does not arise. These principles also help to justify one of the most controversial premises of Wallace’s argument, ‘branching indifference’. Absent any  a priori reason to align the metaphysics with the physics in some other way, we can adopt the proposed principles on grounds of theoretical utility. The upshot is that Everettians can, after all, make clear sense of objective probability.

SUBJECT TO EMBODIMENT : Rethinking Embodiment, Presence and the Body

by Nedine Kachornnamsong

Gothenburg University, ISSN: 1651-4769

With an objective to expand knowledge of physicality as an artistic tool, this paper explores the terms of... more With an objective to expand knowledge of physicality as an artistic tool, this paper explores the terms of phenomenological embodiment from the different perspective that is commonly applied in art theory. By presenting current researches from the field of new media development, the concept of embodiment is broadened from theory and practice of minimal art. The sense of  presence and the body in relation to human experience is also investigated for a better understanding in how we perceive and interact with the world. By conducting a research-in-practice, the outcome of the finding is also implemented into an interactive installation which focuses on an embodied experience.

Contextual-value approach to the generalized measurement of observables

by Justin Dressel

Physical Review A 85, 022123 (2012).

We present a detailed motivation for and definition of the contextual values of an observable, which were introduced... more We present a detailed motivation for and definition of the contextual values of an observable, which were introduced by Dressel et al. [Phys. Rev. Lett. 104 240401 (2010)]. The theory of contextual values is a principled approach to the generalized measurement of observables. It extends the well-established theory of generalized state measurements by bridging the gap between partial state collapse and the observables that represent physically relevant information about the system. To emphasize the general utility of the concept, we first construct the full theory of contextual values within an operational formulation of classical probability theory, paying special attention to observable construction, detector coupling, generalized measurement, and measurement disturbance. We then extend the results to quantum probability theory built as a superstructure on the classical theory, pointing out both the classical correspondences to and the full quantum generalizations of both Lüder's rule and the Aharonov-Bergmann-Lebowitz rule in the process. As such, our treatment doubles as a self-contained pedagogical introduction to the essential components of the operational formulations for both classical and quantum probability theory. We find in both cases that the contextual values of a system observable form a generalized spectrum that is associated with the independent outcomes of a partially correlated and generally ambiguous detector; the eigenvalues are a special case when the detector is perfectly correlated and unambiguous. To illustrate the approach, we apply the technique to both a classical example of marble color detection and a quantum example of polarization detection. For the quantum example we detail two devices: Fresnel reflection from a glass coverslip, and continuous beam displacement from a calcite crystal. We also analyze the three-box paradox to demonstrate that no negative probabilities are necessary in its analysis. Finally, we provide a derivation of the quantum weak value as a limit point of a pre- and postselected conditioned average and provide sufficient conditions for the derivation to hold.

An Investigation of Fifth Grade Students’ Conceptual Development of Probability through Activity Based Instruction: A Quasi-Experimental Study

by Halil Eksi

Ramazan GÜRBÜZ, Hakan ÇATLIOĞLU, Osman BİRGİN,
Emrullah ERDEM
Educational Sciences: Th eory & Practice
10 (2) • Spring 2010 • 1053-1068

Th is study aims to compare the eff ects of activity-based instruction and traditional instruction
on fifth grade... more Th is study aims to compare the eff ects of activity-based instruction and traditional instruction
on fifth grade primary school students’ conceptual development of probability.
Th e study was conducted through quasi-experimental method and carried out with 50 5th
grade primary school students, 25 for experimental group and 25 for control group. Th e
Conceptual Development Test consisting of 12 open-ended questions was administered
to the students in the study before and after the treatment. Data were analyzed using independent
samples t-test, and analysis of covariance (ANCOVA). It was determined that
activity-based instruction was more eff ective in helping students develop the probability
concepts than did traditional one.

Convex imprecise previsions

by Renato Pelessoni

Co-authored with Paolo Vicig. Published in Reliable Computing, 9 (6), 465-485, 2003.

In this paper centered convex previsions are introduced as a special class of imprecise previsions, showing that they... more In this paper centered convex previsions are introduced as a special class of imprecise previsions, showing that they retain or generalise most of the relevant properties of coherent imprecise previsions but are not necessarily positively homogeneous. The broader class of convex imprecise previsions is also studied and its fundamental properties are demonstrated, introducing in particular a notion of convex natural extension which parallels that of natural extension but has a larger domain of applicability. These concepts appear to have potentially many applications. In this paper they are applied to risk measurement, leading to a general definition of convex risk measure which corresponds, when its domain is a linear space, to the one recently introduced in risk measurement literature.

Dave Sayers' Ten Tips for Down and Out Academics

by Dave Sayers

Inaugural message to the veritable tea party of the damned which is 'Linguists Outside Academia', http://groups.google.com/group/ling-outside.

Have you finished your PhD but not found academic employment? Are you in a fixed‐term academic post, or moving between... more Have you finished your PhD but not found academic employment? Are you in a fixed‐term academic post, or moving between those sorts of jobs? Do you intend to pursue a career in academia, despite your better judgement and the weary look of experienced academics? If so, this guide might be for you...

Envelope theorems and dilation with convex conditional previsions

by Renato Pelessoni

Co-authored with Paolo Vicig. Published in the Proceedings of the Fourth International Symposium on Imprecise Probabilities and Their Applications, Pittsburgh, pp. 266 - 275, 2005.

This paper focuses on establishing envelope theorems for convex conditional lower previsions, a recently investigated... more This paper focuses on establishing envelope theorems for convex conditional lower previsions, a recently investigated class of imprecise previsions larger than coherent imprecise conditional previsions. It is in particular discussed how the various theorems can be employed in assessing convex previsions. We also consider the problem of dilation for these kinds of imprecise previsions, and point out the role of convex previsions in measuring conditional risks.

Direct algorithms for checking consistency and making inferences from conditional probability assessments

by Renato Pelessoni

Co-authored with Peter Walley and Paolo Vicig. Published in the Journal of Statistical Planning and Inference, 126 (1), 119-151, 2004. Preliminary version attached.

We solve two fundamental problems of probabilistic reasoning: given finitely many conditional probability assessments,... more We solve two fundamental problems of probabilistic reasoning: given finitely many conditional probability assessments, how to determine whether the assessments are mutually consistent, and how to determine what they imply about the conditional probabilities of other events? These problems were posed in 1854 by George Boole, who gave a partial solution using algebraic methods. The two problems are fundamental in applications of the Bayesian theory of probability; Bruno de Finetti solved the second problem for the special case of unconditional probability assessments in what he called ‘the fundamental theorem of probability’. We give examples to show that previous attempts to solve the two problems, using probabilistic logic and similar methods, can produce incorrect answers. Using ideas from the theory of imprecise probability, we show that the general problems have simple, direct solutions which can be implemented using linear programming algorithms. Unlike earlier proposals, our methods are formulated directly in terms of the assessments, without introducing unknown probabilities. Our methods work when any of the conditioning events may have probability zero, and they work when the assessments include imprecise (upper and lower) probabilities or previsions. The main methodological contribution of the paper is to provide general algorithms for making inferences from any finite collection of (possibly imprecise) conditional probabilities.

Williams coherence and beyond

by Renato Pelessoni

Co-authored with Paolo Vicig, published in International Journal of Approximate Reasoning, Volume 50, Issue 4, April 2009, pp. 612–626.

In this paper we discuss the consistency concept of Williams coherence for imprecise conditional previsions,... more In this paper we discuss the consistency concept of Williams coherence for imprecise conditional previsions, presenting a variant of this notion, which we call W-coherence. It is shown that W-coherence ensures important consistency properties and is quite general and well-grounded. This is done comparing it with alternative or anyway similar known and less known consistency definitions. The common root of these concepts is that they variously extend to imprecision the subjective probability approach championed by de Finetti. The analysis in the paper is also helpful in better clarifying several little investigated aspects of these notions.

Making sense of surprise: an investigation of the factors influencing surprise judgments.

by Rebecca Maguire

An information-theoretic approach to statistical dependence: Copula information

by Rafael Calsaverini

Co-authored with Renato Vicente. Available at arXiv: http://arxiv.org/abs/0911.4207

We discuss the connection between information and copula theories by showing that a copula can be employed to... more We discuss the connection between information and copula theories by showing that a copula can be employed to decompose the information content of a multivariate distribution into marginal and dependence components, with the latter quantified by the mutual information. We define the information excess as a measure of deviation from a maximum-entropy distribution. The idea of marginal invariant dependence measures is also discussed and used to show that empirical linear correlation underestimates the amplitude of the actual correlation in the case of non-Gaussian marginals. The mutual information is shown to provide an upper bound for the asymptotic empirical log-likelihood of a copula. An analytical expression for the information excess of T-copulas is provided, allowing for simple model identification within this family. We illustrate the framework in a financial data set.

The “rationality wars” in psychology: Where they are and where they could go.

by Thomas Sturm

Published in:  Inquiry, 55 (2012), 66-81.

Current psychology of human reasoning is divided into several different approaches. For instance, there is a major... more Current psychology of human reasoning is divided into several different approaches. For instance, there is a major dispute over the question whether human beings are able to apply norms of the formal models of rationality such as rules of logic, or probability and decision theory, correctly. While researchers following the “heuristics and biases” approach argue that we deviate systematically from these norms, and so are perhaps deeply irrational, defenders of the “bounded rationality” approach think not only that the evidence for this conclusion is problematic but also that we should not, at least not very often, use formal norms in reasoning. I argue that while the evidence for heuristics and biases is indeed questionable, the bounded rationality approach has its limits too. Most especially, we should not infer that formal norms play no role in a comprehensive theory of rationality. Instead, formal and bounded rules of reasoning might even be connected in a more comprehensive theory of rationality.

Schroyens, W. (2009). On Is and Ought: Levels of analysis and the descriptive versus normative analysis of human reasoning. Behavioral and Brain Sciences, 32 (1), 101-102.

by Walter Schroyens

Which Me am I? Branching, Fission, Probability, and Pizza in Everettian Quantum Mechanics

by Jon Lawhead

A note on the St. Petersburg paradox

by Shrisha Rao

3D Object Reconstruction: Formal Analysis and Practical Applications

by Laura Papaleo

Cartwright, Capacities, and Probabilities

by Gurol Irzik

"When Do I Get My Money?" A Probabilistic Theory Of Knowledge. PhD thesis 2011

by Jonny Blamey

PhD thesis KCL 2011. Examiners Jon Williamson and Luc Bovens, final supervisor David Papineau.

The important claim in this thesis is that it is rational to vary your degree of belief relative to what is at stake.... more The important claim in this thesis is that it is rational to vary your degree of belief relative to what is at stake. This allows a probabilistic theory of knowledge that can answer scepticism and avoid Gettier problems. It provides a decision theory that explains the Allais paradox, the Ellsberg paradox and gives an empirically adequate account of decisions under risk, whilst providing a probability measure that conforms to the Kolmogorov axioms. The probabilistic theory of knowledge thus vindicates the intuition that the same evidence can yeild knowledge at low stakes but not at high stakes.
At the heart of the theory is the Stake Size variation principle. This is a development of Ramseys theory of probability. The Stake Size Variation Principle give a measure of evidential support that is commensurable with measures for good. This allows a two dimensional account of evidential strength in terms of the point probability and the evidential value. The evidential value then determines how resilient the point probability is in the face of changes in stake size, as well as changes in evidence. The SSVP thus equates knowledge with value via the expectation principle.
There are two main arguments for the SSVP.
Firstly there is the contextualist argument that the scope of alternatives that need to be eliminated by the evidence can expand with an expansion of the stakes. In otherwords, it is wise to consider more possibilities when the stakes are high before settling in certainty.
Secondly there is the information theoretic argument based on the work of Kelly Jnr, which shows the inevitability of a downward drift and eventual ruin if one bets at the objective probability. This is due to the logarhythmic nature of growth.
The thesis concludes with a discussion of inductive certainty, showing how the SSVP can give an adequate account of how inductive certainty is possible.