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INTRODUCTION

Jon Williamson

in Bayesian Nets and Causality: Philosophical and Computational Foundations

This chapter describes the central claims of the book. From a philosophical point of view, the book argues for an objective Bayesian interpretation of probability and an epistemic interpretation of ... More

This chapter describes the central claims of the book. From a philosophical point of view, the book argues for an objective Bayesian interpretation of probability and an epistemic interpretation of causality, and claims that these offer a firm foundation for causal modelling. From the computational point of view, the book investigates the relationship between Bayesian nets and maximum entropy methods, and develops a general computational framework for probabilistic and causal reasoning.Less

OUTLOOK

Francesco Calogero

in Isochronous Systems

In Chapter 8 five directions of future research are tersely mentioned in order of increasing importance. The investigation of Diophantine relations, such as those reported in Appendix C. The ... More

In Chapter 8 five directions of future research are tersely mentioned in order of increasing importance. The investigation of Diophantine relations, such as those reported in Appendix C. The investigation of the behavior in a quantal context of systems described by isochronous Hamiltonians. The investigation of isochronous systems outside the phase space sectors where they behave isochronously. The investigation of the behavior of the isochronous many-body problems treated in Section 5.5, in particular, when the corresponding unmodified system is a realistic molecular dynamics model characterized by a chaotic evolution yielding—in the context of statistical mechanics and thermodynamics—a steadily increasing entropy—which, however, must eventually decrease when the corresponding Ω-modified system returns to its initial state, as entailed by its isochronous character. And, last but by no means least, the investigation of isochronous systems in an applicative context—be it physics, chemistry, biology, medicine, economy, and so on.Less

An introduction to maximum entropy

Peter Main

in Crystal Structure Analysis: Principles and Practice

Maximum entropy methods are used in crystallography for estimating reflection phases among other applications. This chapter explains the basic concepts behind maximum entropy, including entropy ... More

Maximum entropy methods are used in crystallography for estimating reflection phases among other applications. This chapter explains the basic concepts behind maximum entropy, including entropy itself and its relationship to probability and information. They are illustrated by a simple non-scientific example of estimating missing information based on reasonable assumptions.Less

Introduction to Information Theory

M. Vidyasagar

in Hidden Markov Processes: Theory and Applications to Biology

This chapter provides an introduction to some elementary aspects of information theory, including entropy in its various forms. Entropy refers to the level of uncertainty associated with a random ... More

This chapter provides an introduction to some elementary aspects of information theory, including entropy in its various forms. Entropy refers to the level of uncertainty associated with a random variable (or more precisely, the probability distribution of the random variable). When there are two or more random variables, it is worthwhile to study the conditional entropy of one random variable with respect to another. The last concept is relative entropy, also known as the Kullback–Leibler divergence, which measures the “disparity” between two probability distributions. The chapter first considers convex and concave functions before discussing the properties of the entropy function, conditional entropy, uniqueness of the entropy function, and the Kullback–Leibler divergence.Less

 The Origins of Life and the Cosmos as Evolutionary Themes

Paul F. Lurquin and Linda Stone

in Evolution and Religious Creation Myths: How Scientists Respond

This chapter demonstrates that the origin of the universe was probabilistic, not preordained, because Heisenberg's uncertainty principle would have been violated at the level of the Big Bang. The ... More

This chapter demonstrates that the origin of the universe was probabilistic, not preordained, because Heisenberg's uncertainty principle would have been violated at the level of the Big Bang. The appearance of structure in the universe was made possible by the decoupling of matter and radiation about 400,000 years after the Big Bang. Structure appeared — and continues to appear — without violating the second law of thermodynamics because entropy is not increasing as fast as it could, and this keeps the universe in a state of disequilibrium, not equilibrium. Life appeared as a consequence of this disequilibrium, possibly involving a sequence of events such as formation of an organic prebiotic soup (or synthesis of organic compounds in hydrothermal vents), the appearance of an RNA world, and the formation of lipid-bound hypercycles.Less

OBJECTIVE BAYESIANISM

Jon Williamson

in Bayesian Nets and Causality: Philosophical and Computational Foundations

Objective Bayesianism imposes two norms on degrees of belief: degrees of belief should be constrained by empirical information and they should otherwise be as equivocal as possible. The origins of ... More

Objective Bayesianism imposes two norms on degrees of belief: degrees of belief should be constrained by empirical information and they should otherwise be as equivocal as possible. The origins of objective Bayesianism are explained, with the work of Jakob Bernoulli and Laplace presented at some length. A contemporary reading of the two norms is developed in detail. Then, it is shown how Bayesian nets can be used to represent objective Bayesian degrees of belief (this leads to what are now called objective Bayesian nets).Less

LANGUAGE CHANGE

Jon Williamson

in Bayesian Nets and Causality: Philosophical and Computational Foundations

This chapter discusses the situation in which an agent's language or domain may change. It is argued — against a principle of language invariance — that language contains implicit knowledge and so ... More

This chapter discusses the situation in which an agent's language or domain may change. It is argued — against a principle of language invariance — that language contains implicit knowledge and so degrees of belief should be expected to change when language changes. A conservativity principle, which demands that new degrees of belief be as close as possible to old degrees of belief, is rejected. An overview is given of formal methods for revising beliefs as language changes, and maximum entropy methods are advocated.Less

Energy flow

David M. Wilkinson

in Fundamental Processes in Ecology: An earth systems approach

The second law of thermodynamics is central to understanding ecology, although it is ignored by most ecology text books. It follows from the second law that all organisms must draw free energy from ... More

The second law of thermodynamics is central to understanding ecology, although it is ignored by most ecology text books. It follows from the second law that all organisms must draw free energy from their environment and return waste products back to their environment. Microorganisms often play a central role in decomposition of these waste products, but in spite of their importance are ignored by most food web studies. The possible relevance of developing ideas on maximum entropy production (MEP) to global ecology is also discussed.Less

Outlook

Vlatko Vedral

in Introduction to Quantum Information Science

This book has discussed the foundations of quantum information science as well as the relationship between physics and information theory in general. It has considered the quantum equivalents of the ... More

This book has discussed the foundations of quantum information science as well as the relationship between physics and information theory in general. It has considered the quantum equivalents of the Shannon coding and channel capacity theorems. The von Neumann entropy plays a role analogous to the Shannon entropy, and the Holevo bound is the analogue of Shannon's mutual information used to quantify the capacity of a classical channel. Quantum systems can process information more efficiently than classical systems in a number of different ways. Quantum teleportation and quantum dense coding can be performed using quantum entanglement. Entanglement is an excess of correlations that can exist in quantum physics and is impossible to reproduce classically (with what is termed “separable” states). The book has also demonstrated how to discriminate entangled from separable states using entanglement witnesses, as well as how to quantify entanglement, and looked at quantum computation and quantum algorithms.Less

Epilogue

Robert Alicki and Mark Fannes

in Quantum Dynamical Systems

This chapter presents some theories relating quantum dynamical entropy and the open problem of a rigorous description of transport phenomena in macroscopic bodies.

Density Stratification

Gary A. Glatzmaier

in Introduction to Modeling Convection in Planets and Stars: Magnetic Field, Density Stratification, Rotation

This chapter examines the effects of large variations in density with depth, that is, density stratification. It first describes anelastic models for 2D cartesian box and 2D cylindrical annulus ... More

This chapter examines the effects of large variations in density with depth, that is, density stratification. It first describes anelastic models for 2D cartesian box and 2D cylindrical annulus geometries, using entropy and pressure as working thermodynamic variables or using temperature and pressure, for both convectively unstable and stable regions. In particular, it considers anelastic approximation and how to formulate the anelastic equations, as well as the anelastic form of mass conservation, momentum conservation with entropy as a variable, internal energy conservation with entropy as a variable, and temperature as a variable. It also discusses possible choices for a reference state, focusing on polytropes, before explaining modifications to the numerical method and presenting the numerical simulations using the anelastic model.Less

The Limits of Science: What Do We Really Know?

Paull Nunez

in Brain, Mind, and the Structure of Reality

This chapter considers the general limits of scientific knowledge, and identifies probability and entropy as important measures of human ignorance; scientific predictions are shown to be severely ... More

This chapter considers the general limits of scientific knowledge, and identifies probability and entropy as important measures of human ignorance; scientific predictions are shown to be severely limited in complex systems. It argues that fundamental limits on computer power may preclude the creation of artificial consciousness, even if natural brains actually do create minds, as assumed by most contemporary scientists.Less

The Dying Cosmos: Qoheleth’s Misanthropic Principle

William P. Brown

in The Seven Pillars of Creation: The Bible, Science, and the Ecology of Wonder

Though not a creation account per se, Ecclesiastes 1:2-11 features a portrait of creation that is entirely unique in the biblical corpus. Qoheleth, the ostensible author of most of Ecclesiastes, ... More

Though not a creation account per se, Ecclesiastes 1:2-11 features a portrait of creation that is entirely unique in the biblical corpus. Qoheleth, the ostensible author of most of Ecclesiastes, describes creation in terms of wearying, ever-repeating cycles, from wind and water to life and death. Qoheleth’s world is devoid of anything “new,” a creation without pause and effect with human life beset by uninterrupted toil. The final chapter of Ecclesiastes, moreover, alludes to the eventual dissipation of creation. All is “vanity” (hebel); creation is a static, closed whole. Science, too, points to the ultimate dissolution of the universe, death by entropy. At the same time, science reveals that the cycles of nature Qoheleth so disparaged are in fact life-sustaining. Science both underscores and reorients Qoheleth’s “empirical” view of the world. Though creation seems pointless, it nevertheless provides sustaining moments of joy for human well-being.Less