In the first of two chapters on probability in scientific inquiry, the basic ideas of probability theory are introduced through examples involving games of chance. The chapter then focuses on the Bayesian approach to probability, which adopts the stance that probabilities should be understood as expressions about the degrees of belief. The Bayesian approach as a general framework for probability is explained through examples involving betting that extend beyond games of chance, which also allows the introduction of the idea of probabilistic coherence as a condition of rational partial belief. We are then finally ready for Bayes’s theorem, a theorem of the probability calculus that plays a central role in the Bayesian account of learning from evidence. That account is illustrated with a historically motivated example from the history of paleontology. The chapter considers objections to the Bayesian approach and the resources Bayesians may draw on for answering those objections.

In practice, much of statistical reasoning in science relies on probabilities subject to interpretation as relative frequencies. This chapter explains how probability can be understood in terms of relative frequencies and the uses scientists and philosophers have devised for frequentist probabilities. Particularly prominent in those uses are error probabilities associated with particular approaches to hypothesis testing. The approaches pioneered by Ronald Fisher and by Jerzy Neyman and Egon Pearson are outlined and explained through examples. The chapter then explores the error-statistical philosophy advocated by Deborah Mayo as a general framework for thinking about how we learn from empirical data. The error-statistical approach utilizes a frequentist framework for probabilities to articulate a view of severe testingof hypotheses as the means by which scientists increase experimental knowledge. Error statistics represents an important alternative to Bayesian approaches to scientific inquiry, and this chapter considers its prospects and challenges.

This chapter comes in two related but distinct parts. The first presents general trends in the neurosciences and considers how these impact upon psychiatry as a clinical science. The second picks up a recent and important development in neuroscience which seeks to explain mental functions such as perception and has been profitably extended into explanations of psychopathology. The second part can be viewed as a working example of the first’s overarching themes.

Chapter 5 explores the consequences of decoherence. We live in a Universe that is fundamentally quantum. Yet, our everyday world appears to be resolutely classical. The aim of Chapter 5 is to discuss how preferred classical states, and, more generally, classical physics, arise, as an excellent approximation, on a macroscopic level of a quantum Universe. We show why quantum theory results in the familiar “classical reality” in open quantum systems, that is, systems interacting with their environments. We shall see how and why, and to what extent, quantum theory accounts for our classical perceptions. We shall not complete this task here—a more detailed analysis of how the information is acquired by observers is needed for that, and this task will be taken up in Part III of the book. Moreover, Chapter 5 shows that not just Newtonian physics but also equilibrium thermodynamics follows from the same symmetries of entanglement that led to Born’s rule (in Chapter 3).

Chapter 3 describes how quantum entanglement leads to probabilities based on a symmetry, but—in contrast to subjective equal likelihood based solely on ignorance—it is an objective symmetry of known quantum states. Entanglement-assisted invariance (or envariance for short) relies on quantum correlations: One can know the quantum state of the whole and use this to quantify the resulting ignorance of the states of parts. Thus, quantum probability is, in effect, an objective consequence of the Heisenberg-like indeterminacy between global and local observables. This derivation of Born’s rule is based on the consistent subset of quantum postulates. It justifies statistical interpretation of reduced density matrices, an indispensable tool of decoherence theory. Hence, it gives one the mandate to explore—in Part II of this book—the fundamental implications of decoherence and its consequences using reduced density matrices and other customary tools.

Deterministic and probabilistic mathematical theories have in common that they construct mathematical representations of real-world phenomena. On a basic level this can be regarded as a type of explicit problem-solving. This involves presenting the problem in ‘abstract form’ in symbols (often numbers, letters, or geometrical elements). These symbols are then manipulated in accordance with precise rules: Strings of symbols in sets of equations come to represent ideas. The construction of mathematical theories involves testing whether experimental observations fit the postulated ‘mathematical rules.’ If they do not fit then the ‘mathematical rules’ may be refined, extended, or new ones may be formulated. Newly mathematically formalized ideas are validated by testing whether they align with observations but also by examining whether they are consistent with other, previously established, mathematical rules.

Part of the motivation for this book was its role in solving open problems in regular variation – in brief, the study of limiting relations of the form f(λx)/f(x) → g(x) as x → ∞ for all λ > 0 and its relatives. This was the subject of the earlier book Regular Variation by N. H. Bingham, C. M. Goldie and J. L. Teugels (BGT). So to serve as prologue to the present book, a brief summary of the many uses of regular variation is included, to remind readers of BGT and spare others needing to consult it. Topics covered include: probability (weak law of large numbers, central limit theorem, stability, domains of attraction, etc.), complex analysis (Abelian, Tauberian and Mercerian theorems, Levin–Pfluger theory), analytic number theory (prime divisor functions; results of Hardy and Ramanujan, Erdős and Kac, Rényi and Turán); the Cauchy functional equation g(λμ) = g(λ)g(μ) for all λ; μ > 0; dichotomy – the solutions are either very nice (powers) or very nasty (pathological – the ‘Hamel pathology’).

Chapter 5 examines the normal distribution, its relationship to z-scores, and its applicability to probability theory and statistical inference. z-scores or standardized scores are values depicting how far a particular score is from the mean in standard deviation units. Different proportions of the normal curve area are associated with z-scores. The conversions of raw scores to z-scores and z-scores to raw scores are illustrated. Nonnormal distributions which differ markedly from normal curve characteristics are also described. The importance of the normal curve as a probability distribution, along with a brief introduction to probability, is discussed.

This is a revision of John Trimmer’s English translation of Schrödinger’s famous ‘cat paper’, originally published in three parts in Naturwissenschaften in 1935.

This is a reprinting of Schrödinger’s famous pair of papers delivered at the Cambridge Philosophical Society in late 1935 and 1936, wherein he first coins the term ‘entanglement’ to describe interacting quantum systems. The first paper (1935) is given here in full; section 4 of the second paper (1936) is reprinted as an appendix.

Discusses statistical methods, covering random variables and variates, sample and population, frequency distributions, moments and moment measures, probability and stochastic processes, discrete and continuous probability distributions, return periods and quantiles, probability density functions, parameter estimation, hypothesis testing, confidence intervals, covariance, regression and correlation analysis, time-series analysis.

Forecasting elections is a high-risk, high-reward endeavor. Today’s polling rock star is tomorrow’s has-been. It is a high-pressure gig. Public opinion polls have been a staple of election forecasting for almost ninety years. But single source predictions are an imperfect means of forecasting, as we detailed in the preceding chapter. One of the most telling examples of this in recent years is the 2016 US presidential election. In this chapter, we will examine public opinion as an election forecast input. We organize election prediction into three broad buckets: (1) heuristics models, (2) poll-based models, and (3) fundamentals models.

Network science is a broadly interdisciplinary field, pulling from computer science, mathematics, statistics, and more. The data scientist working with networks thus needs a broad base of knowledge, as network data calls for—and is analyzed with—many computational and mathematical tools. One needs good working knowledge in programming, including data structures and algorithms to effectively analyze networks. In addition to graph theory, probability theory is the foundation for any statistical modeling and data analysis. Linear algebra provides another foundation for network analysis and modeling because matrices are often the most natural way to represent graphs. Although this book assumes that readers are familiar with the basics of these topics, here we review the computational and mathematical concepts and notation that will be used throughout the book. You can use this chapter as a starting point for catching up on the basics, or as reference while delving into the book.