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This chapter gives a survey of the basic concepts of estimating functions, which are used in subsequent chapters. The concept of unbiasedness for estimating functions is introduced as a generalization of the concept of an unbiased estimator. Godambe efficiency, also known as the Godambe optimality criterion, is introduced by generalizing the concept of minimum variance unbiased estimation. Within the class of estimating functions which are unbiased and information unbiased, the score function is characterized as the estimating function with maximal Godambe efficiency. Extensions to the multiparameter case are given, and the connection to the Riesz representation theorem is described briefly. This chapter also discusses a number of examples from semiparametric models, martingale estimating functions for stochastic processes, empirical characteristic function methods and quadrat sampling; the estimating equations in some of these examples have possibly more than one solution.

In the introduction to his book Principles of Mechanics, Heinrich Hertz emphasised that one of the advantages of his geometric formulation of his mechanics is that it throws a bright light upon William Rowan Hamilton's method of treating mechanical problems by the aid of characteristic functions. Hertz developed the geometric version of the Hamilton formalism in the first kinematic book and then applied these results to the motion of free holonomic systems, and finally to the motion of unfree systems. Thus, Hertz was able to express the analytical equations of the Hamilton formalism for a conservative system, without taking the hidden system into account except through the force function U. The geometry that made his theory for the straightest distance so appealing no longer holds in his description of conservative systems. However, it is possible to introduce a different metric in configuration space, so that the geometric part of the theory also applies to conservative systems.

This chapter considers methods for describing the quantum statistics of a single mode of the electromagnetic field. Some of these methods may be extended to multimode fields, and this is discussed in the last section of the chapter. The moment generating function is developed for studying the photon number statistics of a single field mode. The quantum properties of optical phase are described using the optical phase operator. The characteristic functions and quasi-probability distributions provide a complete statistical description of the field. These rely on the properties the coherent states and the Glauber displacement operator.

This chapter reviews probability theory, limiting the discussion to the study of random events as opposed to random processes, the latter being a sequence of random events extended over a period of time. The goal is to raise the level of approach by demonstrating the usefulness of delta functions. The chapter presents a calculation of the chi-squared distribution (important in statistical decision making) with delta functions. The normalisation condition of the probability density in chi-square leads to a geometric result; namely, the volume of a sphere in n dimensions can be determined without ever transferring to spherical coordinates. This chapter also discusses the first and second laws of gambling, along with distribution functions, stochastic variables, expectation values for single random variables, characteristic functions and generating functions, measures of dispersion, joint events, conditional probabilities and Bayes' theorem, sums of random variables, fitting of experimental observations, and multivariate normal distributions.

This chapter begins with a review of elementary probability theory, conditional probability, and statistical independence. It explains the notion of a random variable and its distribution function or probability density function. It then introduces the concepts of mathematical expectation and variance and discusses several distributions often met in practice: the binomial, Gaussian, and Poisson distributions. The characteristic function of a random variable is defined and applied to the determination of the distribution of the sum of independent random variables. The central limit theorem is described but not proved.

Despite the ex post facto identification of a number of “anticipations” of game-theoretic results through history, it is generally agreed that modern game theory’s founding work was mathematician John von Neumann and economist Oskar Morgenstern’s 1944 book, Theory of Games and Economic Behavior. Yet to a student of game theory trained in recent decades, the book must seem antique in terms of its notations, style of presentation, and terminology. This chapter is therefore principally devoted to explicating the text of von Neumann and Morgenstern’s book, emphasizing the diverse nature of its contents: a dynamic, set-theoretic depiction of games in “extensive form;” the matrix “normal form” of the game and the celebrated “minimax theorem,” with its rich connections to topology and the theory of fixed points; the “characteristic function form” of games and definition of “solutions” as non-dominated sets of imputations; and finally, the von Neumann – Morgenstern theory of utility, which constructed a measure of utility from axioms of preference ordering. These pieces of the theory were not just selectively appropriated and used by different groups of individuals after 1944, but they were also the outgrowth of varied research interests of the book’s authors in the years preceding its publication.

This chapter formally presents the representation of the quantum density operator by phase space distribution functions of phase space variables, leading to expressions for quantum correlation functions in terms of phase space integrals in which phase space variables for each mode—c-numbers for bosons, Grassmann numbers for fermions—replace mode annihilation and creation operators. The emphasis is on double-phase-space normally ordered (positive P type) normalised distribution functions, with distinct phase space variables for annihilation and creation operators; however, symmetrically ordered (Wigner type) distribution functions are also considered, along with unnormalised B distribution functions, which lead to phase space integrals for Fock state populations and coherences. Characteristic functions are first defined and then shown to be related to distribution functions via phase space integrals. The existence and symmetry properties of distribution functions—which are non-unique and non-analytic for bosons—is demonstrated using Bargmann coherent-state projectors.

There are three strands of justification for the account of combinatorial unity presented in the last chapter. The first is negative: the other extant accounts of unity are, I have argued, either partial or face insuperable problems. The second is positive: the account satisfies our three desiderata. The third reason is that the account withstands a host of likely objections: philosophical, formal, and linguistic. To give substance to this last strand of justification is the job of the last two chapters. In the present chapter, the focus will be on philosophical and formal matters. In particular, the account on offer will be distinguisged from similar accounts recently offered by Jeff King and Scott Soames. The next chapter will consider matters arising in linguistics about the status of Merge.

Since simple random walk is a process with independent increments, its properties are represented in the most simple way by using the techniques based on characteristic functions. This chapter introduces the necessary mathematical instruments, and then use them to discuss general expressions for the distribution of the walker's displacement after a given number of steps in one dimension and in higher dimensions. It moreover discusses moments of displacement, provided these moments exist. The chapter then considers a simple approach to the central limit theorem, and discusses situations, when this breaks down (corresponding to the cases when the second moment of step lengths diverges).

Just as Fokker–Planck processes are a generalisation of thermal noise, Langevin processes constitute a generalisation of shot noise. In general, Langevin processes permit a larger class of problems to be solved than Fokker–Planck processes and procedures. They provide more intuition, but the class of such processes that can be reduced exactly to analysis is limited. However, approximations (typically adiabatic ones) are easier to envision and to make in the Langevin language. Langevin methods, at least for linear or quasilinear systems, have the simplicity of the circuit equations of electrical engineering. The noise source may arise from thermal reservoirs as in Johnson noise, or shot noise from the discreteness of particles. Once the noise is represented as a voltage source with known moments, the physical nature of the source is no longer important. This chapter discusses the simplicity of Langevin methods, proof of delta correlation for Markovian processes, homogeneous noise with linear damping, conditional correlations, generalised characteristic functions, generalized shot noise, and systems possessing inertia.

This chapter begins with a look at convolutions and the distribution of sums of random variables. It briefly surveys complex number theory before defining the characteristic function and studying its properties, with a range of examples. The concept of infinite divisibility is introduced. The important inversion theorem is treated and finally consideration is given to characteristic functions in conditional distributions.

Wilson statistics, described in Chapter 2, aims at calculating the distribution of the structure factor P(F) ≡ P(|F|, φ) when nothing is known about the structure; the positivity and atomicity of the electron density (both promoted by the positive nature of the atomic scattering factors fj) are the only necessary assumptions. Wilson results may be synthesized as follows: . . . the modulus R = |E| is distributed according to equations (2.7) or (2.8), while no prevision is possible about φ, which is distributed with constant probability 1/(2π). . . . In other words, knowledge of the R moduli does not provide information about a phase; this agrees well with Section 3.3, according to which experimental data only allow an estimate of s.i. (and also s.s. if the algebraic form of the symmetry operators has been fixed). Let us now consider P(Fh1, Fh2 ) ≡ P(|Fh1 |, |Fh2 |, φh1, φh2 ), the joint probability distribution function of two structure factors. If the two structure factors are uncorrelated (i.e. no relation is expected between their moduli and between their phases), P will coincide with the product of two Wilson distributions (2.7) or (2.8), say, . . . P(Fh1, Fh2 ) ≡ P(|Fh1 |, φh1 ) · P(|Fh2 |, φh2 ) = 1/4π2 P(|Fh1 |)P(|Fh2 |), . . . which is useless (because the two Wilson distributions are useless) for solving the phase problem; indeed, the relation does not provide any phase information. The question is now: if two structure factors are correlated, may their joint probability distribution function be used for solving the phase problem? Let us first use a simple example to show how much additional information (i.e. that is not present in the two elementary distributions) may be stored in a joint probability distribution function; then we will answer the question. Let us suppose that the human population of a village has been submitted to statistical analysis to define how weight and height are distributed.

This chapter introduces the fundamentals of weak convergence for real sequences. Definitions and examples are given. The Skorokhod representation theorem is proved and the chapter then considers the preservation of weak convergence under transformations. Next, the role of moments and characteristic functions is considered. In the leading case of random sums, the criteria for weak convergence and the concept of a stable distribution are studied.

In this chapter, the Hamilton–Jacobi formulation is discussed in two parts: from a generating function perspective and as a variational principle. The Poincaré–Cartan 1-form is derived and solutions to the Hamilton–Jacobi equations are discussed. The canonical action is examined in a fashion similar to that used for analysis in previous chapters. The Hamilton–Jacobi equation is then shown to parallel the eikonal equation of wave mechanics. The chapter discusses Hamilton’s principal function, the time-independent Hamilton–Jacobi equation, Hamilton’s characteristic function, the rectification theorem, the Maupertius action principle and the Hamilton–Jacobi variational problem. The chapter also discusses integral surfaces, complete integral hypersurfaces, completely separable solutions, the Arnold–Liouville integrability theorem, general integrals, the Cauchy problem and de Broglie–Bohm mechanics. In addition, an interdisciplinary example of medical imaging is detailed.

Thus far we have only studied representations of the source. Now we add the channel, pushing us into the frameworks of estimate then examine estimation bounds for understanding rigid object recognition involving the low-dimensional matrix groups. Minimum-mean-squared error bounds are derived for recognition and identification.