Chapter 7 covers PROBABILITY FUNDAMENTALS and includes the following specific topics, among others: the discrete case, additive rules of probability, complement rule of probability, multiplicative rule of probability,conditional probability, Bayes’ theorem, the Law of Large Numbers.

Chapter 7 covers probability fundamentals and includes the following specific topics, among others: the discrete case, additive rules of probability, complement rule of probability, multiplicative rule of probability, conditional probability, Bayes’ theorem, and the law of large numbers.

Abstract: In this chapter, we survey some recent developments in the field of geophysical inversion. We aim to provide an accessible general introduction to the breadth of current research, rather than focusing in depth on particular topics. We hope to give the reader an appreciation for the similarities and connections between different approaches, and their relative strengths and weaknesses.

Various statistical problems of assessing legal evidence are covered. Poisson's attempts to model the probability of a jury coming to the correct decision are considered. Various versions of the famous Island Problem and possible Bayesian solutions are covered in some detail.

A standard probability formalism is introduced, including a definition of the probability density function (PDF) and its first four moments. Most basic PDFs such as the uniform and Gaussian PDFs, are defined. The fundamentals of the Bayes’ formula derivation and its formulation in terms of PDFs are also presented. More importantly, data assimilation is described as a recursive Bayes’ formula, which connects the standard Bayes’ formula from different analysis times by using transition PDFs. A basic introduction to Shannon information theory is presented, followed by a definition of uncertainty in terms of entropy, and therefore establishing a mathematical basis for interpreting data assimilation in terms of information processing that is used throughout this book. The multivariate Gaussian data assimilation framework, most often used in practice, is described. Common analysis solutions that include maximum a posteriori and minimum variance methods are derived, which include a formulation of the cost function and posterior probability.

Rational beliefs need not be truth-tracking nor adhere strictly to basic logical or statistical-inference principles. They must simply be appropriate to the attainment of the individual’s purposes in specific contexts. We analyze various “errors” that behavioral economists have supposedly discovered in belief formation. Specifically, we examine supposed errors related to imperfect logical deduction, the conjunction fallacy (including the famous “Linda problem”), availability bias, overconfidence bias, and distorted salience. In each case we show that the standard behavioral analysis is too simple and that many of their behaviors qualify as inclusively rational. The fundamental error of behavioral analysts is to expect that universal, abstract methods of belief formation will be appropriate for guiding all concrete choices.