● Darwin’s theory of evolution by natural selection was influenced by his reading Thomas Malthus’s Essay on Population. The status of Malthusian ideas in evolutionary theory is discussed. ● An organism’s fitness is its ability to survive and reproduce. ● Natural selection influences the evolution of height (for example) precisely when individuals with different heights differ in their fitnesses. ● Fitness can be represented as a mathematical expectation, but mathematical variance matters too. ● It is almost always impossible to estimate the fitness of a single organism, but the fitnesses of traits that are possessed by multiple organisms can often be estimated. ● Two coextensive traits must have the same fitness, even if one of them helps organisms to survive and reproduce while the other does not. ● Natural selection does not inevitably lead the average fitness of organisms in a population to improve, even when the external environment is stable.

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.