We trace the evolution of statistics from its origins in the seventeenth century, contrasting twentieth-century data sets and approaches with those seen frequently today. This discussion motivates the approach taken in the book, which focuses on the bias--variance tradeoff as opposed to unbiased estimation, and on finite-sample guarantees as opposed to asymptotics, which can often be misleading in problems whose difficulty is controlled by several parameters. We introduce the decision-theoretic framework that underpins our assessment of the performance of statistical procedures, and discuss the concepts of minimaxity and adaptation. One of our perspectives is to seek to understand the behaviour of methods with respect to as wide a class of data generating mechanisms as possible, and in particular under model misspecification --- where an idealised model that motivates a procedure fails to hold. We discuss the use of the book for teaching and prerequisites for students, concluding with notation used throughout the book.

Anthropomorphic learning in behavioural data science refers to a hybrid modelling paradigm that integrates human-centric principles from decision theory with data-driven methodologies of machine learning to simulate, predict and optimise human-like behaviours in artificial systems. This chapter introduces and explores this hybrid approach, examining its theoretical underpinnings, methodological framework and practical applications. Anthropomorphic learning seeks to bridge the normative rigour of decision theory – grounded in models of preference, choice and rationality – with the empirical flexibility of machine learning, particularly in settings marked by uncertainty, complexity and interactivity. The chapter distinguishes anthropomorphic learning from conventional hybrid modelling techniques and critiques its promise through a detailed analysis of recent implementations in user modelling, digital choice architectures and autonomous agent systems. It also discusses its limitations and ethical implications, especially concerning transparency, replicability and human interpretability. By embedding agency, intentionality and bounded rationality into algorithmic structures, anthropomorphic learning represents a compelling frontier for advancing both human-aligned AI and behavioural data science.

Not everything we know about numbers can be interpreted as true of what they represent, so this chapter explores how far we can go in applying the numbers to real problems, and how we can be sure they are meaningful.

The people who started decision theory, how they identified the key quantifiable ingredients of good decisions: values, trade-offs and probabilities, then turned this mathematical beginning into the applied discipline of decision analysis.

Computational neuroimaging is defined broadly as the use of neuroimaging to investigate the localization and representation of parameters in formal mathematical models. We focus upon models of behavior and neural processing that have been adopted widely in behavioral sciences and cognitive neuroscience, including reinforcement learning, predictive coding, decision theory (drift diffusion and evidence accumulation), population receptive field models, and encoding models (including artificial neural networks). The aim is not to explain all the technical details of the various models, but illustrate and discuss the added value of combining such models with neuroimaging.

This chapter overviews three recently developed posterior test statistics for hypothesis testing based on posterior output. These three statistics can be viewed as the posterior version of the trinity of test statistics based on maximum likelihood (ML), namely, the likelihood ratio (LR) test, the Lagrange multiplier (LM) test, and the Wald test. The asymptotic distributions of the test statistics are discussed under repeated sampling. Furthermore, based on the Bernstein–von Mises theorem, the equivalence of the confidence interval construction between the set of posterior tests and their frequentist counterparts is developed, giving the posterior tests a frequentist asymptotic justification. The three statistics are applicable to many popular financial econometric models, including asset pricing models,copula models, and so on. Studies based on simulated data and real data in the context of several financial econometric models are carried out to illustrate the finite sample behavior and the usefulness of the test statistics.

In this chapter, we discuss the decision-theoretic framework of statistical estimation and introduce several important examples. Section 28.1 presents the basic elements of statistical experiment and statistical estimation. Section 28.3 introduces the Bayes risk (average-case) and the minimax risk (worst-case) as the respective fundamental limits of statistical estimation in Bayesian and frequentist settings, with the latter being our primary focus in this part. We discuss several versions of the minimax theorem (and prove a simple one) that equates the minimax risk with the worst-case Bayes risk. Two variants are introduced next that extend a basic statistical experiment to either large sample size or large dimension: Section 28.4 on independent observations and Section 28.5 on tensorization of experiments. Throughout this chapter the Gaussian location model (GLM), introduced in Section 28.2, serves as a running example, with different focus at different places (such as the role of loss functions, parameter spaces, low versus high dimensions, etc.). In Section 28.6, we discuss a key result known as Anderson’s lemma for determining the exact minimax risk of (unconstrained) GLM in any dimension for a broad class of loss functions, which provides a benchmark for various more general techniques introduced in later chapters.