In this chapter we introduce Bayesian inference and use it to extend the frequentist models of the previous chapters. To do this, we describe the concept of model priors, informative priors, uninformative priors, and conjugate prior-likelihood pairs . We then discuss Bayesian updating rules for using priors and likelihoods to obtain posteriors. Building upon priors and posteriors, we then describe more advanced concepts including predictive distributions, Bayes factors, expectation maximization to obtain maximum posterior estimators, and model selection. Finally, we present hierarchical Bayesian models, Markov blankets, and graphical representations. We conclude with a case study on change point detection.

Review of mathematical and statistical concepts includes some foundational materials such as probability densities, Monte Carlo methods and Bayes’ rule are covered. We provide concept reviews that provide additional learning to the previous chapters. We aim to generate first an intuitive understanding of statistical concepts, then, if the student is interested, dive deeper into the mathemetical derivations. For example, principal component analysis can be taught by deriving the equations and making the link with eigenvalue decomposition of the covariance matrix. Instead, we start from simple two- and three-dimensional datasets and appeal to the student’s insight into the geometrical aspect: the study of an ellipse, and how we can transform it to a circle. This geometric aspect is explained without equations, but instead with plots and figures that appeal to intuition starting from geometry. In general, it is our experience that students in the geosciences retain much more practical knowledge when presented with material starting from case studies and intuitive reasoning.

This chapter explicates how to apply heuristic Bayesian reasoning in qualitative case study research. Steps include defining a set of mutually exclusive hypotheses to compare, assessing prior odds, identifying evidence, evaluating likelihood ratios for the evidence, and updating via Bayes’ rule to obtain posterior odds for the hypotheses.

This chapter explicates how to apply explicit Bayesian analysis in qualitative case study research, which involves quantifying probabilities and leveraging the mathematical apparatus of Bayesian probability theory. This approach allows us to more effectively communicate our judgments and make more systematic inferences from complex or ambiguous evidence.

What is critical thinking? To paraphrase the Enlightenment philosopher Immanuel Kant, it is the emergence from one’s self-imposed nonage. Nonage is the inability to use one’s mind without another’s guidance. This inability is self-imposed if its cause lies not in the limits of one’s mind but in the lack of courage to use it independently, without others’ guidance. Yet, in the age of powerful algorithms that play better chess and Go than humans, recommend the music and books we like, predict criminal behavior, and even find us the ideal romantic partner, why would we still need to think critically? Would it not be more economical to cease wasting time on thinking and reflecting, and just click and like? I argue that we need more, not less, critical thinking in the digital age. I discuss several tools for critical thinking, including asking the right questions and detecting misleading statistics, and illustrate these by online dating sites, HIV tests, cancer diagnosis, big data predictive analytics, the Social Credit System, and more. Advances in technology require risk-literate people who can control digital media rather be controlled by it.