  Bayesian inference gets its name from *Bayes's theorem*, expressing posterior
probabilities for hypotheses about a data generating process as the
(normalized) product of prior probabilities and a likelihood function. But
Bayesian inference uses all of probability theory, not just Bayes's theorem.
Many hypotheses of scientific interest are *composite hypotheses*, with the
strength of evidence for the hypothesis dependent on knowledge about auxiliary
factors, such as the values of nuisance parameters (e.g., uncertain background
rates or calibration factors). Many important capabilities of Bayesian methods
arise from use of the law of total probability, which instructs analysts to
compute probabilities for composite hypotheses by *marginalization* over
auxiliary factors. This tutorial targets relative newcomers to Bayesian
inference, aiming to complement tutorials that focus on Bayes's theorem and how
priors modulate likelihoods. The emphasis here is on marginalization over
parameter spaces -- both how it is the foundation for important capabilities,
and how it may motivate caution when parameter spaces are large. Topics covered
include the difference between likelihood and probability, understanding the
impact of priors beyond merely shifting the maximum likelihood estimate, and
the role of marginalization in accounting for uncertainty in nuisance
parameters, systematic error, and model misspecification.
