Poll

What characterizes “Bayesian Analysis”?

All these popular answers are misleading and/or very incomplete:

MAP estimators (maximum a posteriori)
- MAP is seldom used by expert Bayesians (mode is misleading in high dimensions)
posterior means
- the posterior mean is often undefined (e.g. Bayesian analysis over combinatorial objects)
Bayes rule
- Bayes rule is intractable in most practical situations (we use MCMC/variational methods)
models where some unknown quantities are treated as random
- true for Bayesian models, but also for many non-Bayesian models, e.g., random effect models

Bayesian Analysis: statistical discipline centered around the use of Bayes estimators

Bayes estimators: for data \(X\), unobserved \(Z\), loss \(L\), and possible actions \({\mathcal{A}}\), the Bayes estimator is defined as:

\[{\textrm{argmin}}\{ {\mathbf{E}}[L(a, Z) | X] : a \in {\mathcal{A}}\}\]

The primary objective of this course is to understand Bayes estimators:

Why they are so powerful
Their limitations (model misspecification, computational challenges)
Important special cases (posterior means, credible intervals, MAP)
How to use it in practice
- how to build models
- how to approximate conditional expectations (MCMC methods)
A bit of theory (asymptotics)

Christian Robert, The Bayesian Choice, 2nd edition.