# What?

## Poll: What characterizes “Bayesian Analysis”?

- MAP estimators (maximum a posteriori)
- posterior means
- Bayes rule
- models where some unknown quantities are treated as random
- none of the above

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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 such as graphs)

- 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

**So… what is Bayesian Analysis?**

### Preview of key definitions

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

**Bayes estimators:** for data \(Y\), unobserved \(X\), loss \(L\), and possible actions \(A\), the Bayes estimator is defined as:

\[\operatorname{arg\,min}\{ \mathbb{E}[L(a, X) | Y] : a \in A \}\]

**Note:** you are not expected to understand this equation at this point!

### This course

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 do it in practice
- how to build models
- how to approximate conditional expectations.