# Poll

### What characterizes “Bayesian Analysis”?

1. MAP estimators (maximum a posteriori)
2. posterior means
3. Bayes rule
4. models where some unknown quantities are treated as random
5. none of the above

# What characterizes “Bayesian Analysis”?

1. MAP estimators (maximum a posteriori)
• MAP is seldom used by expert Bayesians (mode is misleading in high dimensions)
2. posterior means
• the posterior mean is often undefined (e.g. Bayesian analysis over combinatorial objects)
3. Bayes rule
• Bayes rule is intractable in most practical situations (we use MCMC/variational methods)
4. 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

# Preview of key definitions

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:

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

## 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 use it in practice
• how to build models
• how to approximate conditional expectations (MCMC methods)
• A bit of theory (asymptotics)