## Outline

### Topics

- Review of basic probability theory concepts: outcome, event, sample space
- Intuition from the Bayesian perspective

### Rationale

- One definition of Bayesian inference: applying
*probability theory* to *statistical inference problems*
- Therefore, it is critical to understand probability to learn Bayesian inference
- This week, we will help you “reload in memory” some of the most important bits of probability theory used in this course

## Definitions

**Sample space**, denoted \(S\), a set.
- Example: \(S = \{1, 2, 3, 4\}\) (see Figure).

- Each element \(s\) of \(S\) is called an
**outcome**, \(s \in S\).
- Example: each of the 4 points.

- A set of outcomes \(E \subset S\) is called an
**event**.
- Example: \(E = \{s \in S : s \text{ is odd}\}\) (red in the Figure).

## Intuition: Bayesian view

- In Bayesian statistics, an outcome will describe the state of the world.
- We do not know which outcome is the true state of the world.
- We
**observe** partial information on the state of the world/outcome.
- We rule out the outcomes that are not consistent with the observation…
- …but there will be several outcomes left!
- We will deal with this situation using probability theory.

## Intuition: randomized algorithms

- An algorithm is “randomized” if it has access to virtual dices/coins.
- In practice this is done using pseudorandom number generators.
- In this context an outcome is a random seed, i.e. the initialization of the pseudorandom number generator.