# Conditioning

## Outline

### Topics

- Intuition on conditioning
- A conditional probability is a probability.

### Rationale

- Conditioning is the workhorse of Bayesian inference!
- Used to define models (as when we assigned probabilities to edges of a decision tree)
- And soon, to gain information on latent variables given observations.

## Conditioning as belief update

**Key concept:** Bayesian methods use probabilities to encode *beliefs*.

We will explore this perspective in much more details next week.

## A conditional probability is a probability

The “updated belief” interpretation highlights the fact that we want the result of the conditioning procedure, \(\mathbb{P}(\cdot | E)\) to be a probability when viewed as a function of the first argument for any fixed \(E\).

## Intuition behind conditioning

- For a query even \(A\), what should be the updated probability?
- We want to remove from \(A\) all the outcomes that are not compatible with the new information \(E\). How?
- Take the intersection: \(A \cap E\)
- We also want: \(\mathbb{P}(S | E) = 1\) (last section)
- How?
*Renormalize*: \[\mathbb{P}(A | E) = \frac{\mathbb{P}(A \cap E)}{\mathbb{P}(E)}\] - Intersection can also be denoted using a comma, for example \(\mathbb{P}(A \cap E) = \mathbb{P}(A, E)\)