# Posterior distributions

## Outline

### Topics

- Notion of posterior distribution.
- Examples
- How to use a posterior?

### Rationale

The posterior distribution appears in the second step of the Bayesian Recipe and is therefore encountered (at least implicitly) in all full Bayesian problems.

## Definition

The conditional PMF of the unknowns \(X\) given the observation \(Y\) is the called the **posterior PMF**.^{1}

### Notation

We use \(\pi(x) = \mathbb{P}(X = x | Y = y)\) for the posterior PMF.

**Recall:**

\[\pi(x) = \frac{\gamma(x)}{Z},\]

where \(\gamma(x) = \mathbb{P}(X = x, Y = y)\) is the un-normalized posterior and \(Z = \mathbb{P}(Y = y)\) is the normalization constant.

## Examples

### Example 1: coins in a bag

Visualization of the prior PMF and how it arises from the decision tree and random variable \(X\) (showing 3 flips instead of 4):

- Recall that the probability of each path is the product of the edge labels.

Visualization of the **posterior** PMF and how it arises from the decision tree and random variable \(X\):

- Recall we zero out the contribution of the paths not compatible with the observation (Heads, Heads, Heads).
- This gives a list of numbers that do not sum to one, so we renormalize them.

### Example 2: rocket insurance

You will construct prior and posterior PMFs in question 2 of this week’s exercises.

## What to do with a posterior?

- Show a visualiation (posterior PMF).
- Compute a
**summary**of the PMF:**point estimate:**single “best guess”, or**a credible region:**a set of “guesses”.

- More generally: decision theory / Step 3 of the Bayesian Recipe.

## Footnotes

if the unknown quantity is continuous, the posterior will be expressed using a density. A term that captures both the continuous and discrete case is “distribution” i.e. “posterior distribution”.↩︎