# Notes for Bayesian Statistics

Fall, 2023

## Discrete prior

## Continuous prior via conjugacy

### Binomial-Beta model

### Poisson-Gamma model

### Normal-normal model (known variance)

Likelihood:

$$Y_i \sim \mathcal{N}(\mu, \sigma^2)$$

Prior:

$$\mu \sim \mathcal{N}(m, s^2)$$

Posterior:

$$
\begin{align}
p(\mu \mid y_1, \ldots, y_n) & \propto p(y_1, \ldots, y_n \mid \mu) p(\mu) \\
&= \left( \prod_{i=1}^n p(y_i \mid \mu)\right) p(\mu) \\
&= \left( \prod_{i=1}^n \frac{1}{\sqrt{2\pi\sigma^2}} e^{-\frac{1}{2\sigma^2} (y_i - \mu)^2} \right)\frac{1}{\sqrt{2\pi s^2}} e^{-\frac{1}{2 s^2} (\mu - m)^2} \\
&\propto \left( \prod_{i=1}^n e^{-\frac{1}{2\sigma^2} (y_i - \mu)^2} \right) e^{-\frac{1}{2 s^2} (\mu - m)^2} \\
&= e^{-\frac{1}{2\sigma^2} \sum_i (y_i - \mu)^2 -\frac{1}{2 s^2} (\mu - m)^2} \\
&=
\end{align}
$$

### Normal-normal model (unknown variance)

## Posterior predictive

Simulating from posterior predictive

Model checking

## Tuning priors