\[
p(\theta\mid y, X) = \frac{p(y \mid X, \theta) * p(\theta)}{p(y)} = \frac{p(y \mid X, \theta) * p(\theta)}{\int p(y \mid X, \theta) * p(y) \space d\theta} \propto p(y \mid X, \theta) * p(\theta)
\]

- Bayesian inference is an approach to figuring out the updated \(\boldsymbol{p(\theta)}\) after observing \(\boldsymbol{y}\) and \(\boldsymbol{X}\)
- When \(\boldsymbol{p(y \mid X, \theta)}\) is evaluated at each value of \(\boldsymbol{y}\), it is called a likelihood function - this is our data generating process
- An MCMC algorithm draws from an implied probability distribution \(\boldsymbol{p(\theta \mid y, X)}\)
- In Stan we specify: \[
\log[p(\theta) * p(y \mid X, \theta)] = \log[p(\theta)] + \sum_{i=1}^{N}\log[p(y_i \mid x_i, \theta)]
\]