Introduction to Stan

Model the posterior distribution \[ \begin{aligned}[t] p(\theta|X) &= \frac{p(X | \theta) p(\theta)}{p(X)} \\ &\propto p(X | \theta) p(\theta) \\ \end{aligned} \]

Two steps:

1. Modeling: Define \( p(X | \theta) \) and \( p(\theta) \) that describe the data-generating process.
2. Computation: Estimate the posterior \( p(\theta | X) \)

Many methods:

1. Analytic, Exact Solutions : Conjugate distributions
2. Function Approximations : Laplace (MAP), Variational Bayes, Expectation Propogation
3. Sampling : Gibbs, Metropolis Hastings, Hamiltonian Monte Carlo (HMC), Importance Sampling

Location \( \mu \) of normal distribution with known scale \( \sigma \):

• Prior: \( \mu \sim N(\mu_0, \sigma_0) \)
• Likelihood: \( x \sim N(\mu, \sigma^2) \)
• Posterior: \( \mu | x \sim N\left(\left[\frac{\mu_0}{\sigma_0^2} + \frac{\sum_{i = 1}^n x_i}{\sigma^2}\right] \middle/ \left[ \frac{1}{\sigma_0^2 } + \frac{n}{\sigma_0^2} \right], \left( \frac{1}{\sigma_0^2} + \frac{n}{\sigma^2} \right)^{-1}\right) \)

Why not always ?

• Few known examples
• Most are univariate or simple distributions

1. Modeling: Probabilistic programming language to define models
2. Computation: Hamiltonian Monte Carlo

(Mostly) black-boxes computation so you can focus on modeling

Visualizations of two MCMC algorithms:

• Random Walk Metropolis Hastings
• Hamiltonian Monte Carlo (NUTS)

Creating and Estimating Models with Stan