Stan Fundamentals

Stan Fundamentals

When to Use This Skill

• Writing new Stan models from scratch
• Understanding Stan program structure
• Learning Stan syntax and conventions
• Translating models from other languages to Stan
• Optimizing existing Stan code

Program Structure

Stan models have up to 7 blocks in this exact order:

functions { }           // User-defined functions
data { }                // Input data declarations
transformed data { }    // Data preprocessing
parameters { }          // Model parameters
transformed parameters { } // Derived parameters
model { }               // Log probability
generated quantities { }  // Posterior predictions

All blocks are optional. Empty string is valid (but useless) Stan program.

Type System Quick Reference

Scalars

int n;                    // Integer
real x;                   // Real number
complex z;                // Complex number

Vectors and Matrices

vector[N] v;              // Column vector
row_vector[N] r;          // Row vector
matrix[M, N] A;           // Matrix

Arrays (Modern Syntax)

array[N] real x;          // 1D array of reals
array[M, N] int y;        // 2D array of integers
array[J] vector[K] theta; // Array of vectors

Constrained Types

real sigma;              // Non-negative
real p;         // Probability
simplex[K] theta;                 // Sums to 1
ordered[K] c;                     // Ascending
corr_matrix[K] Omega;             // Correlation
cov_matrix[K] Sigma;              // Covariance
cholesky_factor_corr[K] L_Omega;  // Cholesky correlation

Key Distributions

Continuous (SD parameterization!)

y ~ normal(mu, sigma);      // sigma is SD
y ~ student_t(nu, mu, sigma);
y ~ cauchy(mu, sigma);
y ~ exponential(lambda);
y ~ gamma(alpha, beta);
y ~ beta(a, b);
y ~ lognormal(mu, sigma);

Discrete

y ~ bernoulli(theta);
y ~ binomial(n, theta);
y ~ poisson(lambda);
y ~ neg_binomial_2(mu, phi);
y ~ categorical(theta);

Multivariate

y ~ multi_normal(mu, Sigma);        // Sigma is COVARIANCE
y ~ multi_normal_cholesky(mu, L);
y ~ lkj_corr(eta);

Essential Patterns

Vectorization

// GOOD - Efficient
y ~ normal(mu, sigma);

// BAD - Slow
for (n in 1:N) y[n] ~ normal(mu[n], sigma);

Non-Centered Parameterization

parameters {
  vector[J] theta_raw;
}
transformed parameters {
  vector[J] theta = mu + tau * theta_raw;
}
model {
  theta_raw ~ std_normal();
}

Target Syntax

// These are equivalent:
y ~ normal(mu, sigma);
target += normal_lpdf(y | mu, sigma);

Common Priors

// Location parameters
mu ~ normal(0, 10);

// Scale parameters
sigma ~ exponential(1);
sigma ~ cauchy(0, 2.5);  // half-Cauchy when sigma has lower=0

// Probabilities
theta ~ beta(1, 1);  // Uniform on (0,1)

// Regression coefficients
beta ~ normal(0, 2.5);

// Correlation matrices
Omega ~ lkj_corr(2);  // eta=2 favors identity

R Integration (cmdstanr)

library(cmdstanr)
mod  400
- [ ] ESS_tail > 400
- [ ] Zero divergences
- [ ] Not hitting max_treedepth
- [ ] Prior predictive produces sensible values
- [ ] Posterior predictive matches data pattern

## Key Differences from BUGS

| Feature | Stan | BUGS/JAGS |
|---------|------|-----------|
| Normal | `normal(mu, sigma)` SD | `dnorm(mu, tau)` precision |
| MVN | `multi_normal(mu, Sigma)` cov | `dmnorm(mu, Omega)` precision |
| Execution | Sequential (order matters) | Declarative (order doesn't matter) |
| Sampling | HMC/NUTS | Gibbs/Metropolis |

## Source & license

This open-source skill is cataloged on AgentStack and links to its original source — we do not rehost the code.

- **Author:** [choxos](https://github.com/choxos)
- **Source:** [choxos/BiostatAgent](https://github.com/choxos/BiostatAgent)
- **License:** MIT

Install and usage instructions live in the source repository linked above.