Advanced Bayesian Modeling in Stan

Common Roadblocks We’ll Solve Together

?
Bridging domain physics with Bayesian likelihoods (what actually goes into p(y|θ)?).
?
Working with grouped/binomial data via sufficient statistics without losing information.
?
Writing clean, stable Stan that samples fast and stays numerically sane.
?
Explaining parameters stakeholders understand (e.g., a golfer’s precision, σ).
?
Designing PPCs that actually validate model assumptions.

Free Preview • Physics-Constrained Demo

Golf Putting: Direction, Velocity, and Probability of a Make

What you’ll build

Binomial model for made/attempts with a physics-informed success probability: P(x) = 2·Φ(asin((R − r)/x) / σ) − 1, where σ encodes directional variability.

Files / Where to look

golf_angle.stan, bayes2_code_notes.r • Load data (x, n, y; radii r, R) and run the Stan model.

Stan model (excerpt)

// golf_angle.stan
data {
  int J;                 // number of observed distances
  vector[J] x;           // distances (ft)
  array[J] int n;        // attempts
  array[J] int y;        // successes
  real r;                // radius ball
  real R;                // radius hole
}
transformed data {
  vector[J] angle_rad = asin( (R - r) ./ x );
}
parameters {
  real<lower=0> sigma_rad;
}
model {
  // prior
  sigma_rad ~ std_normal();

  // likelihood
  vector[J] p = 2 * Phi( angle_rad / sigma_rad ) - 1;
  y ~ binomial(n, p);
}

R workflow (snippet)

data_list <- list(
  J = length(x),
  x = x,
  n = n_attempts,
  y = y_made,
  r = ball_radius,
  R = hole_radius
)

mod <- cmdstanr::cmdstan_model("golf_angle.stan")
fit <- mod$sample(data = data_list)

posterior::summarise_draws(fit$draws(c("sigma_rad")))

Why it matters