When a chain runs but the geometry says no

The geometry

A wide mouth narrows into a difficult neck

Let y be normal with standard deviation 3 and x conditional on y be normal with variance exp(y). Negative y compresses x into a narrow neck while positive y permits a wide range. One Euclidean step size must traverse both regions.

DifferentiableLogDensity funnel = (state, gradient) -> {
  double y = state[0], x = state[1];
  double inverseVariance = Math.exp(-y);
  gradient[0] = -y / 9.0 - 0.5
      + 0.5 * x * x * inverseVariance;
  gradient[1] = -x * inverseVariance;
  return -0.5 * y * y / 9.0 - 0.5 * y
      - 0.5 * x * x * inverseVariance;
};

Review

Connect symptoms to evidence

McmcDiagnosticReport report = McmcDiagnostics.analyze(
    new String[] {"y", "x"}, chains);
ChartSpec pairs = DiagnosticGraphs.pairs("y", 0, "x", 1, chains);
ChartSpec energy = DiagnosticGraphs.energy(30, chains);
String review = InferenceHtmlReport.render(
    "Funnel review", report, null, pairs, energy);

Response order

Fix meaning and geometry before tuning

1. Verify the targetCheck gradients, initial density, units, constraints, and prior implications.
2. ReparameterizeFor hierarchical models, sample standardized latent effects and reconstruct their scale.
3. Then tuneA higher target acceptance may reduce residual integration error after geometry improves.

Eight schools

Use a non-centered hierarchy

// Sample mu, logTau, and standard-normal z[j].
double tau = Math.exp(logTau);
double thetaJ = mu + tau * z[j];

This formulation decouples latent effects from a weakly identified group scale. The named eight-schools implementation in the example catalog includes the complete analytic gradient, and its test compares that gradient with finite differences.