Why Normal Instead of Cauchy Priors? 🤔

Why Normal Instead of Cauchy Priors? 🤔#

The Problem with Cauchy Priors#

1. Heavy Tails = Extreme Values#

Cauchy(0,1): Has infinite variance, very heavy tails
Problem: Allows extreme parameter values that are unrealistic
Example: sigma_scaledRI ~ cauchy(0, 0.1) can generate noise levels of 10+ (completely unrealistic for scaled RI)

2. Poor Hierarchical Performance#

// BAD: Cauchy hyperpriors
sigma_t0_culmeso ~ cauchy(0, 1);    // Can be huge!
t0_crtp ~ normal(t0_culmeso, sigma_t0_culmeso);  // Unstable hierarchy

Issue: Huge sigma values → weak hierarchical shrinkage
Result: Coretop parameters don’t learn from culture+mesocosm data

3. Sampling Difficulties#

Cauchy: Hard for Stan to sample efficiently
Symptoms: Slow convergence, divergent transitions, high Rhat
Cause: Extreme tail behavior confuses NUTS sampler

Why Normal Priors Are Better#

1. Realistic Constraints#

// GOOD: Informative normal priors
sigma_scaledRI_cul ~ normal(0, 0.1);        // Half-normal (lower=0): realistic noise levels
sigma_t0_culmeso ~ normal(0, 5);            // Reasonable temperature variation
sigma_k_culmeso ~ normal(0, 0.2);           // Respects k ∈ [0,1] bounds

2. Better Hierarchical Shrinkage#

// Proper hierarchical structure
t0_culmeso ~ normal(30, 10);                    // Culture+mesocosm center
sigma_t0_culmeso ~ normal(0, 5);                // Reasonable scale
t0_crtp ~ normal(t0_culmeso, sigma_t0_culmeso); // Coretop inherits info

Result: Purple line (coretop) properly positioned relative to orange/blue lines

3. Computational Efficiency#

Faster sampling: Normal distributions are Stan’s “native” format
Better convergence: Avoids extreme values that break sampler
Fewer divergences: Smoother posterior geometry

Scientific Justification#

For Observation Noise (`sigma_scaledRI`)#

// OLD: sigma_scaledRI ~ cauchy(0, 0.1)
// NEW: sigma_scaledRI ~ normal(0, 0.1)   // half-normal, lower=0

Expectation: Scaled RI measurements precise to ~0.01-0.05 units
Cauchy problem: Allows noise > 1.0 (larger than signal!)
Normal solution: Concentrates mass around realistic values

For Hierarchical Scales (`sigma_*_culmeso`)#

// OLD: sigma_t0_culmeso ~ cauchy(0, 1)  
// NEW: sigma_t0_culmeso ~ normal(0, 5)

Expectation: Coretop temperatures vary ±2-10°C from culture+mesocosm
Cauchy problem: Allows ±100°C variations (nonsensical)
Normal solution: Reasonable biological variation

The “Weakly Informative” Myth#

Cauchy ≠ Weakly Informative#

Common misconception: “Cauchy(0,1) is non-informative”
Reality: Cauchy is strongly informative toward extreme values
Better approach: Use normal priors scaled to domain knowledge

True Weak Information#

// Domain-appropriate weak priors
sigma_t0_culmeso ~ normal(0, 5);     // "Could be 1-10°C variation"
sigma_k_culmeso ~ normal(0, 0.2);    // "k usually 0.1-0.8, so σ < 0.5"

Expected Improvements in Your Models#

1. Faster Stan Sampling#

Fewer divergent transitions
Better Rhat values (< 1.01)
Higher effective sample size

2. Sensible Parameter Estimates#

Observation noise: 0.01-0.1 range (realistic)
Hierarchical scales: Reasonable biological variation
No extreme outlier estimates

3. Proper Hierarchical Behavior#

Coretop estimates informed by culture+mesocosm data
Purple line positioned correctly relative to other data
Better uncertainty quantification

Summary#

Normal priors provide the right balance of:

✅ Flexibility: Allow parameter exploration
✅ Constraint: Prevent unrealistic values
✅ Efficiency: Fast, stable Stan sampling
✅ Interpretability: Match scientific expectations

Cauchy priors are often:

❌ Too permissive: Allow extreme unrealistic values
❌ Computationally expensive: Slow sampling, divergences
❌ Scientifically inappropriate: Don’t match domain knowledge
❌ Hierarchically weak: Poor information sharing

Practical Guidelines for Prior Choice#

Use Normal Priors When:#

You have domain knowledge about reasonable parameter ranges
Working with hierarchical models (always!)
Parameter bounds are important (e.g., k ∈ [0,1])
Computational efficiency matters

Consider Other Distributions When:#

Beta: For bounded [0,1] parameters (like proportions)
Gamma/Lognormal: For positive-only parameters with skew
Student-t: When you need slightly heavier tails than normal
Half-normal: For positive scale parameters

Avoid Cauchy Unless:#

You specifically need very heavy tails
You’re modeling rare extreme events
Other distributions have failed for theoretical reasons

References and Further Reading#

Gelman et al. (2017): “Prior distributions for variance parameters in hierarchical models” - Argues against uniform/Cauchy hyperpriors
Stan User’s Guide: Section on “Priors for Hierarchical Models”
McElreath (2020): Statistical Rethinking - Chapter on weakly informative priors
Betancourt (2017): “A Conceptual Introduction to Hamiltonian Monte Carlo” - On computational considerations

Prior-standardization note for the TEXAS Stan models. The current observation-noise prior is the half-normal normal(0, 0.1) on sigma_proxyObs_* (see the variance-partitioning update, 2026-04-08).