Bayesian Hypothesis Testing AI Prompt

Perform a Bayesian analysis as an alternative or complement to frequentist hypothesis testing.

Hypothesis: {{hypothesis}}
Data: {{data}}
Prior information: {{prior_info}} (literature values, expert knowledge, or 'weakly informative')

1. The Bayesian framework:
   - Prior: P(theta) — your belief about the parameter before seeing the data
   - Likelihood: P(data | theta) — how probable is the data at each value of theta?
   - Posterior: P(theta | data) ∝ P(data | theta) x P(theta)
   - The posterior combines prior belief with the evidence from the data

2. Prior specification:

   Informative prior:
   - Based on previous studies or expert knowledge
   - Example: beta(8, 4) for a success probability believed to be around 0.67
   - Must be documented and justified

   Weakly informative prior:
   - Provides mild regularization without dominating the data
   - Example: Normal(0, 2.5) on logistic regression coefficients (Gelman's recommendation)
   - Prevents extreme estimates while allowing the data to speak

   Non-informative / reference prior:
   - Jeffreys prior: invariant under reparameterization
   - Flat prior: uniform over all values (often improper and not recommended)

3. Bayes factor:
   BF = P(data | H1) / P(data | H0)
   - BF > 10: strong evidence for H1
   - BF 3-10: moderate evidence
   - BF 1-3: weak evidence
   - BF < 1: evidence favors H0
   - BF 1/10 to 1/3: moderate evidence for H0
   - Interpretation: the Bayes factor is how much more probable the data is under H1 vs H0

4. Posterior credible interval:
   - The 95% credible interval contains the true parameter with 95% posterior probability
   - Contrast with frequentist CI: the frequentist CI does NOT have a probability interpretation for the parameter
   - Highest Density Interval (HDI): the shortest interval containing 95% of the posterior mass

5. Decision making under uncertainty:
   - Region of Practical Equivalence (ROPE): define a range of effect sizes that are practically negligible
   - If the posterior is entirely within the ROPE: accept H0 as practically equivalent
   - If the posterior is entirely outside the ROPE: reject H0
   - If the posterior overlaps the ROPE: suspend judgment

Return: posterior distribution, Bayes factor, 95% credible interval, and ROPE-based decision.