Bayesian Methods AI Prompts

Specify and interpret a Bayesian hierarchical (multilevel) model for this data. Data structure: {{data_structure}} (units nested in groups: students in schools, measurements in subjects) Outcome: {{outcome}} Level-1 predictors: {{l1_predictors}} (measured at the unit level) Level-2 predictors: {{l2_predictors}} (measured at the group level) Group variable: {{group_variable}} Number of groups: {{n_groups}} 1. Why hierarchical models: - Observations within groups are not independent (ICC > 0) - Using a single-level model underestimates standard errors (overestimates precision) - Hierarchical models borrow strength across groups (partial pooling): - Complete pooling: ignores group membership (too simple) - No pooling: estimates each group separately (too noisy for small groups) - Partial pooling: shrinks group estimates toward the overall mean (optimal) 2. Model specification: Level 1 (within groups): y_ij = alpha_j + beta_j * x_ij + epsilon_ij epsilon_ij ~ Normal(0, sigma^2) Level 2 (between groups): alpha_j ~ Normal(mu_alpha, tau_alpha^2) [random intercepts] beta_j ~ Normal(mu_beta, tau_beta^2) [random slopes, if specified] Hyperpriors (Bayesian): mu_alpha ~ Normal(0, 10) tau_alpha ~ HalfNormal(0, 1) [must be positive; HalfNormal is a good weakly informative prior] 3. Intraclass correlation coefficient (ICC): ICC = tau^2 / (tau^2 + sigma^2) - ICC = 0: no clustering; single-level model is fine - ICC = 0.10: 10% of variance is at the group level; standard errors inflated by DEFF = 1 + (m-1) x 0.10 - ICC > 0.20: strong clustering; hierarchical model is essential 4. Cross-level interactions: - Does the effect of x_ij (Level 1) vary with w_j (Level 2)? - Include: beta_j = gamma_10 + gamma_11 * w_j + u_1j - This is the hallmark of hierarchical modeling: testing whether context moderates individual-level effects 5. Convergence diagnostics (for MCMC estimation): - R-hat (Gelman-Rubin): should be < 1.01 for all parameters - Effective sample size (ESS): should be > 100 (preferably > 400) for each parameter - Trace plots: chains should mix well, with no trends or stuck periods - Posterior predictive check: does the model reproduce the observed data distribution? Return: hierarchical model specification, ICC calculation, cross-level interaction interpretation, and MCMC convergence assessment.

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.