bayesian-stats

When the user invokes `/bayesian-stats`, help them convert frequentist statistical tests into Bayesian equivalents. If an argument is provided (e.g., `/bayesian-stats t-test`), look up that specific test. Otherwise, ask which frequentist test they want to convert. ## Mappings | Frequentist Test | Bayesian Equivalent | Python Library | |-----------------|--------------------|--------------| | Paired t-test | Bayesian paired t-test with JZS prior → BF₁₀ | `pingouin.bayesian_ttest(x, y, paired=True)` | | Independent t-test | Bayesian independent t-test with JZS prior → BF₁₀ | `pingouin.bayesian_ttest(x, y, paired=False)` | | Wilcoxon signed-rank | Bayesian paired t-test (robust alternative) or Bayesian sign test via PyMC | `pingouin` for approximate BF, `pymc` for full model | | Fisher's exact / Chi-squared | Beta-Binomial model with Beta(1,1) priors on each group's success rate | Analytical or `pymc`: `pm.Beta("p", 1, 1)` per group | | Mixed-effects logistic regression | Bayesian mixed-effects model | `bambi`: `bmb.Model("y ~ condition + (1|question)", data, family="bernoulli")` | | ANOVA / F-test | Bayesian ANOVA | `pingouin.bayesian_anova(data, dv, between)` or `bambi` | | Pearson correlation | Bayesian correlation | `pingouin.bayesian_corr(x, y)` | | Bootstrap CI | Posterior credible interval from MCMC | `pymc` model → `arviz.summary()` for HDI | ## Code Snippets ### Bayesian Paired t-test (replaces Wilcoxon / paired t-test) ```python import pingouin as pg bf = pg.bayesian_ttest(x, y, paired=True, r=0.707) # JZS prior, Cauchy scale r=√2/2 print(f"BF₁₀ = {bf:.3f}") ``` ### Beta-Binomial (replaces Fisher's exact) ```python import pymc as pm import arviz as az with pm.Model(): p_a = pm.Beta("p_a", 1, 1) # Condition A success rate p_b = pm.Beta("p_b", 1, 1) # Condition B success rate pm.Binomial("obs_a", n=n_a, p=p_a, observed=k_a) pm.Binomial("obs_b", n=n_b, p=p_b, observed=k_b) delta = pm.Deterministic("delta", p_b - p_a) trace = pm.sample(4000) az.summary(trace, var_names=["delta"], hdi_prob=0.95) az.plot_posterior(trace, var_names=["delta"], ref_val=0) ``` ### Bayesian Mixed-Effects (replaces frequentist mixed-effects) ```python import bambi as bmb import arviz as az model = bmb.Model("correct ~ advice_source * question_category + (1|question_id)", data, family="bernoulli") results = model.fit(draws=4000) az.summary(results, var_names=["advice_source", "question_category", "advice_source:question_category"]) ``` ## Interpreting Bayes Factors (BF₁₀) | BF₁₀ | Evidence | |-------|----------| | > 100 | Extreme evidence for H₁ | | 30–100 | Very strong evidence for H₁ | | 10–30 | Strong evidence for H₁ | | 3–10 | Moderate evidence for H₁ | | 1–3 | Anecdotal evidence for H₁ | | 1/3–1 | Anecdotal evidence for H₀ | | 1/10–1/3 | Moderate evidence for H₀ | | 1/30–1/10 | Strong evidence for H₀ | | < 1/30 | Very strong evidence for H₀ | Key advantage: BF < 1/3 provides **evidence for the null**, not just "failure to reject." This is impossible with p-values. ## Best Practices 1. **Always report both** frequentist (p-values, CIs) and Bayesian (BF₁₀, posterior credible intervals) results. Reviewers expect p-values; Bayesian results add rigor. 2. **Sensitivity analysis**: Rerun Bayes factors with different prior scales (e.g., Cauchy r = 0.5, √2/2, 1.0). If conclusions are robust across priors, the result is more credible. 3. **Small samples**: Bayesian methods handle small n more gracefully — posteriors are properly wide when data is limited, rather than giving misleading p-values. 4. **Combining experiments**: Posteriors from one experiment become priors for the next. This is how evidence accumulates across studies. 5. **No multiple comparison correction needed**: Bayesian updating naturally handles multiplicity (though model comparison via Bayes factors still requires care). ## Required Packages ``` uv add pingouin pymc bambi arviz ```