STEP: 1528 / Test: 27/27 PASS / Lean 4: 7 theorem zero sorry / Type: STEP 1487 データサイエンス 3rd tool (47 defer 16 番目、 姉妹 tool 19)
❶ Beta-Binomial conjugate posterior: prior Beta(α, β) + observations (s successes, f failures) → posterior Beta(α+s, β+f)。 posterior mean + 95% CI → verdict。
❂ 7 case 決定表: strong evidence → TRUE / strong counter → FALSE / CI contains 0.5 → NEITHER / minimal shift → BOTH (non-informative) / posterior shifted → TRUE (moderate)。
| Condition | Verdict |
|---|---|
| invalid (α≤0, β≤0, s>trials) | NEITHER |
| posterior mean ≥ 0.9 (default) | TRUE (strong_evidence) |
| posterior mean ≤ 0.1 | FALSE (strong_counter_evidence) |
| 95% CI contains 0.5 | NEITHER (uncertain_posterior) |
| |posterior - prior| < 0.05 | BOTH (non_informative_update) |
| posterior shifted (moderate) | TRUE (posterior_shifted) |
// Strong evidence (100/100 successes with uniform prior)
bayesPosterior({ successes: 100, trials: 100 });
// { verdict: 'TRUE', posteriorMean: ~0.99, ciLower: ~0.97, ciUpper: 1.0 }
// Balanced data → NEITHER (CI contains 0.5)
bayesPosterior({ successes: 5, trials: 10 });
// { verdict: 'NEITHER', posteriorMean: 0.5, ciLower: ~0.25, ciUpper: ~0.75 }
// Custom prior Beta(2, 3) + 8/10 successes → posterior (10, 5)
bayesPosterior({ priorAlpha: 2, priorBeta: 3, successes: 8, trials: 10 });
// { verdict: 'TRUE', posteriorMean: ~0.667, reason: 'posterior_shifted' }
posterior_alpha_shift : posterior_alpha - prior_alpha = successes posterior_beta_shift : posterior_beta - prior_beta = failures posterior_no_evidence_alpha : 0 successes → posterior_alpha = prior_alpha uniform_prior_all_successes : Beta(1,1) + n successes → posterior (n+1, 1) no_data_no_update : successes=failures=0 → posterior = prior bayes_balanced_gives_neither : equal posterior → NEITHER bayes_strong_alpha_gives_true : posterior_a > 10 * b → TRUE
❶ Beta-Binomial のみ、 Normal-Normal / Gamma-Poisson 未 embed。
❂ Normal approx for CI (小 sample で 精度低下、 実 exact Beta CDF は 計算重い)。
❸ 「世界初」 なし = Bayes 1763 / Laplace 1812 既知 D-FUMT₈ mapping。
❹ 累計 defer 33 → 32。