Update app.py

app.py

@@ -180,11 +180,17 @@ def analyze(trial_type,
 
 ### 機率表（對應論文 Table 1 & 2）
 
-| 階段 | Beta 參數 | θ ≤ R₀ | R₀ < θ ≤ R₁ | θ > R₁ |
-|------|-----------|--------|-------------|--------|
-| Prior | Beta({prior_a:.1f}, {prior_b:.1f}) | {prior_prob_lt_r0:.3f} | {prior_prob_r0_r1:.3f} | {prior_prob_gt_r1:.3f} |
-| Stage 1 posterior | Beta({post1_a:.1f}, {post1_b:.1f}) | {post1_prob_lt_r0:.3f} | {post1_prob_r0_r1:.3f} | {post1_prob_gt_r1:.3f} |
-| Stage 2 posterior | Beta({post2_a:.1f}, {post2_b:.1f}) | {post2_prob_lt_r0:.3f} | {post2_prob_r0_r1:.3f} | {post2_prob_gt_r1:.3f} |
+| 階段 | Beta 參數 | π（平均值） | θ ≤ R₀ | R₀ < θ ≤ R₁ | θ > R₁ |
+|------|-----------|-------------|--------|-------------|--------|
+| Prior | Beta({prior_a:.1f}, {prior_b:.1f}) | **{beta_mean(prior_a, prior_b):.3f}** | {prior_prob_lt_r0:.3f} | {prior_prob_r0_r1:.3f} | {prior_prob_gt_r1:.3f} |
+| Stage 1 posterior | Beta({post1_a:.1f}, {post1_b:.1f}) | **{beta_mean(post1_a, post1_b):.3f}** | {post1_prob_lt_r0:.3f} | {post1_prob_r0_r1:.3f} | {post1_prob_gt_r1:.3f} |
+| Stage 2 posterior | Beta({post2_a:.1f}, {post2_b:.1f}) | **{beta_mean(post2_a, post2_b):.3f}** | {post2_prob_lt_r0:.3f} | {post2_prob_r0_r1:.3f} | {post2_prob_gt_r1:.3f} |
+
+### 貝氏更新過程（練習用）
+
+| 事前分佈 | Stage I 結果 | Stage I 後分佈 | Stage II 結果 | Stage II 後分佈 |
+|----------|-------------|---------------|--------------|----------------|
+| Beta({prior_a:.1f},{prior_b:.1f}), π={beta_mean(prior_a, prior_b):.2f} | 成功:{s1_success}, 失敗:{s1_total - s1_success} | Beta({post1_a:.1f},{post1_b:.1f}), π={beta_mean(post1_a, post1_b):.2f} | 成功:{s2_success}, 失敗:{s2_total - s2_success} | Beta({post2_a:.1f},{post2_b:.1f}), π={beta_mean(post2_a, post2_b):.3f} |
 
 ### 結果解讀
 - **P(θ > R₁) = {post2_prob_gt_r1:.1%}**：藥物反應率超過理想值的機率