Bayesian Average Rating 貝氏平均評分
Released已發布Apply Bayesian averaging to rank items by combining observed ratings with prior expectations. Use this skill when the user needs to rank items with varying review counts, build a 'top rated' list that handles low-sample items fairly, or implement IMDB-style weighted rating — even if they say 'weighted average rating', 'IMDB formula', or 'ranking with prior'.
演算法技能：Bayesian Average Rating 分析與應用。
Overview概述
Bayesian average combines an item's observed average rating with a prior (global average), weighted by review count. Formula: BR = (C × m + Σrᵢ) / (C + n) where m=global mean, C=confidence parameter, n=item reviews, Σrᵢ=sum of item ratings. Items with few reviews are pulled toward the global mean.
When to Use使用時機
Trigger conditions:
- Ranking items by continuous ratings (1-5 stars) with varying review counts
- IMDB-style "Top 250" lists that balance quality and popularity
- Any rating aggregation where new items shouldn't dominate with few high ratings
When NOT to use:
- For binary (upvote/downvote) data (use Wilson Score instead)
- When all items have similar review counts (simple average is sufficient)
Algorithm 演算法
IRON LAW: The Prior Protects Against Small-Sample Extremes
Without a prior, a single 5-star review makes an item "the best."
The Bayesian average adds C "phantom votes" at the global mean m,
shrinking small-sample items toward average. C controls shrinkage
strength: higher C = more conservative (more phantom votes).
Typical C = median review count across all items.
Phase 1: Input Validation
Compute: global mean rating (m) across all items, choose C (phantom vote count). Collect per item: review count (n), average rating, or sum of ratings. Gate: m computed, C selected, item data available.
Phase 2: Core Algorithm
- Global mean: m = Σ(all ratings) / Σ(all review counts)
- Bayesian average per item: BR = (C × m + n × avg_rating) / (C + n)
- Rank items by BR descending
- For items with n >> C, BR ≈ avg_rating (data dominates). For n << C, BR ≈ m (prior dominates).
Phase 3: Verification
Check: items with very few reviews should be near global mean. Items with many reviews should be near their actual average. Ranking is intuitive. Gate: Shrinkage behavior confirmed, top items have both high ratings AND sufficient reviews.
Phase 4: Output
Return ranked items with Bayesian scores.
Output Format輸出格式
{
"rankings": [{"item": "Movie_A", "bayesian_avg": 8.7, "raw_avg": 9.1, "reviews": 5000, "shrinkage": 0.04}],
"metadata": {"global_mean": 6.8, "confidence_C": 500, "items_ranked": 10000}
}
Examples範例
Sample I/O
Input: m=7.0, C=100. Item A: avg=9.5, n=5. Item B: avg=8.5, n=500. Expected: BR_A = (100×7 + 5×9.5)/(105) = 7.12. BR_B = (100×7 + 500×8.5)/(600) = 8.25. B ranks higher.
Edge Cases
| Input | Expected | Why |
|---|---|---|
| n=0 | BR = m (global mean) | No data, fully prior-driven |
| n=100000 | BR ≈ raw average | Massive sample overwhelms prior |
| All items same n | Equivalent to simple average ranking | Uniform shrinkage, ordering preserved |
Gotchas注意事項
- C selection is subjective: Common choices: median review count, minimum reviews for "reliable" rating (IMDB uses top 25,000 voters with min votes). No universally correct value.
- Rating scale matters: A 4.0 on a 5-point scale means something different than 4.0 on a 10-point scale. Normalize or use the same scale.
- Category-specific priors: A 4.0 average in "horror movies" might be exceptional, while 4.0 in "Studio Ghibli" might be below average. Consider category-level priors.
- Temporal bias: Old items accumulate reviews. Unless you weight recent reviews more, established items permanently dominate "top" lists.
- Review gaming: Bayesian average doesn't prevent review manipulation — it only mitigates small-sample extremes. Pair with fraud detection.
References參考資料
- For IMDB weighted rating formula, see
references/imdb-formula.md - For multi-dimensional Bayesian rating, see
references/multi-dimensional.md