/algo-rank-bayesian

name: "\"algo-rank-bayesian\""
description: "\"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'.\"."
allowed-tools: Bash, Read, Write, Edit, Glob, Grep

Bayesian Average Rating

Overview

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

**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

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

1. Global mean: m = Σ(all ratings) / Σ(all review counts) 2. Bayesian average per item: BR = (C × m + n × avg_rating) / (C + n) 3. Rank items by BR descending 4. 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

{
  "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

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

• **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.

Scripts

| Script | Description | Usage | |--------|-------------|-------| | `scripts/bayesian_avg.py` | Rank items using Bayesian average to handle small-sample extremes | `python scripts/bayesian_avg.py --help` |

Run `python scripts/bayesian_avg.py --verify` to execute built-in sanity tests.

References

• For IMDB weighted rating formula, see `references/imdb-formula.md`
• For multi-dimensional Bayesian rating, see `references/multi-dimensional.md`