"algo-rank-bayesian"

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.