TeeGrade

Model at a glance

Let μᵢ denote a latent, unobserved quality estimate for course i. In production, TeeGrade uses a tie-aware Bradley-Terry-Davidson pairwise model with priors. A compact way to read that pipeline is: the pairwise model defines the learned signal μᵢ^pairwise; available evidence determines how strongly that signal pulls away from the prior; the result is the final latent score Sᵢ; and Sᵢ is then mapped to the displayed TeeGrade scale. TeeRank is simply the ordering induced by those displayed scores.

Conceptual pipeline

1. 01

   Prior

   Tier ratings and broad sentiment provide μᵢ^prior, the regularizing baseline.

2. 02

   Pairwise signal

   A tie-aware Bradley-Terry-Davidson comparison model defines the learned component μᵢ^pairwise.

3. 03

   Latent score

   Available evidence determines how the model blends prior and pairwise signal into Sᵢ.

4. 04

   Displayed TeeGrade

   The latent score is mapped monotonically onto the public 0–10 TeeGrade scale.

5. 05

   TeeRank

   Displayed scores are ordered from high to low; no separate manual ranking is assigned.

Pairwise model and likelihood

Compact Bradley-Terry-Davidson pairwise view

Likelihood view

• μᵢ is the latent quality estimate for course i.
• σ(·) denotes the logistic link used in the compact win/loss view.
• D denotes the observed set of pairwise comparisons.
• μ̂ denotes the estimated latent score vector under the model.

Each head-to-head outcome is treated as a noisy observation of relative preference. In production, this pairwise component is modeled with a tie-aware Bradley-Terry-Davidson likelihood. For readability, the equations below show the simpler win/loss view and omit the explicit tie term. Conceptually, the learned latent score vector can be viewed as the value that maximizes the likelihood of the observed comparison set under this model, without tying the page to any specific solver or update procedure.

Prior, blend, and display

At a high level, the appendix uses several connected quantities:

Prior, evidence weight, latent score, and display mapping

• μᵢ^prior is a regularizing baseline derived from tier ratings and broad sentiment.
• μᵢ^pairwise is the comparison-driven signal inferred from observed head-to-head results under the tie-aware Bradley-Terry-Davidson component.
• σ² denotes an abstract prior uncertainty scale around μᵢ^prior and acts as a regularizing anchor when evidence is sparse.
• wᵢ = nᵢ / (nᵢ + k) is a compact evidence-weighting sketch, where nᵢ is the amount of comparison evidence for course i and k controls how quickly pairwise signal dominates.
• Sᵢ denotes the final latent score before the public display mapping is applied.
• g(·) denotes an abstract monotone mapping from latent score to the displayed 0–10 TeeGrade scale.

The pairwise model above produces the learned component μᵢ^pairwise. In the production system, the final latent score is estimated as a MAP solution under both prior and pairwise evidence. The shrinkage expression below is a compact public approximation of that behavior: with sparse evidence, the prior anchors more strongly; with richer evidence, the pairwise signal carries more of the score. A monotone display mapping then places the result on the public 0–10 TeeGrade scale.

Confidence, stability, and ordering

Uncertainty and ordering

• more comparison evidence narrows the plausible range around a course's latent score
• well-observed courses are less sensitive to small amounts of new evidence
• ordering is computed directly from TeeGrade rather than assigned by hand

Uncertainty is about evidence, not prestige. As comparisons accumulate, the plausible range around a course's latent score narrows, the ordering becomes harder to move with small amounts of new data, and TeeRank remains a direct ordering of the displayed TeeGrade values rather than a separately curated list.