# Bayesian Massey Model — Specification

A hierarchical Bayesian generalization of Massey's least-squares ranking method,
designed to plug into the Chatterjee, Datta, Hou & Mandal (2024) credible-ranking
machinery. This is the margin-of-victory analog; the win/loss (Colley) analog
follows the same structure with a Bernoulli likelihood instead of Gaussian.

## 1. Notation

- Teams: $i = 1, \dots, m$
- Games: $k = 1, \dots, N$, with home team $h(k)$ and away team $a(k)$
- Observed margin: $Y_k = \text{score}_{h(k)} - \text{score}_{a(k)}$
- Team-level covariates: $\mathbf{x}_i \in \mathbb{R}^p$ (standardized; column 1 is an
  intercept)
- Latent team strength: $\theta_i$ (points scale — the Massey rating)
- Home-court advantage: $\gamma$
- Game-level residual SD: $\sigma$
- Between-team ("linking") SD: $A$
- Linking-model coefficients: $\beta \in \mathbb{R}^p$

## 2. Likelihood (game level)

$$
Y_k \mid \boldsymbol{\theta}, \gamma, \sigma \;\overset{ind}{\sim}\; N\big(\theta_{h(k)} - \theta_{a(k)} + \gamma,\; \sigma^2\big), \quad k = 1, \dots, N
$$

This *is* the classical Massey model — the normal equations Massey solves
($\mathbf{M}\mathbf{r} = \mathbf{p}$) are exactly the MLE of this Gaussian linear
model with a flat prior on $\theta$. Everything below is the Bayesian layer added
on top.

## 3. Linking model (the Fay-Herriot analog)

$$
\theta_i \mid \boldsymbol{\beta}, A \;\overset{ind}{\sim}\; N(\mathbf{x}_i^\top \boldsymbol{\beta}, \, A), \quad i = 1, \dots, m
$$

Two variants, mirroring Chatterjee et al.'s UB / HB distinction:

- **M0 (UB analog)** — $\mathbf{x}_i = [1]$ only (single intercept). Every team
  shrinks toward one common league mean. No external information used; this is
  the closest analog to "no covariates," but note it is *not* a flat improper
  prior the way Chatterjee's actual UB model is (see Section 5).
- **M1 (HB analog)** — $\mathbf{x}_i$ includes real covariates: conference
  strength/dummy, preseason rating, recruiting index, prior-season SRS/NET,
  returning-production metric, etc.

## 4. Hyperpriors

Weakly informative, tune scales to the sport (values below are a college-basketball
starting point — margin residual SD is typically 10-12 points; adjust upward for
football, e.g. $\sigma \sim N^+(0, 17^2)$).

$$
\beta_j \sim N(0, 2.5^2), \qquad
A \sim N^+(0, 2.5^2), \qquad
\sigma \sim N^+(0, 10^2), \qquad
\gamma \sim N(0, 5^2)
$$

$N^+$ denotes half-normal (support restricted to positives).

## 5. Identifiability — an important structural difference from Chatterjee's setting

In Chatterjee et al., each $\theta_i$ has its *own* independent direct observation
$Y_i \sim N(\theta_i, D_i)$, so a flat improper prior is fine — every entity is
individually identified by its own data. That's not true here: $\theta$ is only
identified through the *network* of games, and classical Massey needs an
artificial fix (replace one row of the linear system with a sum-to-zero
constraint) because the Massey matrix is a graph Laplacian — singular whenever
the schedule graph is disconnected or poorly connected (e.g., isolated
conferences with few common opponents).

The proper hierarchical prior above ($A$ finite, estimated from data rather than
fixed at $\infty$) fixes this automatically: it's a Bayesian ridge penalty that
regularizes exactly the directions in which the schedule graph under-identifies
$\theta$. No artificial sum-to-zero constraint is needed — and since we only ever
care about the *induced ranking*, invariance to a constant additive shift in all
$\theta_i$ doesn't affect anything downstream anyway.

**Validity check worth running:** as $A \to \infty$ (very weak shrinkage) under
M0, the posterior mean of $\theta$ (up to a location shift) should converge to
the classical Massey point solution. Confirming this is a good sanity check that
the model is implemented correctly and nests Massey as a limiting case.

## 6. Sampling

Fit via MCMC (Stan). Use a **non-centered parameterization** for $\theta$ to avoid
funnel geometry when $A$ is small:

$$
\theta_i = \mathbf{x}_i^\top \boldsymbol{\beta} + A \cdot z_i, \qquad z_i \sim N(0,1)
$$

Run enough chains/iterations to check $\hat{R} \approx 1$ and adequate effective
sample size on $\theta$, $A$, $\sigma$, and $\gamma$ before trusting downstream
rank inferences — poor mixing on $A$ in particular will distort the credible sets.

## 7. Downstream: Chatterjee-style credible rank distributions

Given $S$ post-warmup posterior draws $\theta^{(1)}, \dots, \theta^{(S)}$:

1. **Cartesian credible set** $\mathcal{S}_{C,\alpha}$: per-team empirical
   $(\alpha/2, 1-\alpha/2)$ quantile interval from the marginal draws of
   $\theta_i$; take the Cartesian product across teams.
2. **Elliptical credible set** $\mathcal{S}_{E,\alpha}$: Mahalanobis distance
   $M(\theta^{(s)}, \bar\theta, \hat\Sigma) \le c_\alpha$ using the posterior mean
   $\bar\theta$ and covariance $\hat\Sigma$ of the draws, with $c_\alpha$ chosen so
   the empirical coverage is $1-\alpha$.
3. For every draw retained in the chosen credible set, compute the induced rank
   vector (rank teams by $\theta$ descending).
4. Tabulate these into the $m \times m$ empirical marginal credible distribution
   (EMCD) matrix $\mathbf{R}_{Bayes}$ — column $j$ is team $j$'s posterior
   probability mass function over possible ranks.
5. Report per team: plausible rank range and probabilities, posterior mean rank,
   $AMD_j$ (posterior mean absolute deviation of rank), and $EC_j$ (effective
   cardinality = exp(entropy) of the rank distribution) exactly as defined in
   Chatterjee et al. Section 3.7.
6. Benchmark against the plain point-estimate Massey ranking plus a bootstrap
   confidence set (resample games, resolve the Massey system each time) as the
   frequentist comparison — this stands in for their KWW/MRSW baseline.

## 8. Practical notes / extensions

- Standardize covariates before building $\mathbf{X}$ so the $N(0, 2.5^2)$ prior
  on $\beta$ is a reasonable weakly-informative default.
- Repeated matchups (teams playing each other more than once) need no special
  handling — just multiple rows in the games table.
- Consider a Student-$t$ likelihood in place of Normal if blowout games are
  producing residual outliers.
- Natural extensions: team-specific home advantage $\gamma_i$; separate $A$'s per
  conference; adding a possession-pace or tempo covariate at the game level.