Forward Compiled-Additive-Prior HPM x3

This experiment moves the learned additive portion of canonical linear HPM x3 from the carried lineup function into the following season's player prior. It tests the model boundary directly: additive player-profile signals should be player state; only genuinely non-additive unit shape should remain context.

For completed season \(t\), the linear context fit is

\[ C_t(H,A)= \sum_k \beta_{k,t} \left(\sum_{i\in H}z_{i,t,k}-\sum_{j\in A}z_{j,t,k}\right) +C^{\mathrm{shape}}_t(H,A), \]

where the eight \(z_k\) coordinates are three-point attempts and makes, assists, turnovers, usage, steals, blocks, and offensive-rebound claim. The six remaining x3 coordinates are lineup-shape terms: shooting depth, credible shooter count, top-two assists, usage concentration, shooting-by-usage, and shooter-by-passing.

The prior for season \(t+1\) is then

\[ \mu_{i,t+1}=g_t(r_{i,t}, \mathrm{age}, \mathrm{exposure}, \ldots) +\sum_k \beta_{k,t}\bigl(z_{i,t+1,k}-\bar z_{t,k}\bigr), \]

using only the prior-season fitted \(\beta_t\) and the target season's lagged player profile. \(\bar z_{t,k}\) is a centering convention; it cancels from any five-versus-five margin.

To avoid double counting, the carried context is only

\[ C^{\mathrm{shape}}_t(H,A)=C_t(H,A)- \sum_k \beta_{k,t} \left(\sum_{i\in H}z_{i,t+1,k}-\sum_{j\in A}z_{j,t+1,k}\right). \]

For fixed player coefficients and \(\beta_t\), this is an exact accounting identity. The experiment can differ from ordinary HPM only because the enhanced player prior changes later RAPM refits and subsequent learned state.

This is distinct from Context-Reattributed RAPM. It does not project total context back onto players using another regression; it transfers only terms that are algebraically player-additive in the fitted HPM equation.

Outputs

Each training run stores:

season_compiled_additive_prior_coefficients.parquet: the eight raw prior-season \(\beta\) coefficients carried into each target season;
season_player_priors.parquet: the resulting centered player priors;
season_player_prior_metadata.parquet: transfer coverage and centering metadata;
season_context_models.joblib: the full completed-season 14-term context fit, from which the six-term residual context is evaluated.

The frozen three-season result will determine whether this state allocation improves prediction relative to NAIL-RAPM v1.0.

Frozen Three-Season Result

The completed replay pools 584,970 regular-season possessions from 3,284 games and 39,967 playoff possessions from 238 games. The transfer reached every eligible player in the three target seasons: 547 in 2023-24, 547 in 2024-25, and 562 in 2025-26.

Cohort	Metric	Canonical x3	Additive-prior x3	Candidate minus canonical
Regular season	Possession RMSE	1.197977	1.198079	+0.000102
Regular season	Possession MAE	1.141387	1.141438	+0.000051
Regular season	Eligible-game RMSE	14.1119	14.3203	+0.2084
Regular season	Full-game RMSE	14.3517	14.5908	+0.2391
Regular season	Winner accuracy	68.33%	67.45%	-0.88 pp
Regular season	Team NetRtg RMSE	3.4307	3.7410	+0.3103
Regular season	Pythagorean-win RMSE	7.3460	8.0203	+0.6743
Playoffs	Possession RMSE	1.192752	1.192679	-0.000073
Playoffs	Possession MAE	1.137640	1.137448	-0.000192
Playoffs	Eligible-game RMSE	16.6636	16.5899	-0.0738

This is a clear regular-season rejection. The modest pooled-playoff improvement does not compensate for worse regular game, team, and win forecasts, so the canonical compiled-linear x3 model remains the predictive reference. The experiment does, however, validate the accounting boundary: all additive terms can be moved into player state exactly, but that allocation is not presently the better recursive forecasting state.

Artifact: artifacts/models/analysis/compiled_additive_prior_hpm_x3_frozen/frozen_multiseason_backtest/2023-24_to_2025-26/frozen_multiseason_backtest-2023-24-to-2025-26-20260816T155428Z-44dcf39f.