B7: learned Bayesian planar factor posterior (unpublished research)¶

Status: native implementation and reproducible pilot are experimental. No 1.2 release, root export, selected architecture, or scientific inferential qualification is authorized. Stable 1.1.0 is unchanged.

The parent B7 fit_bayesian_planar_score_baseline() conditions on known population functions. The follow-on experimental eyetrajectoriespy.bayesian.fit_bayesian_planar_factor() instead uses a Gaussian shared-factor model to learn the population mean and joint x/y covariance from irregular coordinate observations.

Generative model¶

For independent participant i, coordinate d in {x,y} and its own observed timestamp t_idj:

Here B(t) is a fixed cubic B-spline dictionary; each coordinate has Gaussian priors on mean and loading coefficients, and epsilon_idj ~ N(0, sigma_d^2) with fixed, supplied coordinate-specific noise scales. Exact Gaussian conditional updates alternate mean, loadings and shared participant scores. Posterior x/y cross-covariance follows from the shared factors, not from interpolation.

Native asynchronous input¶

IrregularTrajectorySet can encode separate coordinate observation times as the union of actual timestamps, using NaN for the unobserved coordinate. Only observed likelihood rows are included. Neither coordinate is silently interpolated or aligned to the other, and all timestamps must lie within the declared evaluation-grid support. No repeated participant_id is permitted by this model.

import numpy as np
from eyetrajectoriespy.bayesian import fit_bayesian_planar_factor

result = fit_bayesian_planar_factor(
    gaze,  # native IrregularTrajectorySet, unique participant_id metadata
    dimensions=("x", "y"),
    evaluation_grid=np.linspace(0, 1, 27),
    observation_noise_sd=(0.04, 0.06),
    n_components=2,
    n_basis=5,
    n_chains=2,
    n_draws=60,
    warmup=90,
    random_state=2026,
)
summary = result.posterior_population_frame()

The joint_population_covariance_draws tensor has shape (chain, draw, 2G, 2G) and channel-major ordering [x(t_1)..x(t_G), y(t_1)..y(t_G)]. Do not feed this tensor into a time-major multivariate recovery helper without an explicit permutation. The loading_function_draws shape is (chain, draw, component, grid, coordinate). Individual loading columns are nonidentified under sign/rotation; compare only reconstructed functions, covariance and appropriate subspaces.

The summary displays marginal posterior means and pointwise 5th/95th quantiles for mean x, mean y, their variances and synchronous-time x/y cross-covariance. They are not scientifically calibrated intervals.

Prior/likelihood pilot¶

python scripts/run_b7_learned_planar_truth_pilot.py --replicates 4 --draws 35 --warmup 60 --out build/b7-truth

The pilot generates actual Gaussian parameters, shared scores and noise from the declared priors/likelihood. It refits the learned x/y model for rank-1 and rank-2 populations, with paired and separately timestamped asynchronous observations. It retains exceptions and evaluates pointwise mean and cross-covariance 90% coverage with exact binomial Monte Carlo intervals. CI uses only two repetitions per condition, an engineering smoke pilot and not scientific coverage evidence.

Outstanding scientific requirements¶

A complete B7 programme still requires posterior convergence and multivariate SBC rank studies; near-tied subspace calibration; measurement-noise and prior misspecification; native PACE and independent external comparison; held-out independent participant prediction; multi-session/participant hierarchy; and residual serial covariance. Channel-specific Gaussian observation noise is fixed, not inferred. Identifiable individually ordered eigenfunctions are not learned. The factor rank and basis are not selected by posterior inference.

Engineering success must never set scientific or publication flags true. See B6/F1/F5 calibration protocol and the scientific issue B7 #239.