Uncertainty-guided family sampling

1. Integration point: attach the controller to an active-learning loop, replay buffer, or Monte Carlo proposal kernel. Maintain one reservoir per family and periodically update sampling priorities from the current neural predictor. For a neural network, dropout or an ensemble of K heads supplies u_c^2; for a Monte Carlo chain, use the score to bias proposals while retaining importance weights so the target distribution remains unchanged. 2. Pseudocode: initialize uniform exploration; collect samples and fit the family-residual model; for each family compute the ensemble mean of s_cP_cE_c and its variance u_c^2; form q_c^mix using the displayed score; draw the next batch or choose proposals from this distribution; apply inverse-proposal weights when estimating training moments; refit and repeat. 3. Computed from the paper’s mechanism: family-wise active importance sampling directed toward under-sampled permutation regions. Estimated empirically: P_c, E_c, u_c, effective sample size, and acceptance rates. 4. First cheap experiment: create a synthetic signed-family regression problem with 32 families, exponentially small P_c for rare families, and known family energies. Compare uniform sampling, magnitude-only prioritization, and uncertainty-weighted sampling at equal budget. The prediction is that active sampling reduces numerator variance according to sum_c u_c^2/n_c and reaches a fixed energy error with at least a twofold lower budget. Sweep epsilon; performance should deteriorate as epsilon approaches zero if a family is initially mispredicted, providing a direct mode-collapse test.