lin-0127

9.10 Uncertainty and stochastic consistency

Suppose an interaction profile is estimated from \(K\) minibatches by \(\widehat\Psi _K\), with target population profile \(\Psi _{\mathbb E}\). A useful certificate separates variance and bias:

\[ \mathbb E \left\| \widehat\Psi _K-\Psi _{\mathbb E} \right\| ^2 \le \frac{C}{K}+b_K^2, \qquad b_K\to 0. \]

Same-minibatch and independently crossed directional estimates can converge to the same population profile while having different variance. The estimator design is therefore part of the admission contract.

Three outcomes should remain available:

  • accept when the repair clears the structural and empirical margins;

  • reject when it violates a hard condition or is clearly inferior;

  • abstain when support, power, identifiability, or model capacity is insufficient.

Abstention is not a failure of learning. It is the correct result when the evidence does not license a model-changing claim.