claimroutine

Template slices and bad-slot probability

Claim·§6.2 Classification and suppression of common variation·inside propCommon variation is negligible

Statement (paper)

Fix a common level. For each template θ\theta of positive probability, let πθ=P(θ)\pi_\theta=\Prob(\theta) and bθ=P(bad edge∣θ)b_\theta=\Prob(\text{bad edge}\mid\theta) under the joint law in . In either the center-star or the point--template-star experiment, each slot has the conditional single-edge law P(p,q∣θ)\Prob(p,q\mid\theta), so a pair with at least one bad slot has probability at most 2bθ2b_\theta; moreover, since all coefficients in a common score are bounded by 1+λ1+\lambda, charging a pair with a bad slot costs at most an absolute multiple of its probability.

In plain English

Condition on a template θ\theta with probability πθ\pi_\theta and bad-edge probability bθb_\theta. In the center-star or point–template-star experiment, each slot follows the single-edge law P(p,q∣θ)\Prob(p,q\mid\theta), so a pair with at least one bad slot has probability at most 2bθ2b_\theta, and since common-score coefficients are bounded by 1+λ1+\lambda, charging such pairs costs at most an absolute constant times their probability.

Why it's here

This sets up the slice-by-slice bookkeeping for the common-variation proof: each template slice θ\theta is analyzed separately, and the pairs with a bad slot, whose probability is at most 2bθ2b_\theta, are absorbed into an error O(bθ)O(b_\theta) that is later summed against πθ\pi_\theta. The conditional single-edge law and the transfer lemma's joint law make the bad-edge probability controllable per slice.
Classified routine:
Direct bookkeeping consequence of the joint law in com-transfer: a union bound over the two slots gives probability at most 2b_θ, and the bounded coefficients (at most 1+λ) of the common score turn this into a cost of O(b_θ).

Uses

lemTransfer from sampled histories to actual point

Used by

claimSlice score lower bound via classification

Sources

PaperOpen
Section
§6.2 Classification and suppression of common variation
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sections/05-common-levels.tex:259
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