Theorem: Calibrated finite-model source-weight inference

Under Assumption ass:statistics, for $0<\alpha <1$ with $\alpha +\beta <1$, define

Then, uniformly over the promised source and calibration family,

Thus

is a confidence interval with that coverage.

For $0<\epsilon <1$, a sufficient finite shot budget is

This is a certificate for the named model parameter, not a Galactic-source identification.

Proof. Conditional on the calibration record, the bounded independent scores obey

For completeness, a bounded variable $W\in [0,1]$ has

Indeed the second derivative of its log moment-generating function is the variance of a tilted law on $[0,1]$, at most $1/4$. Integration from zero proves the bound. Independence, exponential Markov inequalities, and optimization in $\lambda$ give the stated two-sided concentration bound. This is the bounded-variable form of Hoeffding's inequality Hoeffding1963.

On $G$, the law allowance gives

Lemma lem:calibration then gives, on the concentration event,

Projection onto $[0,1]$ cannot increase distance from $\theta \in [0,1]$, so division by $b_0$ proves the radius. The bound $|{\hat{\theta }}_0-\theta |\le 1$ permits its cap at one. Integrating the conditional statistical failure probability and adding $\mathrm{Pr}(G^c)\le \beta$ proves (eq:source-coverage). Rearranging the radius requirement proves (eq:shot-budget).