Repeated-observation protocols of the force branch: adopted IID and second-moment sampling hypotheses and the sample mean

Fix an operational observable model of the force branch (Operational observable model of the force branch: parameters, preparations, observation times, readout, units and additive error) in a fixed inertial frame, with parameter set $\mathrm{\Theta }$, preparation settings $Z$, prediction map $\mathrm{\Phi }:\mathrm{\Theta }\times Z\to \mathbb{R}$, probability space $(\mathrm{\Omega },\mathfrak{F},\mathbb{P})$, error domain $V\subseteq \mathbb{R}$ and error random element $\epsilon$; the readout is scalar here ($d=1$), since the mathematical sampling interface used below is stated for real random variables, and vector readouts are treated coordinatewise where needed.

A repeated-observation protocol for this model consists of

1. a repetition count $n\ge 1$ and one error random element ${\epsilon }_k:\mathrm{\Omega }\to \mathbb{R}$ for each $k=1,\dots ,n$, with ${\epsilon }_k(\omega )\in V$ for every $\omega$, the error of the $k$-th repetition on the common probability space (Probability measures and probability spaces, Random elements and real random variables);
2. the observations $Y_k:=\mathrm{\Phi }(\theta ,z)+{\epsilon }_k$ at a fixed pair $(\theta ,z)\in \mathrm{\Theta }\times Z$: the same parameter value and the same preparation are declared for every repetition, so that the repetition changes only the error.

The protocol satisfies the IID sampling hypothesis when the family $({\epsilon }_1,\dots ,{\epsilon }_n)$ — equivalently the family $(Y_1,\dots ,Y_n)$ — is independent and identically distributed in the sense of Identical distribution and IID families; it satisfies the second-moment hypothesis when $E\left[Y_1^2\right]<+\mathrm{\infty }$. Under both hypotheses the common mean and variance $\mu :=E[Y_1]\in \mathbb{R},{\sigma }^2:=\mathrm{Var}(Y_1)\in [0,+\mathrm{\infty })$ are defined by Moments, variance, and covariance on a probability space, the sample mean is $\bar{Y}:=\frac{1}{n}{\sum }_{k=1}^n{Y}_k,$ and the mathematical consequences of the two hypotheses are those of the sampling model Repeated-observation sampling models with IID and finite-variance hypotheses, namely the conclusions of Finite IID sample-mean variance and Chebyshev bound: $E\mid \bar{Y}-\mu {\mid }^2={\sigma }^2\mathrm{/}n$ and, for every $a>0$, $\mathbb{P}(\mid \bar{Y}-\mu \mid \ge a)\le \frac{{\sigma }^2}{n{a}^2}\mathrm{.}$

The equivalence uses the same constant $c:=\mathrm{\Phi }(\theta ,z)$ in every repetition. For any finite selection of indices and Borel sets $B_k$, $\{Y_k\in B_k\}=\{{\epsilon }_k\in B_k-c\}$. Since translation and its inverse preserve Borel sets, independence factorizes these intersection probabilities for the $Y_k$ exactly when it does for the ${\epsilon }_k$. Each observation law is the pushforward of its error law under $u\mapsto u+c$; applying the inverse translation shows that identical distribution is also equivalent. This argument requires the fixed common prediction; random or varying predictions need separate hypotheses.

Well-definedness and conventions

The IID and second-moment hypotheses are adopted model hypotheses about the repetitions, not consequences of the protocol and not properties of the data. Independence of the errors fails when successive repetitions share a systematic effect, when the apparatus drifts between repetitions or when the same recorded data are reused; identical distribution fails when the preparation or the conditions change between repetitions. A use of the displayed bounds must state both hypotheses, the declared law (if any) of $\epsilon$, and the conditions under which the repetitions are claimed to be comparable.

The sample mean $\bar{Y}$ is a random variable of the model. The definition attaches no observed value to $\mu$ or ${\sigma }^2$; in particular ${\sigma }^2$ is model data and not a measured uncertainty, as Scope of the sampling, propagation and inference interfaces records for the mathematical interface. A physical claim that a stated procedure satisfies the IID sampling hypothesis requires separate empirical evidence and is not part of this item.