Local-response representations of a binary correlation model: one setting-independent specification space and per-station response laws

Let $\mu$ be a binary correlation model on the finite setting sets $I,J$ with outcome labels $\mathrm{\Sigma }=\{-1,+1\}$ (Binary correlation models: setting-indexed tables of two-valued outcome laws, marginals and the CHSH functional). No choice principle is used in this definition.

(a) Local-response representation. A local-response representation of $\mu$ is a tuple consisting of a probability space $(\mathrm{\Lambda },{\mathrm{\Sigma }}_{\mathrm{\Lambda }},\mathbb{P})$ (Measure spaces, Probability measures and probability spaces) and two families of response laws $({\alpha }_i{)}_{i\in I},({\beta }_j{)}_{j\in J},$ where for each $i\in I$ the function ${\alpha }_i:\mathrm{\Lambda }\times \mathrm{\Sigma }\to [0,1]$ has the two properties ${\sum }_{\epsilon \in \mathrm{\Sigma }}{\alpha }_i(\lambda ,\epsilon )=1(\lambda \in \mathrm{\Lambda }),\lambda \mapsto {\alpha }_i(\lambda ,+1)\text{ is measurable}$ (A measurable function between measurable spaces), and the family $({\beta }_j{)}_{j\in J}$ has the same two properties; and the representation identity ${\mu }_{ij}(\epsilon ,\delta )={\int }_{\mathrm{\Lambda }}{\alpha }_i(\lambda ,\epsilon ){\beta }_j(\lambda ,\delta )d\mathbb{P}(\lambda )={\mathbb{E}}_{\mathbb{P}}\text{\hspace{0.17em}⁣}[{\alpha }_i(\cdot ,\epsilon ){\beta }_j(\cdot ,\delta )]$ holds for every $(i,j)\in I\times J$ and every $(\epsilon ,\delta )\in \mathrm{\Sigma }\times \mathrm{\Sigma }$. The integrand is a product of two bounded measurable $[0,1]$-valued functions, so the expectation exists as a finite number (Expectation of a nonnegative or integrable random variable, Random elements and real random variables). The functions ${\alpha }_i(\cdot ,-1)=1-{\alpha }_i(\cdot ,+1)$ and likewise for ${\beta }_j$ are measurable as well. The variable $\lambda$ is called the specification of the representation; it is a mathematical index, and no physical interpretation is asserted.

(b) Response functions (deterministic form). Equip the finite outcome set $\mathrm{\Sigma }=\{-1,+1\}$ with its discrete $\sigma$-algebra $\mathcal{P}(\mathrm{\Sigma })$. A local-response representation has response functions when there are measurable maps $a_i:(\mathrm{\Lambda },{\mathrm{\Sigma }}_{\mathrm{\Lambda }})\to (\mathrm{\Sigma },\mathcal{P}(\mathrm{\Sigma }))$ $(i\in I)$ and $b_j:(\mathrm{\Lambda },{\mathrm{\Sigma }}_{\mathrm{\Lambda }})\to (\mathrm{\Sigma },\mathcal{P}(\mathrm{\Sigma }))$ $(j\in J)$ with ${\alpha }_i(\lambda ,\epsilon )={\mathbf{1}}_{\{a_i=\epsilon \}}(\lambda ),{\beta }_j(\lambda ,\delta )={\mathbf{1}}_{\{b_j=\delta \}}(\lambda ),$ in which case the representation identity reads ${\mu }_{ij}(\epsilon ,\delta )=\mathbb{P}(\{a_i=\epsilon \}\cap \{b_j=\delta \})\mathrm{.}$ The deterministic form is a special case of clause (a): it is obtained by setting the response laws to the Dirac laws at $a_i(\lambda )$ and $b_j(\lambda )$. In this form one and the same specification space and the same single-station functions $a_i$ and $b_j$ serve every setting pair, and the response of a station depends on the specification and on that station's own setting only.

(c) Terminology. The two mathematical features named by local response are: (i) a single specification space carrying the probability law used for every setting pair (setting-independence), and (ii) per-station response laws that do not depend on the distant station's setting. A model that admits such a representation is also called a local model of the tables in the literature; the representation is an existence statement about the tables and asserts no determinism, no mechanism, no dynamics and no causal relation. In particular clause (a) permits probabilistic responses and does not require the response functions of clause (b).

(d) Status. Clauses (a)-(c) fix vocabulary. They assert no physical assumption and no property of any laboratory arrangement; whether a declared physical model of a comparison admits a representation of this form is a mathematical question about its tables, and the physical reading of that question is fixed on the page of this pair.