Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> Beyond binary measurements <s> 1 Random Variables.- 1 Introduction.- 2 The i.i.d. Coupling - Positive Correlation.- 3 Quantile Coupling - Stochastic Domination.- 4 Coupling Event - Maximal Coupling.- 5 Poisson Approximation - Total Variation.- 6 Convergence of Discrete Random Variables.- 7 Continuous Variables - Hitting the Limit.- 8 Convergence in Distribution and Pointwise.- 9 Quantile Coupling - Dominated Convergence.- 10 Impossible Coupling - Quantum Physics.- 2 Markov Chains and Random Walks.- 1 Introduction.- 2 Classical Coupling - Birth and Death Processes.- 3 Classical Coupling - Recurrent Markov Chains.- 4 Classical Coupling - Rates and Uniformity.- 5 Ornstein Coupling - Random Walk on the Integers.- 6 Ornstein Coupling - Recurrent Markov Chains.- 7 Epsilon-Coupling -Nonlattice Random Walk.- 8 Epsilon-Coupling -Blackwell's Renewal Theorem.- 9 Renewal Processes - Stationarity.- 10 Renewal Processes - Asymptotic Stationarity.- 3 Random Elements.- 1 Introduction.- 2 Back to Basics - Definition of Coupling.- 3 Extension Techniques.- 4 Conditioning - Transfer.- 5 Splitting.- 6 Random Walk with Spread-Out Step-Lengths.- 7 Coupling Event - Maximal Coupling.- 8 Maximal Coupling Two Elements - Total Variation.- 9 Hitting the Limit.- 10 Convergence in Distribution and Pointwise.- 4 Stochastic Processes.- 1 Introduction.- 2 Preliminaries - What Is a Stochastic Process?.- 3 Exact Coupling - Distributional Exact Coupling.- 4 Distributional Coupling.- 5 Exact Coupling - Inequality and Asymptotics.- 6 Exact Coupling - Maximality.- 7 Coupling with Respect to a Sub-a-Algebra.- 8 Exact Coupling - Another Proof of Theorem 6.1.- 9 Exact Coupling - Tail a-Algebra - Equivalences.- 5 Shift-Coupling.- 1 Introduction.- 2 Shift-Coupling - Distributional Shift-Coupling.- 3 Shift-Coupling - Inequality and Asymptotics.- 4 Shift-Coupling - Maximality.- 5 Shift-Coupling - Invariant a-Algebra - Equivalences.- 6 E-Coupling - Distributional E-Coupling.- 7 e-Coupling - Inequality and Asymptotics.- 8 E-Coupling - Maximality.- 9 e-Coupling - Smooth Tail a-algebra - Equivalences.- 6 Markov Processes.- 1 Introduction.- 2 Mixing and Triviality of a Stochastic Process.- 3 Markov Processes - Preliminaries.- 4 Exact Coupling.- 5 Shift-Coupling.- 6 Epsilon-Coupling.- 7 Stationary Measure.- 7 Transformation Coupling.- 1 Introduction.- 2 Shift-Coupling Random Fields.- 3 Transformation Coupling.- 4 Inequality and Asymptotics.- 5 Maximality.- 6 Invariant a-Algebra and Equivalences.- 7 Topological Transformation Groups.- 8 Self-Similarity - Exchangeability - Rotation.- 9 Exact Transformation Coupling.- 8 Stationarity, The Palm Dualities.- 1 Introduction.- 2 Preliminaries - Measure-Free Part of the Dualities.- 3 Key Stationarity Theorem.- 4 The Point-at-Zero Duality.- 5 Interpretation - Point-Conditioning.- 6 Application - Perfect Simulation.- 7 The Invariant a-Algebras I and J.- 8 The Randomized-Origin Duality.- 9 Interpretation - Cesaro Limits and Shift-Coupling.- 10 Comments on the Two Palm Dualities.- 9 The Palm Dualities in Higher Dimensions.- 1 Introduction.- 2 The Point-Stationarity Problem.- 3 Definition of Point-Stationarity.- 4 Palm Characterization of Point-Stationarity.- 5 Point-Stationarity Characterized by Randomization.- 6 Point-Stationarity and the Invariant a-Algebras.- 7 The Point-at-Zero Duality.- 8 The Randomized-Origin Duality.- 9 Comments.- 10 Regeneration.- 1 Introduction.- 2 Preliminaries - Stationarity.- 3 Classical Regeneration.- 4 Wide-Sense Regeneration - Harris Chains - GI/GI/k.- 5 Time-Inhomogeneous Regeneration.- 6 Classical Coupling.- 7 The Coupling Time - Rates and Uniformity.- 8 Asymptotics From-the-Past.- 9 Taboo Regeneration.- 10 Taboo Stationarity.- 11 Perfect Simulation - Coupling From-the-Past.- Notes.- References.- Notation. <s> BIB001 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> Beyond binary measurements <s> We present general principles underlying analysis of the dependence of random variables (outputs) on deterministic conditions (inputs). Random outputs recorded under mutually exclusive input values are labeled by these values and considered stochastically unrelated, possessing no joint distribution. An input that does not directly influence an output creates a context for the latter. Any constraint imposed on the dependence of random outputs on inputs can be characterized by considering all possible couplings (joint distributions) imposed on stochastically unrelated outputs. The target application of these principles is a quantum mechanical system of entangled particles, with directions of spin measurements chosen for each particle being inputs and the spins recorded outputs. The sphere of applicability, however, spans systems across physical, biological, and behavioral sciences. <s> BIB002 </s> Contextuality-by-Default: A Brief Overview of Ideas, Concepts, and Terminology <s> Beyond binary measurements <s> Contextuality is a central property in comparative analysis of classical, quantum, and supercorrelated systems. We examine and compare two well-motivated approaches to contextuality. One approach (“contextuality-by-default”) is based on the idea that one and the same physical property measured under different conditions (contexts) is represented by different random variables. The other approach is based on the idea that while a physical property is represented by a single random variable irrespective of its context, the joint distributions of the random variables describing the system can involve negative (quasi-)probabilities. We show that in the Leggett-Garg and EPR-Bell systems, the two measures essentially coincide. <s> BIB003
How could the definition of a maximally connected coupling be generalized to arbitrary random variables? A straightforward way to do this is to extend definition (8) for a coupling T q of a connection R q as This is an approach adopted in BIB003   . It is based on the following mathematical considerations, derived from the discussion of maximal couplings in Thorisson's monograph BIB001 (Section 7 of Chapter 3). Given two sigma-additive measures µ and ν on the same sigma algebra, let us write µ ≤ ν if µ (E) ≤ ν (E) for every measurable set E. Let µ C q be the probability measure associated with R C q . Let X q and Σ q be the set of values and sigma algebra associated with R C q (they are assumed the same for all C, because otherwise one should not consider R C q measurements of one and the same object). For every object q, define µ q as the largest sigma-additive measure such that µ q ≤ µ C q for all contexts C. The measure µ q is the largest in the sense that µ q ≤ µ q for any other measure µ q such that µ q ≤ µ C q for all contexts C. A theorem proved in Ref. BIB001 (Theorem 7.1) guarantees the existence and uniqueness of µ q , for any set of probability measures µ C q C∈C , whatever the indexing set C . That is, µ q is uniquely defined for any connection R q . Note that µ q is not generally a probability measure, so µ q (X q ) can be any number in [0, 1]. Let us denote max eq (R q ) = µ q (X q ) . For ±1-measurements R C q this definition specializes to (8)-  . Consider now a coupling T q for R q . It is defined on the product sigma-algebra C Σ q on the product set C X q . An event E q ∈ C Σ q is called a coupling event if S q ∈ E q implies T C q = T C q for any two C, C ∈ C (assuming, as always, that both C and C involve q). It follows from Theorem 7.2 in Ref. BIB001 that Pr [T q ∈ E q ] ≤ max eq (R q ) , for any q and any choice of E q . Now, it is natural to define a maximal coupling for R q as a coupling T q for which E q can be chosen so that Pr [T q ∈ E q ] = max eq (R q ) . Theorem 7.3 in Ref. BIB001 says that such a maximal coupling always exists. Note that E q in a maximal coupling can always be thought of as the largest measurable subset of the diagonal of the set C X q . Having established this generalized notion of a maximal coupling, the theory of contextuality can now be generalized in a straightforward fashion. Consider a coupling S for the entire system R. The definition of a maximally connected coupling remains unchanged: every marginal S q of a maximally connected coupling S is a maximal coupling for the corresponding connection R q . Our Main Definition could remain unchanged too: a system R is contextual if and only if no coupling S of this system is maximally connected. This can be equivalently presented as follows. For any set P of probability values, let f (P ) be a bounded smooth nonnegative function strictly increasing in all components of P . Thus, for finite systems of random variables f can be chosen as a sum or average, as in BIB002 . Define max eq (R) = max and, if this value exists, cntx (R) = f (max eq (R q ) : q ∈ Q) − max eq (R) . The system is defined as contextual if and only if cntx (R) > 0. We do not know whether max eq (R) exists for all possible systems of random variables. If it does not, however, the definition can be extended by replacing max with sup. This generalization has to be further explored to determine whether it is a good generalization, i.e., whether it provides valuable insights, leads to interesting mathematical developments, and does not yield non-interpretable results when applied to specific systems of measurements.