Is the Moon there if nobody looks: Bell Inequalities and Physical Reality

Various Bell inequalities are trivial algebraic properties satisfied by each line of particular data spreadsheets.It is surprising that their violation in some experiments, allows to speculate about the existence of nonlocal influences in Nature and to doubt the existence of the objective external physical reality. Such speculations are rooted in incorrect interpretations of quantum mechanics and in a failure of local realistic hidden variable models to reproduce quantum predictions for spin polarisation correlation experiments. These hidden variable models use counterfactual joint probability distributions of only pairwise measurable random variables to prove the inequalities. In real experiments Alice and Bob, using 4 incompatible pairs of experimental settings, estimate imperfect correlations between clicks, registered by their detectors. Clicks announce detection of photons and are coded by 1 or -1. Expectations of corresponding ,only pairwise measurable, random variables are estimated and compared with quantum predictions. These estimates violate significantly the inequalities. Since all these random variables cannot be jointly measured , a joint probability distribution of them does not exist and various Bell inequalities may not be proven. Thus it is not surprising that they are violated. Moreover,if contextual setting dependent parameters describing measuring instruments are correctly included in the description, then imperfect correlations between the clicks may be explained in a locally causal way. In this paper we review and rephrase several arguments proving that the violation of various Bell inequalities may neither justify the quantum nonlocality nor allow doubting the existence of atoms, electrons and other invisible elementary particles which are building blocks of the visible world around us including ourselves.


Introduction
The external physical reality existed before we were able to probe it with our senses and experiments. From early childhood we learn that the surrounding us objects continue to exist even we stop looking at them. Another notion imprinted in our genes is the notion of a local causality. If a baby elephant or a baby antelope do not stand up immediately, after their birth, they will die. Several events, which we observe, may be connected by causal chains. Amazing birds' and butterflies' migration patterns and courtship rituals are encoded in their genes.
Our brains, developed during millions of years, allowed us to understand that the external physical reality should be governed by natural laws which we try to discover .We succeeded to explain observable properties of macroscopic objects assuming the existence of invisible atoms and molecules. Later we discovered: electrons, nuclei, elementary particles, resonances and various fields playing an important role in the Standard Model. Various conservation laws are obeyed in Nature as well in macroscopic as in quantum phenomena.
The information about the invisible world is indirect and relative to how we probe it. Invisible charged elementary particles leave traces of their passage in photographic emulsion or in different chambers (sparks, bubble, multi-layer etc.). They produce also clicks on detectors.
We accelerate electrons, protons and ions and by projecting them on various targets we probe more deeply the structure of the matter on smaller and smaller distances. We succeeded to trap electrons and ions and to measure precisely their properties. We construct atomic clocks and trapped ion chips for quantum computing.
Therefore it is surprising that the violation of various Bell-type inequalities [1][2][3][4][5] by some correlations between clicks on the detectors observed in spin polarization correlation experiments (SPCE) [6][7][8][9][10][11] may lead to the conclusions that that there is no objective physical reality , that the electron may be at the same time here and a meter away, that a measurement performed by Alice in distant location may change instantaneously an outcome of Bob's measurement or that apparently random choices of experimental settings in SPCE are predetermined due to the super-determinism.
Such conclusions are unfounded what was pointed out by several authors . The violation of the inequalities confirms only that ``unperformed experiments have no outcomes" [84]. That one may not neglect the interaction of a measuring instrument with a physical system and that the "noninvasive measurability" assumption is not valid. It confirms the existence of quantum observables which can only be measured in incompatible experimental contexts.
It proves also that entangled photon pairs, produced in SPCE, may not be described as pairs of socks ( local realistic hidden variable models-LRHVM ) or as pairs of fair dices ( stochastic hidden variable models-SHVM) [1][2][3][4].
We are unable to create any consistent mental picture of a "photon" . The same problem we have with many other elementary particles but the lack of mental pictures does not mean that these particles do not exist. These invisible particles are building blocks of the visible world around us including ourselves.
Probably a completely new approach is needed to reconcile the quantum theory with the theory of general relativity and it is not sure whether we are smart enough to find it. It is sure that we will not find it, if we accept the quantum magic as the explanation of phenomena which we don't understand.
The question in the title of this article was first asked by Einstein during his promenade with Pauli and after it was rephrased in different contexts by Leggett and Garg [85] and by Mermin [86] . In this paper we defend Einstein's position [87][88][89] and we believe that the Moon continues to exist if nobody looks.
The paper is organised as follows.
In section 3 we define LRHVM and explain why these models cannot reproduce quantum predictions for impossible to implement EPRB experiment.
In section 4 we show how, by incorporating in LRHVM setting dependent parameters, describing measuring instruments, one may explain in a locally causal way correlations between distant outcomes observed in SPCE In section 5 we explain why Bell-1971 model [2,91] and Clauser-Horne model [4] are inconsistent with experimental protocols used in SPCE.
In section 6 we define quantum CHSH inequality [92,93], Tsirelson bound [92] and we reproduce Khrennikov's recent arguments [43] that the violation of quantum CHSH inequality confirms only local incompatibility of some quantum observables. . In section 7 we show that speculations about quantum nonlocality are in fact rooted in the incorrect interpretation of von Neumann / Luders [94][95] projection postulates.
In section 8 we discuss simple experiments with elastically colliding metal balls [54] and we explain an apparent violation of Bell-Boole inequalities in these experiments. These experiments allow to understand better LRHVM and why they fail to describe SPCE.

Experimental spreadsheets and Bell-type inequalities.
Let us examine properties of a spreadsheet with 4 columns containing each N entries ±1. We may have N identical rows or 16 different rows permuted in an arbitrary order. The entries may be coded values representing outcomes of some random experiment (e.g. flipping of three fair coins). They may display the results of some population survey or represent daily variations of some stocks. They also may be created by an artist as a particular visual display. Thus the columns in the spreadsheet may be finite samples of particular discrete time-series of data or they can be devoid of any statistical meaning.
If each line of the spreadsheet contains measured values (a. a, b, b') of jointly distributed random variables (A, A', B, B' ) taking the values ±1 then b=b' or b=-b'and From (1) we obtain immediately CHSH inequality: where p(a, a, b, b') is a joint probability distribution of (A, A', B, B' ) and All these inequalities are deduced using the inequality (1) obeyed by any 4 numbers equal to ±1. The inequalities (2) and (3) are in fact necessary and sufficient conditions for the existence of a joint probability distribution of only pairwise measurable ±1-valued random variables [18,19 ]. The inequalities (2) and (3) are of course also valid if |A|≤1, |A'|≤1|, |B|≤1 and |B'|≤1.

Local Realistic Models for EPR-Bohm Experiment
In physics Bell-CHSH inequalities [2] were derived in an attempt to reproduces quantum predictions for impossible to implement , ideal EPRB experiment [96].
In EPRB experiment a source produces a steady flow of electron-or photon-pairs [60] prepared in a quantum spin-singlet state . One photon is sent to Alice and another to Bob in distant laboratories where they measure photons' spin projections in directions a and b (||a||=||b||=1) and the outcomes "spin up" or "spin down" are coded ±1 . There are no losses and for any pair of experimental settings Alice's and Bob's measuring station output correlated pairs of outcomes.
If Alice and Bob perform their experiments using 4 pairs of settings ( (a , b). (a' , b).
(a , b') and (a' , b') ), then outcomes ±1 are the values of corresponding 4 binary random variables A a , A a' , B b , B b'. In [1][2] these values are determined by some ontic parameters λ (hidden variables) describing pairs of photons when they arrive to Alice's and Bob's measuring stations. Pairwise expectations of measured random variables, in different settings, are all expressed in terms of a unique probability distribution p(λ) defined on an unspecified probability space Λ: If in (1) we replace a= A a (λ)=(A(a, λ), a'= A a' (λ)= (A(a', λ), b= B b (λ)= B(b, λ) and b'= B b' (λ)=B'(b', λ) we obtain: Therefore the expectations (4-6) obey the inequality (2).
Bell used the integration over hidden variables instead of the summation and λ could be anything . In agreement with QM, he insisted that one cannot measure simultaneously or in a sequence different spin projections on the same photon, thus the expectations E(A a A a' B b B b' ) have no physical meaning. Nevertheless the existence of those counterfactual non-vanishing expectations is necessary in order to prove (8) . Namely there exists a mapping: which defines a joint probability distribution p(a. a, b, b') and a non-vanishing counterfactual expectations E(A a A a' B b B b' ) [56,97].
If a joint probability distribution p (a, a', b, b') does not exist the inequalities (2) and (8) cannot be proven. According to QM such joint probability distributions do not exist in EPRB thus for some settings quantum predictions violate CHSH inequalities.
This value violates significantly CHSH inequality and saturates the Tsirelton's bound [92] which we discuss in the section 6.
According to QM : E(A a B a )= -1 and E(A a B -a )= 1 for any vector a. Thus Alice and Bob when measuring spin projections using the settings (a , a) and (a , -a) should obtain perfectly anti-correlated or correlated outcomes respectively. At the same time these outcomes are believed to be produced in an irreducible random way thus one encounters an impossible to resolve paradox : ``a pair of dice showing always perfectly correlated outcomes``.
In order to reproduce perfect correlations in LRHVM one abandons the irreducible randomness and assumes that Alice's and Bob's outcomes are predetermined before measurements are done. Therefore there exists a counterfactual joint probability distributions of all these predetermined outcomes and CHSH inequalities may not be violated [86,[97][98][99].
Fortunately this paradox exists only on paper because the ideal EPRB experiment does not exist and in SPCE we neither observe strict correlations nor anti-correlations between clicks.
In the next section we show how imperfect correlations between clicks in SPCE may be explained in a locally causal way without evoking quantum magic.

Contextual Description of Spin Polarization Correlation Experiments
In SPCE correlated signals/photons, produced by some sources, arrive to Alice's and Bob's measuring stations producing clicks on their detectors. There are black counts, laser intensity drifts, photon registration time delays etc. Detected clicks have their time tags which are different for Alice and Bob. One has to identify clicks corresponding to photons being members of the same entangled "pair of photons" what is a settingdependent complicated task. Correlated clicks are rare events and estimated correlations depend on a photon-identification procedure used. A detailed discussion how data are gathered and coincidences determined in different SPCE may be found for example in [60,80,100].
Even if the all mentioned above difficulties had not existed, QM would have not predicted perfect correlations for real experiments. Settings of realistic polarizers may not be treated as mathematical vectors [47]. but rather as small spherical angles therefore instead of E(A a B b )= -a · b= -cosθ we obtain: In order to estimate correlations Alice and Bob have to choose correlated time windows. They retain only pairs of windows containing 3 types of events: "a click on a detector 1 and a click on a detector 2" or "a click on only one of the detectors" . Therefore in SPCE random variables describing outcomes of these experiments have 3 possible values coded as ±1 or 0.
To make easier a comparison with the notation used in [60] , where more details may be found, we denote different pairs of settings by (x,y),…, (x',y') and ( ) ( | , ).
The efficiency of detectors is not 100% and it is difficult to establish correct coincidences between distant clicks because of time delays. These two problems called efficiency and coincidence-time loopholes were discussed in detail by Larsen and Gill [101] in terms of the sub-domains of hidden variables corresponding to 4 experimental settings. They found that CHSH inequality has to be modified: In our model ( ) 0 p  thus the only constraint for S is a no-signalling bound :|S|≤4.
Our model contains enough free parameters to fit any estimated correlations. For example, if we start with k values of λ 1 , k values of λ 2 and m values for each λ x , λ x' , λ y, ,λ y' we have km pairs of (λ 1 , λ x ) , 3 km functions A x ((λ 1 , λ x ) and 3 km functions B y ((λ 2 , λ y ). Besides we have m-1 free parameters for each p x (λ x ), p x' (λ x' ), p y (λ y ), p y' (λ y' ) and also (  In mathematical statistics we concentrate on observable events: outcomes of random experiments or results of a population survey. Joint probability distributions are used only to describe random experiments producing in each trial several outcomes e.g. rolling several dice or various data items describing the same individual drawn from some statistical population. Probabilistic models describe a scatter of these outcomes without entering into the details how outcomes are created. Hidden variable probabilistic models introduce some invisible "hidden events" which determine subsequent real outcomes of random experiments. In Bell model (4-7) pairs of photons ("beables") are described by λ before measurements take place. Because clicks are predetermined by the values of λ there exists the mapping (9) and the probability distribution of "hidden events" described by p(λ ) may be replaced by a joint distribution p(a, a', b, b').
In contextual model (11)(12)(13)(14)(15)(16)(17) an outcome `a 'click' or 'no-click` is not predetermined and is created in a locally causal way in function of a hidden parameter describing a signal (``photon`) arriving to the measuring station and a hidden parameter describing the measuring instrument in the moment of their interaction. The model (11)(12)(13)(14)(15)(16)(17) gives an insight how apparently random outcomes are created in SPCE.

Experimental protocols and averaging over instrument parameters.
In 1971 Bell [91] pointed out that one may incorporate into his model additional hidden variables describing measuring instruments but it does not invalidate his conclusions because after the averaging over instrument variables the pairwise expectations still have to obey CHSH inequalities. We reproduce his reasoning in the notation consistent with (11)(12)(13)(14)(15)(16)(17) .
If we average over the variables λ x and λ y we obtain : In spite of the fact that the expectations calculated using the equations (11)(12)(13)(14) and (19)(20)(21)(22) have the same values the two sets of the formulas describe different experiments.
In the experiment described by (11)(12)(13)(14) pairs of photons arrive sequentially to measuring instruments which produce in a locally causal way "a click" or "no-click" and a counterfactual Nx4 spreadsheet of all possible outcomes does not exist and may not be used to prove CHSH inequalities. Thus the estimated pairwise expectations may significantly violate (8) what they do.
The equations (19)(20)(21)(22) describe an experiment, impossible to implement, which uses the following two-step experimental protocol: 3. Use all the entries of this spreadsheet to estimate expectations (19)(20)(21)(22) Because the entries of each line of this spreadsheet obey the inequality (1) thus if we could implement this protocol the estimated expectations would obey CHSH for any finite sample.
There is a significant difference between a probabilistic model and a hidden variable model. If we average out some variables in a probabilistic model we obtain always a marginal probability distribution describing some feasible experiment. If we average out some hidden variables in a hidden variable model we may obtain a new hidden variable model which does not correspond to any feasible experiment.
For a similar reason the experimental protocol of SHVM is nconsistent with the protocol used in SPCE. A much more detailed discussion of a subtle relation of probabilistic models with experimental protocols may be found in [56].
where Λ' xy = {λϵ Λ xy |A x (λ 1 , λ x ) ≠0 and B y (λ 2 , λ y ) ≠0} , In a similar way we transform the expectations (12)(13)(14) into conditional expectations. Using these conditional expectations we may not prove CHSH thus our model is able to explain their violations in SPCE. It may also explain in a rational way an apparent violation of no-signalling reported in [79,80,[103][104][105][106][107][108]: and The setting-dependence of these marginal expectations does not prove no-signalling because E(A x ) and E(B y ) defined by (15)(16) do not depend on the distant measurement settings.
Please note that the expectations (26) may not be transformed into a factorized form (21).
Naïve quantum predictions for a singlet state cannot explain the correlations observed in SPCE. One has to use much more complicated density matrices [109] containing free parameters and still some discrepancies between the theoretical predictions and the data persist. More detailed discussion of how the data are analysed in SPCE and how the apparent violation of no-signalling may be explained may found in [60].
Since our description of real data is causally local thus all speculations about quantum nonlocality are unfounded.
In the next section we explain why, contrary to what is believed, probabilistic predictions of QM are not in conflict with local causality.

Quantum mechanics and CHSH inequalities
According to the statistical contextual interpretation [29,52,57,89,[110][111] QM provides probabilistic predictions for experiments performed in well-defined experimental contexts. In these experiments identical preparations of physical systems are followed by measurements of physical observables. A class of identical preparations is described by a state vector | or by a density matrix ρ and a class of equivalent measurements of an observable A is represented by a Hermitian/self-adjoint operatorÂ .
Outcomes of measurements are eigenvalues of these operators. In general outcomes are not predetermined and they are created as a result of the interaction of measuring instruments with physical systems. In the same experimental context only the values of compatible physical observables, represented by commuting operators, may be measured jointly.
In SPCE "photon pairs" , prepared by a source, are described by a density matrix ρ and where , and .
If ρ is an arbitrary mixture of separable states then quantum correlations have to obey CHSH: As we saw in the section 2 for the inequality (31) may be significantly violated for entangled quantum states if specific incompatible pairs of settings are chosen.
The quantum description is contextual because a triplet 11{ , , } AB  depends explicitly on a preparation of "photon pairs" and on observables (A,B) measured using specific experimental settings. Different incompatible experimental settings are therefore described in QM by different specific Kolmogorov models.
In particular Cetto et al. [73] have recently demonstrated that expectations E(AB | ψ), for a singlet state | H   , may be expressed in terms of the eigenvalues of operators Â a   andBb   using specific dedicated probability distributions. We reproduce below their results in our notation:   (11)(12)(13)(14).
In 1982 Fine [18][19] demonstrated that Bell-CHSH inequalities are necessary and sufficient condition for the existence of a joint probability distribution of ±1-valued observables (A,A',B,B').
As we saw in the section 3, QM predicts a significant violation of CHSH inequality : S= 22 .
In 1980 Tsirelson [92] proved that 22 is the greatest value of S allowed by QM:  . Therefore the violation of CHSH proves only the local incompatibility of specific Alice's and Bob's physical observables [43] what has nothing to do with quantum nonlocality.
The local incompatibility of some observables neither allows to doubt the local causality in Nature nor the "objective" existence of elementary particles and atoms. The statistical contextual interpretation of QM (SCI) [57,52,89] is free of paradoxes. According to this interpretation a quantum state vector represents only an ensemble of identically prepared physical systems and after a von Neumann/Lüders projection a new state describes a different ensemble of physical systems. Namely: | i Aa   describes all the systems S 2 such that measurements of the observable A on their entangled partners (systems S 1 ) gave the same outcome A=a i .
The statistical interpretation does not claim that QM provides the complete description of individual physical systems and a question whether quantum probabilities may be deduced from some more detailed description of quantum phenomena is left open [46,52,59,61,[87][88][89][112][113].
Lüders projection and its interpretation have been discussed recently in detail by Khrennikov [44]. We reproduce below few statements form the abstract of his article: "..If probabilities are considered to be objective properties of random experiments we show that the Lüders projection corresponds to the passage from joint probabilities describing all set of data to some marginal conditional probabilities describing some particular subsets of data. If one adopts a subjective interpretation of probabilities, such as Qbism, then the Lüders projection corresponds to standard Bayesian updating of the probabilities. The latter represents degrees of beliefs of local agents about outcomes of individual measurements which are placed or which will be placed at distant locations. In both approaches, probability-transformation does not happen in the physical space, but only in the information space. Thus, all speculations about spooky interactions or spooky predictions at a distance are simply misleading.." In 1998 Ballentine explained in his book that "individual interpretation" of QM is incorrect : " Once acquired , the habit of considering an individual particle to have its own wave function is hard to break .Even though it has been demonstrated strictly incorrect …" . Therefore talking about " passion at the distance", "predictions at the distance", "steering at the distance" may only lead to incorrect mental pictures and create unnecessary confusion.
Claims that QM is a non-local theory are also based on an incorrect interpretation of a two -slit experiment. In this experiment a wave function (representing an ensemble of identically prepared electrons) "passes" by two slits but this does not mean that a single electron may be in two distant places at the same time. If two detectors are placed in front of the slits they never click at the same time thus an electron (but not the electromagnetic field created by an electron) passes only by one slit. According to SCI a wave function is only a mathematical entity and QM does not provide a detailed space-time description how the interference pattern on a screen is formed by the impacts of individual electrons.
Another root of quantum nonlocality is Bell's insistence that the violation of Bell-type inequalities SPCE would mean that a locally causal description of these experiments is impossible [1]: "In a theory in which parameters are added to quantum mechanics to determine the results of individual measurements, without changing the statistical predictions, there must be a mechanism whereby the setting of one measuring device can influence the reading of another instrument, however remote. Moreover, the signal involved must propagate instantaneously, so that such a theory could not be Lorentz invariant".
As we explained in the section 3, Bell's statement is correct only if one is talking about the ideal EPRB which does not exist. The violation of various Bell-type inequalities in SPCE prove only that these experiments may not be described by oversimplified hidden variable models. In LRHVM and in Eberhard model [5] a fate of a photon/electron is predetermined before the experiment is performed. In SPHVM the outcomes, registered in distant measuring stations, are produced in irreducible random way thus the correlations between such outcomes are very limited.
As we explained in the section 4 imperfect correlations in SPCE may be explained in a locally causal way, if instrument parameters are correctly included in a probabilistic model closing so called the contextuality loophole [65][66][67].
Bell-CHSH inequalities may also be violated in social sciences by expectations of ±1-valued random variables, which can only be measured pairwise but not all together. The violation of these inequalities in social sciences has nothing to say about the physical reality and the locality of Nature [16,37,38,[114][115][116]. This is why we may repeat after Khrennikov [43] that we should get rid of the quantum nonlocality which is a misleading notion.
In the next section we discuss simple experiments with colliding elastically metal balls in which the experimental outcomes are predetermined but an apparent violation of Bell and Boole inequalities may be proven [54]. We discuss also the violation of inequalities by the estimates obtained using finite samples.

Apparent violations of Bell-Boole inequalities in elastic collision experiments
Let us consider a simple experiment with metal balls colliding elastically: 1. 4kg metal ball and 1 kg metal ball are placed in some fixed positions P 1 and P 2 on a horizontal perfectly smooth surface.

2.
A device D, with a built in random numbers generator, is imparting on a lighter ball a constant rectilinear velocity with a speed described by a random variable V distributed according to a probability density ( ) 1/10 V fx  for 0≤x≤10 and the ball is sliding without friction and without rotating towards the heavier ball.
We see that Bell ( +sign) and Boule (-sign) inequalities (3) seem to be violated: The violation of (39) is surprising because the outcomes of our experiments are predetermined. However one has to pay attention before checking Bell-Booleinequalities. In spite of the fact that in the settings (A,B) and (B,C) Alice and Bob measure the same physical observables , measured values ±1 are described by 2 different random variables B(V 1 )≠ B(V 2 ) thus the inequalities which are violated are not (39) but the inequalities : Since values of random variables (A (V 1 ) ,B (V 1 ), B(V 2 ) , C(V 2 )) are predetermined for each trial by a value of the initial speed x imparted on the lighter ball thus in spite of a fact that these random variables are not jointly measurable there exists an "invisible" joint probability distribution of these random variables and CHSH inequalities may not be violated Treating measuring stations as black boxes Alice and Bob don't know whether this invisible joint probability does exist and that for each trial the values of measured observables are predetermined . Therefore they display the data obtained in different settings using four Mx2 spreadsheets and they estimate only pairwise expectations of measurable pairs of random variables (A (V 1 ) ,B (V 2 )), (A (V 1 ) ,B (V 2 )), (B(V 1, C(V 2 )) and (B(V 1, B(V 2 )) These estimates may violate the inequality (41) because as we demonstrated in the section 1 only the estimates obtained using all ±1 entries of Nx4 spreadsheets obey strictly CHSH inequality for any finite sample. Alice and Bob don't know that their outcomes are in fact extracted from specific lines of invisible Nx4 spreadsheet and that the columns of Mx2 spreadsheets are simple random samples drawn from the corresponding complete columns of Nx4 spreadsheet. This is why if M and N are large the estimated pairwise expectations may not violate the inequality (41) more significantly that it is permitted by sampling errors.
In collision experiments outcomes are predetermines and the correlations exist due to the energy and momentum conservation. In LRHVM outcomes are predetermined and the correlations exist due to the angular momentum conservation .
There is however a big difference between metal balls and photons in SPCE. In collision experiments metal balls are distinct macroscopic objects with well-defined linear momenta. Measurements of speeds are , with a good approximation, noninvasive thus measuring stations in fact register passively their preexisting values and output specific coded values ±1.
In SPCE we cannot observe and follow pairs of photons moving from the source to the measuring stations. By no means a passage of a photon through a polarization beam splitter (PBS) may be considered as a passive registration of a preexisting "spin up" or spin down" value.
In collision experiments all observables are compatible therefore Alice's modified measuring station might output in each trial values of (A (V 1 ) , B (V 1 ) ) and Bob's modified station values of (B(V 2, C(V 2 )) which might be displayed using a Nx4 spreadsheet. In SPCE it is impossible because the observables (A, A') and ( B, B') are not compatible and their joint probability distribution and Nx4 spreadsheet do not exist.
A problem how significantly finite samples, extracted from a counterfactual spreadsheet Nx4, may violate CHSH inequalities was studied by Gill [117] .
where obs AB  is an estimate of E(AB) etc. More detailed discussion of various finite sample proofs of Bell-type inequalities may be found in [57].
Let us see what happens, if we display all experimental data obtained in SPCE (containing N data items for each pair of settings) in a 4Nx4 spreadsheet and fill randomly remaining empty spaces by ±1. Pairwise expectations estimated using complete columns of this spreadsheet obey strictly CHSH inequality. One may ask a question: why real data being subsets of these columns may violate CHSH more significantly than it is permitted by (42)? The answer is simple :the outcomes obtained in SPCE for each pair of incompatible settings are not simple random samples extracted from corresponding columns of the completed 4Nx4 counterfactual spreadsheet.
With Hans de Raedt we studied in [102] the impact of sample inhomogeneity on statistical inference . In particular we generated two large samples (which were not simple random samples) from some statistical population and we estimated some population parameters and the obtained estimates were dramatically different.
Hans de Raedt et al. [82] generated in a computer experiment quadruplets of raw data (±1,±1, ±1,±1). Subsequent setting -dependent photon identification procedures, mimicking procedures used in real experiments, allowed to create new data samples containing only pairs (±1,±1) for each experimental settings. Because these new data sets were not simple random samples extracted from the raw data thus the estimated values of pairwise expectations, obtained using these setting-dependent samples could violate CHSH as significantly as it was observed in SPCE We personally do not believe that the fate of the photons is predetermined only by the preparation at the source and that the violation of Bell-CHSH inequalities is the effect of unfair sampling during a post selection.
For us clicks registered by distant measuring stations in SPCE and coded by ±1 are of completely different nature then colours and sizes of socks, positions and linear momenta of balls and electrons. Spin projections and clicks do not exist before the measurements are done. Thus one may not describe incoming "pair of photons" by lines of nonexisting Nx4 spreadsheet containing ±1 counterfactual outcomes of impossible to perform experiments.

Conclusions
In this article we explained why the speculations about the quantum nonlocality and the quantum magic are rooted in incorrect interpretations of QM and/or in incorrect "mental pictures" and models trying to explain invisible details of quantum phenomena.
For example a "mental picture " of an ideal EPRB experiment in which twin photon pairs produce, in irreducible random way , strictly correlated or anti-correlated clicks on distant detectors creates the impossible to resolve paradox: "a pair of dice showing always perfectly correlated outcomes" As we explained in the section 3 we do not need to worry because the ideal EPRB experiment does not exist.
In SPCE setting directions are not mathematical vectors but only small spherical angles and we neither see nor follow pairs of entangled photons which produce "click" or "no-click" on Alice's and Bob's detectors . There are black counts, laser intensity drifts etc. Detected clicks have their time tags and correlated time-windows are used to identify and select pairs of clicks created by the photons belonging to the same entangled pair.
Since various photon-identification procedures are setting -dependent thus final postselected data may not be described by the quantum model used to describe the nonexisting ideal EPRB. In SPCE not only we don't have strict correlations or anticorrelations between Alice's and Bob's outcomes but also marginal single counts distributions depend on the distant settings what seems to violate Einsteinian nosignalling. This violation is only apparent because single count distributions estimated using raw data do not depend on the distant settings. [60].
Raw and post-selected data in SPCE may be described in a locally causal way using a contextual model [59][60] in which "a click:" or " a no-click" are determined by setting dependent parameters describing a measuring instrument and parameters describing a signal arriving to the measuring station at the moment of the measurement.
In contrast to LRHVM and SHVM in the contextual model (11)(12)(13)(14)(15)(16)(17) and in QM : the outcomes of 4 incompatible experiments performed in different settings are described by dedicated probability distributions defined on disjoint probability spaces. Only if all the physical observables measured in SPCE are compatible these dedicated probability distributions may be deduced as marginal probability distributions from a joint probability distribution defined on a unique probability space. . [43][44] that the quantum nonlocality is also rooted in incorrect individual interpretation of QM and in incorrect interpretation of Lüders projection postulate.

Khrennikov recently explained in
As we mentioned in the introduction: it would be surprising if the violation of Bell-CHSH inequalities , which are proven using simple algebraic inequalities satisfied by any quadruplet of 4 integer numbers equal to ±1 , might have deep metaphysical implications. In fact such metaphysical implications are quite limited and may be resumed in few words as it was done by Peres [84] "unperformed experiments have no results".
Therefore the violation of various Bell-type inequalities may neither justify the existence of non-local influences nor doubts that atoms, electrons and the Moon are not there when nobody looks.