History and connection to Statistics 

Considering Einstein's doubts about quantum mechanics, Einstein was puzzled by the strange "causality" in certain characteristics of quantum mechanics. In particular, the long distance connection between two particles that had briefly interacted, was a mystery to him. For, after all, the possibility of immediate interaction over a long distance could violate relativity. 

Hidden parameters 

Einstein postulated the existence of extra hidden parameters that could explain the quantum entanglement of particles over long distances. His aim was eventually a return to causality and locality in the philosophical basis of quantum mechanics. The latter was, and still is, a  debated controversy. 

Nowadays, one can read statements like: Aspect demonstrated conclusively that Einstein was wrong about quantum mechanics. It means that there are no Einsteinian extra parameters that may explain entanglement. My paper asks if that really is the case. Let us first look at the set up of Aspect's experiment. 

Aspect's experiment and data analysis 

The key to Aspect's experiment is Bell's correlation formula between measurements. In Bell's function, the Einsteinian variables are introduced as ancillary parameters. The ancillary parameters co-determine the outcome of measurements and are ruled by classical probability laws. 

Interestingly, the hypothetical existence of Einstein's hidden ancillary parameters in Bell's formula also give rise to certain inequalities. Aspect wanted to employ these inequalities in his data analysis. 

Methodology of data gathering and analysis in the experiment 

Aspect set up his experiment without using explicit models for measurement and for the way the hidden ancillary variables influence the measured result. This practical abstraction enabled Aspect to employ an approximate form of Bell's correlation function. And that, in turn, enabled him to employ the inequalities.

Therefore, Aspect's approach is a completely classical probability approach to possible Einstein positive data in empirical reality. Furthermore, the “yes or no” violation of the inequalities in empirical reality is a litmus test for the presence or absence of Einstein data in empirical reality. 

Wait a minute 

But please note. The quantum correlation can violate the Bell inequalities. 

Now, on the one hand Einstein data, in the form of the ancillary extra hidden parameters can mathematically not violate Bell inequalities. On the other hand, if they can not violate the inequalities, then they don't exist in empirical reality when in empirical reality the inequalities are violated with the data gathered in experiment. 

The key here is that Aspect in the hypothesis of existence of Einstein data, on the one hand corners Einstein data as classical probability, while on the other hand, the data must be of non classical probability character, in order to exist.

There could be a statistical wrinkle here. My paper 

https://rdcu.be/ftJkR

demonstrates that, indeed, such is the case. Aspect's experiment is statistically flawed and does not conclusively demonstrate the absence of Einstein data i.e. demonstrate non-locality. 

This isn't a probability function 

To give the reader a bit of a taste of what the paper is about, the following point is added. In the data analysis we take the settings of the measurement instrument parameters, a, for Alice and, b, for Bob. One can define an angle x=ange(a,b) in the plane spanned by a  and b. The angle x is i the interval [0,2π).

Aspect requires in the data analysis that:

sin²(x/2) = Prob[≠,x]

With Prob[≠,x] the probability that, given the angle between Alice's measurement instrument parameter, a, and Bob's measurement instrument parameter b, is x, gives unequal results for Alice's measurement and Bob's measurement. But note, sin²(x/2), isn't monotone non-descending on the complete interval [0,2π). It's not a probability function. 

So, Aspect required from Einstein data to be classical probability in nature, but, not to follow a monotone non-descending function on [0,2π) as a probability function. That's not possible. The hypothesis: "in the experiment there are Einstein positive data possible ", is false by statistical design of the experiment. No data can, at the same time, be classical probability based and not follow a monotone non-descending probability function. The details are in the paper 

https://rdcu.be/ftJkR

Thank you.