Joy Christian wrote:In my opinion Richard Gill owes me 10,000 Euros. I have won his challenge fair and square.
I have generated 2 x N vectors, e_k and -e_k, in this simulation: http://rpubs.com/jjc/16415. They produce a very good approximation to the -cosine curve, with neither "a" nor "b" fixed a priori.
Please do check out the last plot of the simulation to recognize that none of the four correlations, namely
E(0, 45) = - 0.7071...,
E(0, 135) = + 0.7071...,
E(90, 45) = - 0.7071...,
and
E(90, 135) = - 0.7071....,
when calculated separately without replacement, deviate by more than 0.2 from the predicted values. This is more than evident from the last plot of the simulation, with both "a" and "b" completely freely chosen. That is all that is needed to recognize the fact that I have won the 10,000 Euros from Richard Gill.
set.seed(9875)
M <- 10^5 ## Sample size. Next try 10^6, or even 10^7
s <- runif(M, 0, pi)
t <- runif(M, 0, pi)
x <- cos(s)/1.28
y <- (-1 + (2/(sqrt(1 + (3 * t/pi)))))
e <- cxbind(x, y) ## M x 2 matrix; M *rows* of e represent the
## x and y coordinates of points on a circle; y -> -y => e -> -e.
## Note: cbind instead of rbind (glue columns together, not rows).
write.table(e, "JoyVector1.txt", row.names=FALSE, col.names=FALSE) ## save in a file
gill1109 wrote:......
Joy Christian wrote:gill1109 wrote:......
Richard,
Save your energy. I have won already. I have generated the "impossible" 2 x N vectors that produce the strong correlations. This adds a further feather in the cap of my local-realistic framework. A further result in the long list of results supporting my framework.
gill1109 wrote:If you want to follow Michel's advice and create four separate files then I have to revise my offer. N must be at least 10 000 and I can only allow you an error margin of +/- 0.1. Moreover, I will decide which correlation to calculate on which data-set. Four data-sets, four correlations, but I get to choose which a and b pair are combined with which data set.
gill1109 wrote:Sorry, no 10 000 Euro. No feather in your cap. Another in mine...
Or do you want to submit a file and call in the adjudicators?
gill1109 wrote:If you want to follow Michel's advice and create four separate files, then I do not have to revise my offer.
Four non-empty data-sets, four correlations, but for fairness, obviously I get to choose which a and b pair are combined with which data set.
I allow you the same error-margin +/- 0.2.
The value of N is immaterial and whether it is the same or different for all data-sets is matter of no concern to me whatsoever.
(Thanks to Heinera for his sharp insight).
Joy Christian wrote:Maybe you haven't heard. I have already won 10,000 Euros offered by you.
gill1109 wrote:Joy Christian wrote:Maybe you haven't heard. I have already won 10,000 Euros offered by you.
No such thing. You announced your imminent winning of the bet. I haven't seen the computer file yet. Seems it was a kind of premature ejaculation, sorry premature announcement, because I still haven't seen the file.
You also single handedly rewrote the rules we had agreed to, thereby proving to the world that you are a liar and a cheat. It was about time that it was said so clearly. I'm glad that this is now completely plain for all to see.
You can email me when you have posted the file on internet.
## My response to Christian's claim to my prize.
## First I delete all lines from his code except those generating
## the set of directions "e"
set.seed(9875)
N <- 10^5
s <- runif(N, 0, pi)
t <- runif(N, 0, pi)
x <- cos(s)/1.28
y <- -1 + (2/(sqrt(1 + (3 * t/pi))))
e <- rbind(x, y) ## 2 x N matrix; N columns of e represent the
## x and y coordinates of points on a circle:
## Alice's observed directions of angular momentum.
## Bob's observed directions are -e.
## Now I separately compute four correlations according to the
## formulas agreed by Christian
alpha <- 0 * pi / 180
beta <- 45 * pi / 180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
(E_0_45 <- mean(sign(colSums(e * a)) * -sign(colSums(e * b))))
alpha <- 0 * pi / 180
beta <- 135 * pi / 180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
(E_0_135 <- mean(sign(colSums(e * a)) * -sign(colSums(e * b))))
alpha <- 90 * pi / 180
beta <- 45 * pi / 180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
(E_90_45 <- mean(sign(colSums(e * a)) * -sign(colSums(e * b))))
alpha <- 90 * pi / 180
beta <- 135 * pi / 180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
(E_90_135 <- mean(sign(colSums(e * a)) * -sign(colSums(e * b))))
## Just for fun
- E_0_45 + E_0_135 - E_90_45 - E_90_135
[1] -0.68418
[1] 0.68382
[1] -0.31326
[1] -0.31874
[1] 2
gill1109 wrote:I am now going on a short vacation but will be reading my email from time to time.
gill1109 wrote:alpha <- 0 * pi / 180
beta <- 45 * pi / 180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
(E_0_45 <- mean(sign(colSums(e * a)) * -sign(colSums(e * b))))
alpha <- 0 * pi / 180
beta <- 135 * pi / 180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
(E_0_135 <- mean(sign(colSums(e * a)) * -sign(colSums(e * b))))
alpha <- 90 * pi / 180
beta <- 45 * pi / 180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
(E_90_45 <- mean(sign(colSums(e * a)) * -sign(colSums(e * b))))
alpha <- 90 * pi / 180
beta <- 135 * pi / 180
a <- c(cos(alpha), sin(alpha))
b <- c(cos(beta), sin(beta))
(E_90_135 <- mean(sign(colSums(e * a)) * -sign(colSums(e * b))))
## Just for fun
- E_0_45 + E_0_135 - E_90_45 - E_90_135
[/code]
Here are the numerical results which I obtained:
- Code: Select all
[1] -0.68418
[1] 0.68382
[1] -0.31326
[1] -0.31874
[1] 2
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