More fit studies

I had a look at the pulls for 5000 tests of fitting together two toy samples at 10x10 binning. Results are below, I'm also running it over 2-20 bins now but it's taking a while.

Note that I defined the pull differently to the normal convention so a positive mean means that the results are too low. I'll correct that in any further plots.

r_bb, different chi2 variants
r_bb, different likelihood variants
r_cc, different chi2 variants
r_cc, different likelihood variants
r_gg, different chi2 variants
r_gg, different likelihood variants

Barlow-Beeston has a weird double hump which I don't understand, again this is probably my implementation, or the Poisson approximation breaking down. I wouldn't have thought it's the approximation because it's not as bad for r_bb where the number of bins in the peak is not much less than the total entries.

The only chi2 fit without a bias is the one that uses the r_xx values in the error calculation, which "gives nonsense results". We could potentially do the fit recursively using a constant r_xx from the previous iteration, as Klaus suggested. Is there time to implement it though?

The likelihood gives reliable central values, but underestimates the error (the pull sigma is high). I thought about trying the equation Roberval posted last week, but the only difference is the d*Log[d]-d term which will just give a constant shift - hence the fit values and errors will be the same. I could modify the change for which Minuit calculates the errors (the "Up()" method) using (another) toy study, but without a mathematical reason for changing it is that justified?