Question about the log-likelihood function

[ghost]

Hi!

I wondered why ar.chi2err is always zero when you do not estimate errors? I.e.: why is it [-2 log(L) = 2ar.ndatalog(sqrt(2pi)) + x^2/sigma^2] and not [-2 log(L) = 2ar.ndatalog(sqrt(2pi)) + 2log(sigma) +x^2/sigma^2]? Shouldn't [2log(sigma)] be included in the log-likelihood regardless of whether experimental errors are used?

Thanks in advance,

Best, Jo

Originally posted by @JoanneJansen in https://github.com/Data2Dynamics/d2d/issues/42#issuecomment-461152269

[clemenskreutz]

Dear Jo.

I guess, you talk about the following:

% see arCollectRes.m:
fiterrors = ( ar.config.fiterrors == 1  || (ar.config.fiterrors==0 && sum(ar.qFit(ar.qError==1)<2)>0 ) );
 ...
if fiterrors == 1
     ar.chi2fit = ar.chi2 + ar.chi2err;
else
    ar.chi2fit = ar.chi2;
end

Usually people don't fit experimental errors (sigma parameters) and they expect "chi-square", i.e. a positive number. However, 2log(sigma) can be negative which is confusing in this standard setting. I guess that this was the reason why this negative constant was excluded if there are no error parameters to be fitted.

The following function call provides all terms which play a role for calculating the objective function [m,m2]=arGetMerit