# Propagating asymmetric and large errors in a ratio of likelihood-fit parameters

I'm currently trying to propagate the errors of two values:

$A=143^{+28.6}_{-26.6}$ , $B=19^{+10.9}_{-8.7}$ $\rightarrow$ $B/A = 0.133^{+?}_{-?}$

Both values A and B (#counts) are derived from a log-likelihood model fit on poisson data. The asymmetrical errors are due to the low number of events, which means that the Poisson-$\chi^2$ distribution (and thus the errors) is slightly asymmetric. I also know the numerical chi2 (likelihood) distribution for A and B.

Now, there are multiple major problems that I'm facing in the error propagation:

1) The errors are asymmetric. Of course you can propagate the upper/lower error for both separately, but there's no theoretical justification for that. However, sometimes even Barlow recommends it, despite that fact (http://www.slac.stanford.edu/BFROOT/www/Statistics/Report/report.pdf).

2) The errors are large (50%), which breaks assumptions for standard error propagation.

3) Even if you consider the errors as symmetric and small, calculating the ratio of two gaussian distributions can still result in an asymmetric ratio distribution.

The major struggle that I have right now is to incorporate all of those three issues into one method. If this is very time demanding, I could also assume the errors to be symmetric (because the difference isn't sooo big). However, even then, the problem of 2) and 3) still exist.

I've thought about combining the likelihood of A and B to C (C=A/B), but unfortunately that led to nothing.

Note: This is a crosspost from CrossValidated, because I feel that this community should be a better place for the problem (i.e. different error propagation is needed for a mean of some sample data in e.g. biology compared to the values based on a likelihood fit).

• Quoting and combining error bars is not appropriate in this situation. You need to calculate the full probability distribution of C using a Monte Carlo simulation. – Rob Jeffries Jan 14 '17 at 10:17
• Thank you for the comment! I guess that MC-generating A and B based on their probability (=exp(DeltaL)) and then calculating C should be sufficient? – 0vbb Jan 15 '17 at 2:58
• It would be helpful to have more details of your experiment and the meaning of A and B. – sammy gerbil Jan 15 '17 at 3:12
• I am counting events (1D energy histo) in a certain energy range. I know that the physical distribution of these events is the sum of two underlying physical processes. Thus I fit (Binned Likelihood) the counts in the data with two model MC-spectra in a joint fit. As a result, I get A and B based on the fit. However, the A and B that I want to compare (C=A/B) are from different, independent data sets. – 0vbb Jan 15 '17 at 3:20