- #1

- 73

- 4

- Summary:
- Which is the right way to estimate the correlation between parameters estimated with a likelihood function?

Hello community!

I am facing a conceptual problem with the correlation matrix between maximum likelihood estimators.

I estimate two parameters (their names are

I know the best way to evaluate the correlation between two parameters from a likelihood fit is starting from the Hessian matrix. Performing that method I get a correlation of -0.14.

Then I tried a different approach: studying the

On the other hand, if I consider the two trends independently and apply again the definition of correlation I got a correlation of -1 (central and left plot in attachment).

In my mind all these approaches should be equivalent and giving the same value of the correlation, but it is not the case. So there might be some bug in my reasoning. Can someone help me to sort out my ideas, please?

I am facing a conceptual problem with the correlation matrix between maximum likelihood estimators.

I estimate two parameters (their names are

**SigmaBin0**and**qqzz_norm_0**) from a multidimensional likelihood function, actually the number of parameters are larger than the two I am focusing my attention now. I need to evaluate the correlation between that two parameters.I know the best way to evaluate the correlation between two parameters from a likelihood fit is starting from the Hessian matrix. Performing that method I get a correlation of -0.14.

Then I tried a different approach: studying the

**SigmaBin0**vs**qqzz_norm_0**values when one of the parameters is the Parameters Of Interest (POI) and the other is profiled, and viceversa. I mean, I scan the likelihood function along**SigmaBin0**while "watching" the**qqzz_norm_0**profiled values, and then I run another scan along**qqzz_norm_0**while "watching" the**SigmaBin0**profiled values. My expectation is to find the same trend in both cases, but what I find is the right plot in attachment. The*vertical*line is the former case, instead the*horizontal*line is the latter one. If I apply the definition of correlation (ratio of the covariance wrt the standard deviations) I get a correlation of -0.29.On the other hand, if I consider the two trends independently and apply again the definition of correlation I got a correlation of -1 (central and left plot in attachment).

In my mind all these approaches should be equivalent and giving the same value of the correlation, but it is not the case. So there might be some bug in my reasoning. Can someone help me to sort out my ideas, please?