Make cross-correlations between 2 Fisher or 2 Covariance matrices

[fab13]

TL;DR Summary

I am looking for a way to include the MLE(Maximum Likelihood Estimator) in the synthesis of 2 Fisher matrix and so make cross-correlations instead of a simple sum of 2 Fisher matrices.

I have 2 Fisher matrixes which represent information for the same variables (I mean columns/rows represent the same parameters in the 2 matrixes).

Now I would like to make the cross-correlations synthesis of these 2 matrixes by applying for each parameter the well known formula (coming from Maximum Likelihood Estimator method) :

$\dfrac{1}{\sigma_{\hat{\tau}}^{2}}=\dfrac{1}{\sigma_1^2}+\dfrac{1}{\sigma_2^2}\quad(1)$

$\sigma_{\hat{\tau}}$ represents the best estimator representing the combination of a `sample1` ($\sigma_1$) and a `sample2` ($\sigma_2$).

Now, I would like to do the same but for my 2 Fisher matrixes, i.e from a matricial point of view.

1) Firstly, for this, I tried to diagonalize in a simultaneous way each of these 2 Fisher matrix, i.e by finding a common eigenvectors basis for both. Then, I would add the 2 diagonal matrices and I have so a global diagonal Fisher matrix, and after come back in the space of start.

But this method gives the same constraints (by inversing the Fisher matrix) are the same than a classical synthesis between 2 matrices since :

$V^{-1} A V = D_1$

$V^{-1} B V = D_2$,

then what I wanted to do by summing the 2 diagonal matrices $D_1$ and $D_2$ is the same thing than doing $A+B$ :

$A+B = V (D_1+ D_2) V^{-1}$

Finally, I had to give up this track

2) Secondly, I tried to work directly in the space of Covariance matrix. I diagonalized "`simultaneously`" each of the 2 matrices. (Then, I have no more covariances terms).

and I build another covariance matrix by applying the MLE like putting on the diagonal :

$\sigma_{\hat{\tau}}^{2}=\dfrac{1}{\dfrac{1}{\sigma_1^2}+\dfrac{1}{\sigma_2^2}}\quad(2)$

Unfortunately, it doesn't increase the FoM (Figure of Merit, equal to $\dfrac{1}{\sqrt{\text{det(block(2 parameters))}}}$ ), I mean that constraints are not better : as a conclusion, I can't manage to do cross-correlations since I have no gain on constraints.

Moreover, with the first method 1) above, after the building of Fisher matrix and its inversion, I can marginalize over the nuisance parameters and re-invert to get a Fisher matrix where nuisance parameters estimations are encoded into it.

But I can't do the same for method 2). I can fix parameters (remove directly lines/columns) in Fisher matrix but this produces too high FoM (and so too small constraints) since I have less error parameters to estimate.

So, my question is about the research of a way to apply the MLE on Fisher (or maybe directly on Covariance matrix) to make cross-correlations between 2 given Fisher matrices (which represent actually 2 different probes into a cosmology context).

Any suggestion/track/clue about this way to do is welcome.

 

[mmwave]

Thank you for sharing your thoughts and methods for synthesizing cross-correlations between two Fisher matrices. I would like to provide some feedback and suggestions on your approach.

Firstly, it is important to note that the Fisher matrix is a mathematical tool used for estimating the uncertainties in the parameters of a model. It is not a physical quantity and therefore, it cannot be directly manipulated or combined in the same way as physical matrices. This is why your first method, which involves diagonalizing and adding the two Fisher matrices, does not give any improvement in the constraints.

Secondly, the Maximum Likelihood Estimator (MLE) method is a powerful tool for estimating the parameters of a model. However, it is not applicable to Fisher matrices as they are not probability distributions. The MLE formula that you have mentioned in your post is only applicable to the standard deviation of a sample, which is not the same as the uncertainties in the parameters estimated by the Fisher matrix.

In light of these points, I would suggest looking into other methods for combining the two Fisher matrices. One approach could be to use a weighted average of the two matrices, where the weights are based on the relative uncertainties in the parameters estimated by each matrix. This would take into account the relative importance of each probe in constraining the parameters.

Additionally, I would recommend consulting with other experts in the field of cosmology to see if there are any established methods for combining Fisher matrices from different probes. Collaboration and discussion with other scientists can often lead to new insights and solutions.

I hope this helps in your research and I wish you all the best in finding a suitable method for synthesizing cross-correlations between Fisher matrices.