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I have initially posted on physics.stackexchange but I think my issue is more adapted on Cross-Validated (so I am going to delete the initial post on physics.stackexchange).

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

Now I would like to make the cross 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.

For this, I tried to diagonalize each of these 2 Fisher matrix. Then, I add the 2 diagonal matrix and I have so a global diagonal Fisher matrix, but I don't know how to come back in the space of start (since the diagonalization don't give the same combination of eigen vectors for each matrix).

If I could get back in the same time to starting parameters space, I could do matrix products to get the final Fisher matrix by doing :

$$\text{Fisher$_{\text{final,cross}}$} = P.\text{Fisher$_{\text{diag,global}}$}.P^{-1}\quad(2)$$

with $P$ the passing matrixes (composed of eigen vectors) and I could get directly the covariance matrix by inverting $\text{Fisher}_{\text{final,cross}}$

How can I come back from $(2)$ of the $\text{Fisher$_{\text{diag,global}}$}$ diagonal matrix to starting space, i.e the single parameters ?

My difficulties come from the fact that the diagonlization of the 2 Fisher matrixes will produce different passing matrix $P_1$ and $P_2$, that is to say, different eigen vectors, so a different linear combination of variables between both. I have written the passing matrix $P$ above but it is not defined, I think an expression of $P$ as a function of $P_1$ and $P_2$ passsing matrixes is the key point of my issue.

There is surely a linear algebra property whih could allow to circumvent this issue of taking into account the 2 different linear combination of variables while being able to come back in the space of start, i.e space of single parameters which represent the Fisher matrixes.

If someone could help me to perform this operation, I hope that I have been clear. If you have any questions, don't hesitate, I would be glad to give you more informations.

EDIT 1: I take the following notations :

  1. $D$ diagonal matrix is the sum of the 2 diagonalized matrix $D_1$ and $D_2$ (from Fisher1 and Fisher2 initial matrixes) : $D=D_1+D_2$

  2. $P_1$ and $P_2$ are respectively the "passing" matrixes (from Fisher1 and Fisher2 diagonalization) composed of eigen vectors.

  3. $M_1$ is the Fisher1 matrix and $M_2$ is the Fisher2 matrix.

So, I am looking a way to find an endomorphism $M$ that checks :

$$D=P^-1.M.P\quad(3)$$ where $P$ is the unkonwn "passing" matrix.

So, there are 2 unknown quantities in my issue :

  1. The "passing" matrix, i.e the eigenvectors (I am yet trying to build it from $P_1$ and $P_2$ matrixes).

  2. The $M$ matrix which represents this endomorphism.

But in this world of unknown quantities, I know however the eigen values of this wanted endomorphism $M$ (they are equal to the diagonal of matrix $D$).

Would anyone help me to find a way to build this $P$ "passing" matrix from $P_1$ and $P_2$ ? As you can see, a simple sum is not enough.

If an exact building of $P$ from $P_1$ and $P_2$ is not possible, is there a way to approximate it ?

ps : the links that @AJKOER gave are pretty hard to extract useful informations from these papers.

EDIT 2: @AJKOER : thanks for your help. Your idea is a good beginning in my research to carry out a full cross-correlation between my 2 Fisher matrixes, at least for the moment rather an approximation. I say that since in a previous study, I have used antoher method which is also an approximation of an exact formulation of cross-correlation (that will be pretty hard if I have understood your opinion).

For example, with your trick, I can get constraints closed to another method that I have test, so we are on the right way.

This old known test gives for 1 sigma error the following constraints :

wm +/- 0.0012417036832725796
wb +/- 0.0005995521931530912
w0 +/- 0.020152731097000408
wa +/- 0.07473741589196892
h +/- 0.001003370899834334
ns +/- 0.0018175790165196942
s8 +/- 0.0010147066130214034

and with the method of average eigen values into diagonal Fisher matrix1 matrix2, I get :

wm +/- 0.002003104719056934
wb +/- 0.0006792309014032004
w0 +/- 0.023872510159754036
wa +/- 0.08488679494420406
h +/- 0.0008331587961258862
ns +/- 0.002039401547697202
s8 +/- 0.0019000745102872813

Except for the first and last parameters (wm and s8 parameters), as I already said, contraints are closed. I have also a criterion of precision (determinant of the sub-block (w_0,w_a)) which is pretty similar.

But I think I can do better, what are your suggestions ?

A colleague told me to apply a prior but How could I proceed ?

The method of average on eigen values is already a "prior method", isn't it ?

By the way, what does this average from a Fisher information point of view or physics point of view mean ?

I have to keep on studying the papers given by @AJKOER, I have difficulties to assimilate them and extract a more rigorous method !

Any help is welcome.

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My suggestion, look at the literature on pooled covariance matrix estimation, for example, this article 'Partially pooled covariance matrix estimation in discriminant analysis', but not freely available. Relatedly, here is a fully available paper: 'A hierarchical eigenmodel for pooled covariance estimation'.

Also, fully available is this work: A Two-Stage Approach to Synthesizing Covariance Matrices in Meta-Analytic Structural Equation Modeling.

Note: I intended to only present a comment to this question, however, this last citation has a link address that exceeds the comment length limit.

I hope this assists.

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  • $\begingroup$ Thanks for your quick answer. Unfortunately, it's relatively hard for me to extract useful information for my issue. Could you help me to do it ? $\endgroup$ – youpilat13 Sep 18 '20 at 13:36
  • $\begingroup$ Not likely much help, no prior experience. Here is a simple crude idea, which may perhaps be useful. Pool (as in average) the eigenvalues by row for the two the diagonal eigenvalue matrices. Then, based on the readings I provided, where an equivalence of eigenvectors assumption was made, select one set of the two set of eigenvectors, and proceed to derive results. Repeat with the other eigenvector choice. Compare results. To assess whether this suggested procedure adds any value, do a simulation exercise employing the derived Fisher matrix knowing true underlying values and repeat 'n' times. $\endgroup$ – AJKOER Sep 18 '20 at 16:00
  • $\begingroup$ I don't understand very well when you say "Pool (as in average) the eigenvalues by row for the two the diagonal eigenvalue matrices : you mean that I have to put all the eigen values on the first row into "diagonal matrices" ? In the paper you give in your answer above, How to proceed this stuff of 2 stages procedure to perform the cross synthesis ? I mean, how to combine all the matrices I know, i.e $D$, $P_1$, $P_2$, $M_1$ and $M_2$ ? $\endgroup$ – youpilat13 Sep 18 '20 at 19:12
  • $\begingroup$ I meant for each diagonal matrix (D1, D2) of eigenvalues, compute a composite diagonal matrix equal to say, 1/2*D1 + 1/2*D2, so the row j eigenvalue is averaged, to lead to back to a pooled Fisher matrix. Whatever simple (or advanced complex) methodology employed for the dimensionality of your model, does a corresponding simulation exercise with known model parameters and specified noise distribution, and therefrom derived Fisher matrices, appear to actually benefit from an attempted 'pooling' exercise (and does the improvement apparently merit the effort and added theoretical complexity?). $\endgroup$ – AJKOER Sep 19 '20 at 3:06
  • $\begingroup$ Thanks for your suggestion : you can see the results in my EDIT2, this is up-and-coming and can be yet improved in the results to be more closed to the "complex theory of cross-correlations between 2 samples from a Fisher point of view". I have to stufy more your papers but this is technical. $\endgroup$ – youpilat13 Sep 21 '20 at 4:35

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