Combining 2 probes in order to increase Figure of Merit

[fab13]

TL;DR Summary

I would like to find a way to combine 2 probes in cosmology represented each one by a Fisher matrix. This combination should increase the Figure of Merit (FoM) by a significant gain, that is, doing make cross-correlations between the 2 probes.

This post is slightly different from a previous post sent to mathematical forum : this is because I talk about here the MATLAB function "eig" with 2 arguments but this concerns actually the combination between 2 biased tracers in Cosmology context.

I am looking for a common basis of eigenvectors between 2 Fisher or Covariance matrices A and B. Different algorithms exist to "joint diagonalize" 2 matrices like for example qndiag. They allow to find the same eigen vectors that diagonalize both A and B symmetric matrix.

In Matlab, the command `[V, D] = eig(A, B)` should solve the Generalized Eigenvalue Problem `A*V = B*V*D`. Knowing vectors `V`, this equality is true also for all other vectors under the form `V=VD'` with `D'` a chosen diagonal matrix.

I want to get the same basis for matrices A and B because I want to apply the MLE (Maximum Likelihood Estimator) by doing on diagonalized matrix :

1) For Fisher matrix, the sum coming from MLE :

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

2) Or For Covariance matrix, putting on the diagonal the variances :

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

Unfortunately, But in both cases, it doesn't increase the FoM (Figure of Merit, equal to $\dfrac{1}{det(block(2 parameters))}$), I mean that constraints are not better than with a classical synthesis ((where we simply sum sthe 2 Fisher matrices) : 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 (shot noise, intrinsic alignement) 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, I am looking for a way to apply the MLE on Fisher or Covariance matrix to make cross-correlations between 2 given Fisher matrices (which represent actually 2 different probes into a cosmology context : spectroscopic and photometric probes).

And to carry out it, I would like to know if the Generalized eigenvectors and eigenvalues formulated with the Matlab command `[V, D] = eig(A,B) could help me by taking for example (I am not sure) a special or rather an appropriate diagonal matrix `D'`.Indeed, I didn't yet grasp all the usefulness of using `[V, D] = eig(A, B)` in my case : I only understand that it allows to find ```ectors such that `AV` is "parallel" to `BV` (I mean the both are linked by a diagonal matrix).

I guess you think it is about numerical stuff but the underying issue is about Cosmology and the way to combine 2 probes in this context.

I hope you will understand the formulation of my issue.

Regards

 

[kimbyd]

I guess I'm not quite understanding why summing the Fisher matrices isn't what you need here. That's the standard way of combining experiments if you're using Fisher matrices.

 

[fab13]

I thought that it would be higher with cross-correlation (with MLE) from the constraints point of view. For example, when we combine the Weak Lensing with Photometric probes, we have this analytical formula to compute the $(\alpha,\beta)$ element of Fisher cross-correlated :

and we get roughly a factor between x3 and x4 on classical sumed Fishermatrix Weak Lensing and Fisher Photometric.
But the MLE doesn't seem to be the right method to apply between spectro and (WL+GCph+XC).

Could anyone suggest another method to get a factor significant for this final combination ?

 

[kimbyd]

Could you describe the notation in that equation?

 

[fab13]

Of course, the original version expresses the formula of Fisher element like this :

The final expression of the combined Fisher matrix for the angular power spectra, including the contribution of photometric galaxy clustering, cosmic shear and their cross-correlation, is given in the case of the fourth order covariance by
$$
F_{\alpha \beta}^{\mathrm{XC}}=\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{A B C D} \sum_{i j, m n} \frac{\partial C_{i j}^{A B}(\ell)}{\partial \theta_{\alpha}} \operatorname{Cov}^{-1}\left[C_{i j}^{A B}(\ell), C_{m n}^{C D}(\ell)\right] \frac{\partial C_{m n}^{C D}(\ell)}{\partial \theta_{\beta}}
$$
and in the case of the second-order covariance as
$$
F_{\alpha \beta}^{\mathrm{XC}}=\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{A B C D} \sum_{i j, m n} \frac{\partial C_{i j}^{A B}(\ell)}{\partial \theta_{\alpha}}\left[\Delta C^{-1}(\ell)\right]_{j m}^{A B} \frac{\partial C_{m n}^{C D}(\ell)}{\partial \theta_{\beta}}\left[\Delta C^{-1}(\ell)\right]_{n i}^{C D}
$$
The block descriptors $A, B, C, D$ run over the combined probes $L$ and $G$, thus including the three observables described in Eq. (136)
i.e. cosmic shear auto-correlation, cross-correlation between galaxy clustering and cosmic shear, galaxy clustering auto-correlation. The indices in the sum $i$ $j$ and $m n$ run over all unique pairs of tomographic bins $(i \leq j, m \leq n)$ for the cosmic shear auto-correlation and the galaxy clustering auto-correlation, while all pairs of tomographic bins are considered to take into account all the crosscorrelations between galaxy clustering and cosmic shear.

with :

$$\begin{aligned}
&C_{i j}^{\mathrm{LL}}(\ell)=\int_{z_{\min }}^{z_{\max }} \frac{d z}{H(z) r^{2}(z)} \mathcal{W}_{i}^{\mathrm{L}}(z) \mathcal{W}_{j}^{\mathrm{L}}(z) P_{\delta \delta}\left(\frac{\ell+1 / 2}{r(z)}, z\right)\\
&\text { where we have defined the overall weight function } \mathcal{W}_{i}^{\mathrm{L}}(z) \text { , which includes intrinsic alignment contributions }\\
&W_{i}^{\mathrm{L}}=W_{i}^{\gamma}(z)-\frac{\mathcal{A}_{\mathrm{IA}} C_{\mathrm{IA}} \Omega_{\mathrm{m}} \mathcal{F}_{\mathrm{IA}}(z)}{D(z)} W_{i}^{\mathrm{IA}}(z)
\end{aligned}
$$

and :

$$\begin{aligned}
C_{i j}^{\mathrm{GL}}(\ell) &=\int \frac{d z}{H(z) r^{2}(z)} \mathcal{W}_{i}^{\mathrm{G}}(z) \mathcal{W}_{j}^{\mathrm{L}}(z) P_{\delta \delta}\left(\frac{\ell+1 / 2}{r(z)}, z\right) \\
C_{i j}^{\mathrm{GG}}(\ell) &=\int \frac{d z}{H(z) r^{2}(z)} \mathcal{W}_{i}^{\mathrm{G}}(z) \mathcal{W}_{j}^{\mathrm{G}}(z) P_{\delta \delta}\left(\frac{\ell+1 / 2}{r(z)}, z\right)
\end{aligned}$$

 

[kimbyd]

I see. I believe that in order to perform the analysis you're suggesting here you need more than just the Fisher matrices of the two experiments. The Fisher matrix describes how each experiment constrains the model variables, but no information about the distribution of data used.

Summing the Fisher matrices is the correct thing to do when the two data sets are independent. I believe the more complex analysis they're using above accounts for the fact that they are not. So you'd in essence need to follow their analysis to obtain a single Fisher matrix describing the full relationship. Then you'd just diagonalize that.