A Need help about a demo with inverse weighted variance average

AI Thread Summary
The discussion revolves around understanding the transition from equation (3) to equation (4) in the context of cosmological probes using inverse weighted variance averaging. The user seeks clarification on how the factor (2ℓ + 1) appears in the variance expression of the estimator. They note that equation (3) provides an optimal estimator for the ratio of the spectroscopic to photometric coefficients, while equation (4) expresses its variance. The user recalls that the power spectrum Cℓ is defined as the average of the squared coefficients, which may be key to deriving the variance. Assistance is requested to detail the steps needed to prove the relationship between these equations.
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I need help about the understanding of all the steps in a demonstration of the optimal variance by inverse-weighted variance average.
I have a problem of understanding in the following demo :

In a cosmology context with 2 probes (spectroscopic and photometric), let notice ##a_{\ell m, s p}## the spectroscopic and ##a_{\ell m, p h}## the photometric coefficients of the decomposition in spherical harmonics of the distributions of each population. In the absence of any Poisson noise we have:
##\dfrac{a_{\ell m, s p}^{2}}{a_{\ell m, p h}^{2}}=\left(\dfrac{b_{s p}}{b_{p h}}\right)^{2}\quad(1)##
Now let assume the spectroscopic sample is a Poisson realization of density ##N_{s p}## (the galaxy density of the spectroscopic sample) and that we have an unbiased estimator ##\hat{a}_{\ell m, s p}## of ##a_{\ell m, s p}##. We then have the average :
##
\left\langle\dfrac{\hat{a}_{\ell m, s p}^{2}}{a_{\ell m, p h}^{2}}\right\rangle=\dfrac{\left\langle\hat{a}_{\ell m, s p}^{2}\right\rangle}{a_{\ell m, p h}^{2}}=\left(\dfrac{b_{s p}}{b_{p h}}\right)^{2}\quad(2)
##
with its variance :
##
\dfrac{2}{f_{s k y} a_{\ell m, p h}^{4} N_{s p}^{2}}
##
We can therefore build an estimator ##\hat{O}## of ##\left(\dfrac{b_{s p}}{b_{p h}}\right)^{2}## by taking the optimal (inverse-variance weighted) average over all ##\ell## and ##m## :

##\hat{O}=\dfrac{\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{m=-\ell}^{\ell} a_{\ell m, p h}^{2} \hat{a}_{\ell m, s p}^{2}}{\sum_{\ell=\ell_{\min }}^{\ell_{\max }} \sum_{m=-\ell}^{\ell} a_{\ell m, p h}^{4}}\quad(3)##
the variance of which being:

##\sigma_{\hat{o}}^{2}=\left(\sum_\limits{\ell=\ell_{\min }}^{\ell_{\max }}(2 \ell+1) C_{\ell, p h}^{2}\right)^{-1} \dfrac{2}{f_{s k y} N_{sp}^{2}}\quad(4)##

I don't understanding the passing between eq(3) and eq(4). Indeed, I can't make appear from eq(3) the existing factor ##2\ell+1## in eq(4). The goal is to prove the relation eq(4).

If someone could help me to detail the different necessary steps to obtain eq(4), this would be fine.

I recall that in general, ##C_{\ell}=\dfrac{1}{2\ell+1}\sum_{m=-\ell}^{+\ell} a_{\ell m}^{2}##

Best regards
 
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a little up to demonstrate the passing from eq(3) to eq(4) ?
 
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