# #StackBounty: #covariance-matrix #fisher-information #matrix-inverse #constraint How can nuisance parameters in Fisher matrix can deter…

### Bounty: 50

I have a Fisher matrix $$F$$ which has the matrix blocks form like this :

$$F=begin{bmatrix} A & B\ C & D end{bmatrix}$$

The block $$A$$ is the most important block, in the sense the parameters of this block are the parameters that I want really get. The block $$D$$ is called the "nuisance block", that is to say, it consists of the nuisance parameters which deteriorate the informations of the block $$A$$. The blokc $$B$$ and $$C$$ are the correlations blocks between "important block" $$A$$ and "nuisance block" $$D$$.

Question : How can I prove mathematically that, by inversing the Fisher matrix $$F$$, I will get worse constraints (I mean larger variances) on the important parameters (corresponding to $$A$$ covariance matrix), all of this due to the 3 others blocks ?

Track followed :

I know that (1,1) block for inverse of $$F$$ :
$$F=begin{bmatrix} A & B\ C & D end{bmatrix}$$

is $$(A-BD^{-1}C)^{-1}$$ (Schur complement).

From this, if I take the inverse of Fisher matrix $$F$$, I will get for covariance matrix (the inverse of $$F$$), the block (1,1) equal to $$(A-BD^{-1}C)^{-1}$$.

How can we be sure that diagonal of this block $$(A-BD^{-1}C)^{-1}$$ (which represents the variances of important parameters) will be decreased by the terms "$$-BD^{-1}C$$" : we should have positive quantities for the diagonal elements of "-$$BD^{-1}C$$" to make increase the block (1,1) diagonals elements, i.e in order to make worse the constraints, shouldn’t we ?

I make the comparison with the values unmarginalised of block A which have the form into initial Fisher matrix F : a_11 = 1/sigma_1^2, a_22 = 1/sigma_2^2, a_33 = 1/sigma_3^2 … etc. with sigma_i the standard deviation of each important parameters of block A.

I guess it is difficult to prove that nuisance parameters deteriorate the constraints for block (1,1) of covariance matrix but it would be interesting to have a rigorous demonstration.

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