#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.


Get this bounty!!!

Leave a Reply

This site uses Akismet to reduce spam. Learn how your comment data is processed.