(* What is the correct way to marginalize some parameter? Pick out the elements of the covariance matrix or the Fisher matrix? *)
(* Let's see! *)

(* Define a symmetric covariance matrix *)
cij[i_,j_]:=cij[j,i]/;i>j

(* Set up a 3x3 matrix and get the Fisher matrix *)
Cij=Table[cij[i,j],{i,1,3},{j,1,3}];
Fij=Inverse[Cij];

(* Define the (χ^2)*)
chi2={x,y,z}.Fij.{x,y,z};

(* Prove the normalization is 1 *)
norm=Integrate[(1/((((2 Pi))^(3/2))((Det[Cij])^(1/2))))Exp[-chi2/2],{x,-∞,∞},{y,-∞,∞},{z,-∞,∞},GenerateConditions->False];

(* Mathematica needs some help, so square it *)
norm^2//FullSimplify

(* Marginalize over the 3rd parameter *)
marg=Integrate[(1/((((2 Pi))^(3/2))((Det[Cij])^(1/2))))Exp[-chi2/2],{z,-∞,∞},GenerateConditions->False];

(* Mathematica needs some help, so square it *)
(* The result corresponds to 2d Gaussian as expected *)
marg^2//FullSimplify

(* Pick out the elements of 2x2 block matrix for first 2 params from the Covariance matrix *)
Cij21=Cij[[1;;2,1;;2]];
(* Pick out the elements of 2x2 block matrix for first 2 params from the Fisher matrix *)
Cij22=Inverse[Fij[[1;;2,1;;2]]];

(* Only the first case agrees with the expected result! *)
(1/((2 Pi)((Det[Cij21])^(1/2))))Exp[-{x,y}.Inverse[Cij21].{x,y}/2]//FullSimplify
(1/((2 Pi)((Det[Cij22])^(1/2))))Exp[-{x,y}.Inverse[Cij22].{x,y}/2]//FullSimplify