Let $K$ be a positive integer and $C$ be any $K \times K$ non-singular matrix. For positive real numbers $q\_1, \dots, q\_K$, define 
$$\Sigma(q\_1, \dots, q\_K) = CC' + diag(\frac{1}{q\_1}, \dots, \frac{1}{q\_K})$$ 
and 
$$V(q\_1, \dots, q\_K) = I\_k - C'\Sigma^{-1}C.$$


**Question:** Are each of the diagonal entries of $V$ log-convex in the arguments $q\_1, \dots, q\_K$? 


The statistical interpretation is the following: suppose a decision maker is learning about $K$ unknowns $\theta\_1, \dots, \theta\_K$, which are i.i.d. normally distributed according to the prior. She has access to $K$ signals $X\_1, \dots, X\_K$, which are linear combinations of the unknowns (with coefficients $C$) plus i.i.d. noise. Then $V(q\_1, \dots, q\_K)$ represents her conditional covariance matrix given $q\_k$ observations of signal $X\_k$. 


It's not too difficult to show that $V\_{ii}$ is convex. By brute-force computation I found that it is in fact log-convex when $K = 2$. Thus I'm wondering whether that might be true in general (and why). 



Log-convexity of conditional variances
It seems that a stronger claim holds. Let $D$ can be any positive definite matrix (not just a diagonal). 


The $i$-th entry of the matrix in the OP is
\begin{equation\*}
v\_{ii}(D) = 1 - e\_i^TC'(CC' + D^{-1})^{-1}Ce\_i.
\end{equation\*}


**Key idea.** Notice that we can write $v\_{ii}(D)$ as the ratio
\begin{equation\*}
 v\_{ii} = \frac{\det(CC'+D^{-1} - c\_ic\_i^T)}{\det(CC'+D^{-1})} = \frac{\det(A+D^{-1})}{\det(B+D^{-1})} = \frac{\det(I+AD)}{\det(I+BD)},
\end{equation\*}
where $B> A \ge 0$. 


We wish prove that $v\_{ii}$ is log-convex in $D$. Even though $A$ may not invertible, the calculus gets considerably simplified if we assume it is. So we perturb it by a small amount, say $A\_{\epsilon} \gets A + \epsilon I$ and perform the argument in terms of $A\_{\epsilon}$; then we can conclude the general case by a limiting argument (*this should be made rigorous*). We abuse notation a bit and omit the subscript on $A$ below for clarity.


\begin{equation\*}
\log v\_{ii}(D) = \log\frac{\det(I+AD)}{\det(I+BD)} =\log\frac{\det(I+A^{1/2}DA^{1/2})}{\det(I+B^{1/2}DB^{1/2})},
\end{equation\*}
where the latter equivalence follows after similarity transforms on the numerator and denominator.


Computing the first-derivative wrt $D$ we obtain
\begin{equation\*}
\nabla\log v(D) = A^{1/2}(I+A^{1/2}DA^{1/2})^{-1}A^{1/2} - B^{1/2}(I+B^{1/2}DB^{1/2})^{-1}B^{1/2}, 
\end{equation\*}
which simplifies to
\begin{equation\*}
\nabla\log v(D) = (A^{-1}+D)^{-1} - (B^{-1}+D)^{-1}.
\end{equation\*}
Now computing second derivatives wrt we see that the Hessian can be identified with the operator
\begin{equation\*}
 (B^{-1}+D)^{-1}\otimes (B^{-1}+D)^{-1} - (A^{-1}+D)^{-1} \otimes (A^{-1}+D)^{-1} \ge 0.
\end{equation\*}
The final inequality follows because $A \le B \implies A^{-1}+D\ge B^{-1}+D \implies (A^{-1}+D)^{-1} \le (B^{-1}+D)^{-1}$, combined with the fact that Kronecker products preserve matrix monotonicity.


