I am interested in a lower bound on the Bhattacharya distance between two independent multivariate Gaussian distributions. To be precise, consider zero-mean independent Gaussian distributions $p\_1\sim\mathcal{N}(\mathbb{0},\Theta\_1^{-1})$ and $p\_2\sim\mathcal{N}(\mathbb{0},\Theta\_2^{-1})$ (note that $\Theta\_1$, $\Theta\_2$ are $n\times n$ positive definite matrices). Then the Bhattacharya distance between $p\_1$ and $p\_2$ can be expressed as: 


\begin{align\*}
d\_B(p\_1,p\_2)=\frac{1}{2} \ln\left(\frac{|\Theta\_1+\Theta\_2)/2|}{|\Theta\_2|^{1/2}|\Theta\_2|^{1/2}}\right).
\end{align\*}


I would prefer the lower bound is in terms of some matrix norm of $(\Theta\_1-\Theta\_2)$, for e.g., $||\Theta\_1-\Theta\_2||\_1$ (L1 norm) or $||\Theta\_1-\Theta\_2||\_2$ (L2 norm) or $||\Theta\_1-\Theta\_2||\_F$ (Frobenius norm).


One may note that $2d\_B(p\_1,p\_2)=D(p\_1||p)+D(p\_2||p)$, where $p=(p\_1+p\_2)/2$ and $D(\cdot||\cdot)$ is the KL-divergence function, but I could not make use of this property to get any answer. 



Lower bound on Bhattacharya distance between independent Gaussian distributions ?
The KL divergence is given by
$$\hbox{Tr}(\Sigma\_0\Sigma\_1^{-1})- \ln(|\Sigma\_0\Sigma\_1^{-1}|)$$
Let $\mu\_{-}$ be the smallest eigenvalue of $\Sigma\_0\Sigma\_1^{-1}$ and $\mu\_{+}$ the largest, then


$$\mu\_{-} = ||\Sigma\_1 \Sigma\_0^{-1} ||\_2$$


$$\mu\_{+} = ||\Sigma\_0 \Sigma\_1^{-1} ||\_2$$


$$\hbox{Tr}(\Sigma\_0\Sigma\_1^{-1}) > n\mu\_{-}$$
$$-\ln(|\Sigma\_0\Sigma\_1^{-1}|) > -n\ln(\mu\_{+})$$
$$KL > n ||\Sigma\_1\Sigma\_0^{-1}||\_2 -n\ln(||\Sigma\_0\Sigma\_1^{-1}||\_2)$$


With the lower bound on the KL you can get a lower bound on the distance you're looking for. It doesn't seem you can get away with looking at the norm of the difference between the matrices.


