On Likelihood Functions to Minimize KL Divergence in Binary Hypothesis Testing

Kullback-Leibler (KL) divergence is widely used to determine lower bounds on detection error probability for binary hypothesis testing in covert communications and location verification systems. For a Gaussian likelihood function under the null hypothesis ${\mathcal{H}_0}$, it has been proved that it is a Gaussian likelihood function under the alternative hypothesis ${\mathcal{H}_1}$ that minimizes the KL divergence from ${\mathcal{H}_1}$ to ${\mathcal{H}_0}$. Due to the asymmetry of KL divergence, it is still unclear what is the optimal likelihood function under ${\mathcal{H}_1}$ that minimizes the KL divergence from ${\mathcal{H}_0}$ to ${\mathcal{H}_1}$. Surprisingly, in this work we prove that this optimal likelihood function is unachievable.