For discrete probability distributions $P$ and $Q$ defined on the same sample space, $\mathcal{X}$, the Kullback-Leibler divergence is defined as $$ D_{\mathrm{KL}}(P \| Q)=\sum_{x \in \mathcal{X}} P(x) \log \frac{P(x)}{Q(x)} $$

Is there a notion of a higher Kullback-Leibler divergence? In other words, has the following divergence been studied:

$$ D^k_{\mathrm{KL}}(P \| Q)=\sum_{x \in \mathcal{X}} P(x) \left(\log \frac{P(x)}{Q(x)}\right)^k. $$

This is with a view to generalizing Stam's inequality.

For discrete probability distributions $P$ and $Q$ defined on the same sample space, $\mathcal{X}$, the Kullback-Leibler divergence is defined as $$ D_{\mathrm{KL}}(P \parallel Q)=\sum_{x \in \mathcal{X}} P(x) \log \frac{P(x)}{Q(x)} $$

Is there a notion of a higher Kullback-Leibler divergence? In other words, has the following divergence been studied:

$$ D^k_{\mathrm{KL}}(P \parallel Q)=\sum_{x \in \mathcal{X}} P(x) \left(\log \frac{P(x)}{Q(x)}\right)^k. $$

This is with a view to generalizing Stam's inequality.

For discrete probability distributions $P$ and $Q$ defined on the same sample space, $\mathcal{X}$, the Kullback-Leibler divergence is defined as $$ D_{\mathrm{KL}}(P \| Q)=\sum_{x \in \mathcal{X}} P(x) \log \left(\frac{P(x)}{Q(x)}\right) $$

Is there a notion of a higher Kullback-Leibler divergence? In other words, has the following divergence been studied:

$$ D^k_{\mathrm{KL}}(P \| Q)=\sum_{x \in \mathcal{X}} P(x) \log \left(\frac{P(x)}{Q(x)}\right)^k. $$

This is with a view to generalizing Stam's inequality.

For discrete probability distributions $P$ and $Q$ defined on the same sample space, $\mathcal{X}$, the Kullback-Leibler divergence is defined as $$ D_{\mathrm{KL}}(P \| Q)=\sum_{x \in \mathcal{X}} P(x) \log \frac{P(x)}{Q(x)} $$

Is there a notion of a higher Kullback-Leibler divergence? In other words, has the following divergence been studied:

$$ D^k_{\mathrm{KL}}(P \| Q)=\sum_{x \in \mathcal{X}} P(x) \left(\log \frac{P(x)}{Q(x)}\right)^k. $$

This is with a view to generalizing Stam's inequality.