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If the condition $$ \frac{d}{d\theta}\mathbb{E}\Big[\frac{\partial}{\partial\theta}logp_\theta(x)\Big] = \int_{x} \frac{\partial}{\partial\theta}\Big[\frac{\partial}{\partial\theta}logp_\theta(x)p_\theta(x)\Big]dx $$ holds then $\mathbb{E}\Big[\Big(\frac{\partial}{\partial\theta}logp_{\theta}(x)\Big)^2\Big] = -\mathbb{E}\Big[\frac{\partial^2}{\partial\theta^2}logp_{\theta}(x)\Big]$. What is a parallel condition for PMFs?

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