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we start with $\frac{\partial}{\partial \theta} \mathbb E_{q(\mathbf w\mid\theta)}[f(\mathbf w, \theta)]$ using definition of expectation for continuous case: $\mathbb E[X] = \int xp(x) dx$ for the first equation we get: $\frac{\partial}{\partial \theta} \int f(\mathbf w, \theta)q(\mathbf w \mid \theta) d\mathbf w$ we swap $q(\mathbf w \mid \... 1 You cannot do this:$\mathop{\mathbb{E}_\pi }[r(\tau )\bigtriangledown log \pi (\tau )] \\= \mathop{\mathbb{E}_\pi }[r(\tau )] \,\, \mathop{\mathbb{E}_\pi }[\bigtriangledown log \pi (\tau )]$That is because$r(\tau )$and$\bigtriangledown log \pi (\tau )$are correlated by their dependence on$\tau$. In a simpler concrete example, if your expectation ... 1 When we say that we have$N$points that were "drawn from the probability distribution or probability density", this means that every point$x_n$had the correct probability$p(x_n)$of being sampled from the distribution when we were sampling our$n^{th}\$ point. For example, suppose that we wish to compute/estimate the expected value of a distribution ...