Bayes by backprop unbiased monte carlo gradients

Question

I am currently trying to understand a paper on bayesian neural networks whereby the authors use a bayes by backprop approach to learn weight uncertainties in the neural networks.

I am trying to understand the derivation for proposition 1 in the paper. Particularly, I am not sure how

$$\frac{\partial}{\partial\theta}\int f(\boldsymbol{w},\theta)q(\epsilon)d\epsilon = E_{q(\epsilon)}[\frac{\partial f(w,\theta)}{\partial w} \frac{\partial w}{\partial \theta} + \frac{\partial f(w,\theta)}{\partial \theta}]$$

I am not sure why there is an additional $\frac{\partial f(w,\theta)}{\partial \theta}$ inside the expectation ? since I thought $\frac{\partial f(w,\theta)}{\partial w} \frac{\partial w}{\partial \theta} = \frac{\partial f(w,\theta)}{\partial w}$

Answer 1

Okay I think this is why there is an additional term. $f$ is a function of $\textbf{w}$ and $\theta$. Since $\textbf{w}$ is composed of $\theta$, we apply chain rule to obtain $\frac{\partial f}{\partial w}\frac{\partial w}{\partial \theta}$. Since $f$ is also a function of $\theta$. we also differentiate with respect to $\theta$, yielding $\frac{\partial f}{\partial \theta}$