= Combined score and pathwise gradient estimator
{title2=$\nabla_\lambda\mathbb E_{s_\lambda}W_\lambda=\mathbb E_{s_\lambda}[W_\lambda\nabla_\lambda\log s_\lambda+\nabla_\lambda W_\lambda]$}
Under fixed-support regularity permitting <differentiation under the integral sign>, apply the <product rule> to $\int s_\lambda W_\lambda$. The first derivative term is $\int W_\lambda\nabla s_\lambda=\mathbb E[W_\lambda\nabla\log s_\lambda]$ and the second is $\mathbb E[\nabla W_\lambda]$. This proves the displayed identity. For an <evidence lower bound>, take $W_\lambda=\log p(t_\lambda(\varepsilon),x)-\log q_\lambda(t_\lambda(\varepsilon))$, holding noise $\varepsilon$ fixed in the inner derivatives. Parameter-independent continuous noise contributes a <reparameterization gradient>; untransformed discrete choices contribute a <score-function gradient estimator>. A mixed base law cannot be called parameter independent if its discrete factors retain variational parameters.<Gradient Estimation Using Stochastic Computation Graphs by John Schulman and others>[Section 3.1, Theorem 1; Appendix A]{ref}
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