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The discrete base factors in the displayed construction inherit their variational parameters. Accordingly write $s_{\lambda }(\epsilon )$, not a parameter-independent $s(\epsilon )$, for the joint base probability mass function or probability density function. The continuous noise factors can be parameter independent while the discrete factors are $\lambda$ dependent. Calling the entire base law parameterless while retaining its nonzero score function would be inconsistent.

Let $t_{\lambda }$ push the base law forward to $q_{\lambda }$, and defineUse a fixed base measure $\nu$, with counting measure for discrete coordinates and Lebesgue measure for continuous coordinates. Assume locally parameter-independent support, differentiability and an integrable domination permitting differentiation under the integral sign. A valid reparameterization gradient givesApply the product rule before simplifying:This is the required combined score and pathwise gradient estimator. A sample from $s_{\lambda }$ and the integrand on the last line give an unbiased stochastic gradient under the stated regularity conditions. Its first term is a score-function gradient estimator contribution; the remaining terms differentiate the sampled objective along the continuous transformation. The same separation appears in the general stochastic computation graph construction.^[ref]

All derivatives inside the expectation hold $\epsilon$ fixed. They still include both explicit parameter dependence and dependence through $t_{\lambda }$. For a continuous coordinate $z=t_{\lambda }(\epsilon )$,The model joint density $p(z,x)$ is fixed as $\lambda$ changes, but its composition with $t_{\lambda }$ need not be constant. Discrete coordinates are not differentiated as real-valued paths; their changing probabilities enter through the score function. Moving supports or parameter-dependent branch boundaries require additional care and are not justified merely by formally interchanging a derivative and an integral.