Let $\mu$ be a probability distribution on $\mathbb R^d$ with "sufficiently regular" density $p$. Let $f:\mathbb R^d \to \mathbb R$ be a "sufficiently regular" function. Finally, for every $t \ge 0$, define $$ s_f(t) := \mu(f^{-1}((-\infty,t])) = \int_{f^{-1}((-\infty,t])}p(x)dx $$
Question. What is the derivative of $s_f$ w.r.t $t$ ?
As examples for the function $f$, one could consider
- Linear: $f(x) = w^\top x - c$, for some unit-vector $w \in \mathbb R^d$ and scalar $c \in \mathbb R$.
- Quadratic: $f_\pm(x) = \pm (1-\|x\|^2)$.
It seems I should be able to solve my problem in principle using Lemma 3.1 of Malliavin Calculus for non Gaussian differentiable measures and surface measures in Hilbert spaces. However, that paper is hard to parse for a non-expert like myself.
Now, suppose $f$ depends "smoothly" on a parameter $\theta$, i.e let $\Theta$ be a nonempty subset of some $\mathbb R^n$ and suppose $F:\mathbb R^d \times \Theta \to \mathbb R$ is "smooth", and for any $\theta \in \Theta$, define $s_\theta := s_{f_\theta}$, where $f_\theta(x):=F(x,\theta)$ for all $x \in \mathbb R^d$.
Question. For every $t \ge 0$, what is the gradient of $\theta \mapsto s_\theta(t)$.
Examples
- Linear: $f(x) = w^\top x - c$, for some unit-vector $w \in \mathbb R^d$ and scalar $c \in \mathbb R$. Here $\theta = (w,c) \in \mathbb R^{d + 1}=:\Theta$.
- Quadratic: $f(x) = \pm (r^2-\|x\|^2)$, here $\theta = r^2 \in (0,\infty) =: \Theta$.