= Solution
Parameterize the initial product <normal distribution> by $\eta=(m,\rho)$, using
$$
X^{(0)}_\eta=m+\operatorname{diag}(e^{\rho_1},\ldots,e^{\rho_p})\varepsilon,
\qquad \varepsilon\sim N(0,I_p).
$$
Run the given recursion with independent <random variables> with <uniform distributions> to obtain $X_\eta=F_t(X^{(0)}_\eta,Z_{1:t})$. Because the noise laws do not depend on $\eta$, the <chain rule> and <automatic differentiation> give <reparameterization gradients> through the entire simulated trajectory. The variational objective is $J(\eta)=\operatorname{KL}(q_\eta\Vert\pi)$, where $q_\eta=K^t\nu_\eta$ and $\pi=\mu(\cdot\mid y)$.
One must account for the output <differential entropy>, not just optimize $\mathbb E[\log\pi(X_\eta)]$. Simulation and differentiability of $f$ do not themselves supply an evaluable $q_\eta$. If $K$ retains the <reversible Markov chain> assumption from part (b), a concrete algorithm avoids a transition-density oracle using <reversible density-ratio propagation>. Write $\pi=h/Z_\pi$, with $h$ an evaluable unnormalized <posterior density>, and define
$$
A_\eta(x)=\mathbb E_{B\sim K^t(x,\cdot)}\left[\frac{\nu_\eta(B)}{h(B)}\right].
$$
<Detailed balance> implies
$$
q_\eta(x)=h(x)A_\eta(x),\qquad
\boxed{J(\eta)=\log Z_\pi+\mathbb E[\log A_\eta(X_\eta)].}
$$
For example, integrate $\nu_\eta(b)K^t(b,dx)$ and replace $h(b)K^t(b,dx)$ by $h(x)K^t(x,db)$ to obtain the first identity. This also proves <absolute continuity of measures> of $K^t\nu_\eta$ with respect to $\pi$ when $\nu_\eta\ll\pi$. An everywhere finite differentiable <log-posterior> has positive density, so an ordinary product <normal distribution> has this domination.
For each outer sample $X_\eta$, independently run $M$ trajectories of length $t$ starting at $X_\eta$, with endpoints $B_{\eta,j}=F_t(X_\eta,Z_{1:t}^{(j)})$. These are reverse trajectories because of <detailed balance>; the same recursion simulates them. Estimate $A_\eta(X_\eta)$ by
$$
\widehat A_\eta(X_\eta)=\frac1M\sum_{j=1}^M
\frac{\nu_\eta(B_{\eta,j})}{h(B_{\eta,j})}.
$$
Average $\log\widehat A$ over an outer batch. Compute it stably by a logarithmic sum of exponentials. Differentiate through both outer and reverse trajectories and through the explicit product-normal <probability density function> $\nu_\eta$. Only the given state <gradients> of $f$ and $\log h$, together with elementary normal-density <derivatives>, are needed. Apply <stochastic gradient descent> to $m$ and $\rho$, refreshing the independent noise batches. The logarithmic scale parameters keep all <variances> positive.
This is <nested Monte Carlo> optimization of the exact objective in the limit of adequately increasing inner and outer sample sizes. For fixed $M$, $\log\widehat A$ is generally biased, even though $\widehat A$ is an <unbiased estimator>. Finite logarithmic moments, suitable uniform integrability, and justified <differentiation under the integral sign> are needed for consistency and <gradient> interchange. Under those conditions increasing $M$ and controlling the optimization error yields a justified numerical approximation; nonconvexity prevents a generic guarantee of a global optimum.
There is a real assumption issue: part (c) redefines $K$ without explicitly restating reversibility. The algorithm above uses the reversible interpretation inherited from part (b). For an arbitrary differentiable recursion, one additionally needs access to output densities or a simulatable reverse kernel and regularity of the objective. Differentiability alone is insufficient: $f(x,z)=0$ is differentiable and, with a standard-normal target, makes $K^t\nu$ a point mass for $t\geq1$, so every candidate has infinite <Kullback-Leibler divergence>. Thus a universal finite-objective algorithm cannot be justified solely from the literal differentiability assumptions. An invertible recursion with computable density transformations offers a different sufficient route, but invertibility is not stated here.
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