If I allow a circuit family (say, poly size, polylog depth) poly($n$) bits of randomized advice, then I can ask if its output samples from certain distributions or not. However I don't know what the criterion for "solving a sampling problem" in this case should be.
To elaborate: Given a BPP machine $M$, we say $M$ solves the sampling problem $S = (D_x)_{x \in \{0,1\}^*}$ (each $D_x$ a distribution over poly($|x|$) bits) iff for all $x \in \{0,1\}^*$ and for all $\epsilon > 0$ the input $\langle x, 0^{1/\epsilon} \rangle$ implies $$ \|M(x, 0^{1/\epsilon}) - D_x\| \leq \epsilon, $$ where $\|\cdot\|$ is usually the total variation distance. I'm curious if there is an agreed-upon extension of this definition to circuits with randomized advice.
One (flawed) definition is: a circuit family $\{C_{n + r}\}_{n \in \mathbb{N}}$ with $r = \text{poly}(n)$ bits of randomized advice solves the sampling problem $S$ iff for all $x \in \{0,1\}^*$ and for all $\epsilon > 0$ the input $\langle x, 0^{1/\epsilon}, r_1, \dots, r_{\text{poly}(|x|)}\rangle$ implies $$ \|C(x, 0^{1/\epsilon}, r_1, \dots, r_{\text{poly}(|x|)}) - D_x\| \leq \epsilon. $$ However, a definition like this seems pointless because I can always just absorb the $0^{1/\epsilon}$ into the advice string. And plus it's another matter how the number of output bits is influenced by $\epsilon$. Any ideas?