Let $X = (X_1, X_2, \ldots , X_n)$ be an $n$-dimensional random variable, where each $X_i$ is a random variable on finite discrete set $S$. In addition, $X_i$ are independent of each other (but not identically distributed). That is, if we denote $X \sim \mathcal{D}$ and $X_i \sim \mathcal{D}_i$, then $\mathcal{D}$ is just the product of $\{ \mathcal{D}_i \}$.

Similarly, let $Y = (Y_1, Y_2, \ldots , Y_n) \sim \mathcal{D}'$ and $Y_i \sim \mathcal{D}'_i$ independently on $S$ (the same $S$ as $X_i$'s). Then $\mathcal{D}'$ is the product of $\{ \mathcal{D}'_i \}$. $X$ and $Y$ are independent.

Now I don't know $\mathcal{D}$ and $\mathcal{D}'$ precisely but I can sample from them (so that I can sample from all the marginal distributions $\mathcal{D}_i$ and $\mathcal{D}'_i$ as well). I would like to estimate the total variation between $\mathcal{D}$ and $\mathcal{D}'$.

Let $d = \|\mathcal{D} - \mathcal{D}'\|_{TV}$ and $\hat{d}$ be our estimator for $d$. Let $N$ be the number of samples we need. I hope to get an error-confidence bound of the form $$\Pr[ |\hat{d} - d| \ge \epsilon] \le \delta$$ where $\delta$ is at least polynomially small w.r.t. $n, |S|, N$ and $\epsilon$.

I just found a similar question. However, the sample space of $X$ and $Y$ here is $S^n$, which is exponentially large, making the bound in that post not applicable to this question. Besides, here we have $\mathcal{D}$ and $\mathcal{D}'$ being product distributions, which could probably make things easier.

Thank you.