TV over Bernoulli products: the small parameter regime

We study the total variation distance (TV) between two $n$-fold Bernoulli product measures parametrized by $\vec p=(p_1,\ldots,p_n)$ and $\vec q=(q_1,\ldots,q_n)$, respectively, in the \emph{tiny} and \emph{small} regimes. In the tiny regime, we have $p_i,q_i\lesssim 1/n^2$, and in the small regime, $p_i,q_i\lesssim 1/n$. We discover that in the tiny regime, the TV distance behaves as $\|\vec p-\vec q\|_1$, while in the small regime, it behaves as $$\sum_{i=1}^n \Big| p_i\prod_{j\neq i}(1-p_j) - q_i\prod_{j\neq i}(1-q_j) \Big|,$$ both up to absolute constants. Along the way we discover some identities of possible independent interest.