Consider a set of random variables:
$$ \vec{X} = X_1,\ldots,X_n$$
and a function $f(\vec{X})$ used to construct the Doob martingale with elements
$$ B_i = E_{X_{i+1},\ldots,X_{n}}[f(\vec{X}) | X_1,\ldots,X_i] $$
In all uses of the Doob martingale that I've been able to find, the joint distribution $P$ of the random variables seems to be both known and tractable, so that $|B_i - B_{i-1}|$ can be bounded and one can apply Azuma's inequality.
What happens when the distribution $P$ is not known exactly or is intractable to compute? In particular, is there a known way to obtain reasonable concentration guarantees using some other distribution $Q$ that approximates or bounds the true distribution?
I see that this is possible if we can bound the expectations $B_i$ to within an additive or multiplicative constant, so I'm ideally looking for something less restrictive than this—perhaps based on tail bounds on the approximate distribution's error, or upper bounding $D_{KL}(P||Q)$, or having the first and second moments match, or some other condition I might realistically achieve given access to sample realizations of $\vec{X}$ but without knowing the true distribution a priori.
Also, apologies if I'm missing something simple here. I'm new to stochastic processes and any help would be appreciated.