I am drawing a mental blank right now:
I have three quadratic matrices $A_{00}$, $A_{01}$ and $A_{11}$ of size $n$. Those represent binary, joint bivariate distributions, i.e.
$$(A_{00})_{i,j} = P(x_i = 0, x_j = 0),$$
$$(A_{01})_{i, j} = P(x_i = 0, x_j = 1)$$ and
$$(A_{11})_{i, j} = P(x_i = 1, x_j = 1).$$
Those joint distributions are mutually dependent, i.e. $(A_{01})_{i, j} \neq (A_{00})_{i, i}(A_{11})_{j, j}$.
Furthermore, I have all conditional probabilities stored in matrices $B_{00}$, $B_{01}$, $B_{10}$ and $B_{11}$ (where $(B_{00})_{i, j} = P(x_i = 0 | x_j = 0)$, etc.). Not sure if those are needed though.
I want to draw a sample from this joint probability distribution, i.e. I want a $n$-sized vector $v$ with binary entries $0$ and $1$.
I am very uncertain if I can simply take the diagonal entries $(A_{00})_{i, i}$ for $i = 1,\ldots , n$ as probabilities for each vector entry $v_i$ independently, since they are not independent. Or is this actually the way to go?
Has anyone an idea how I can draw a sample from this distribution? Thanks a lot in advance!
EDIT: Maybe it helps to know that the underlying probabilitiy structure is a Markov tree.
I have solved it myself. I drew the root value by its initial univariate probability and iteratively conditioned each descendant's distribution. That worked for me!