Computing $p(d|e_1,e_2)$ from $p(d|e_1)$ and $p(d|e_2)$ I know the probability $p(d|e_1)$ and $p(d|e_2)$, how to compute the $p(d|e_1, e_2)$ if $e_1$ and $e_2$ are independent?
What if $e_1$ and $e_2$ is dependent, how to compute?
 A: Suppose we have probability space $(\Omega, \mathcal{F}, P)$. 
Let events $d,e_1,e_2$ are from $\mathcal{F}$, such that $e_1$ and $e_2$ are independent, $P(e_1\cap e_2) \neq 0$. Therefore $$P(d|e_1 \cap e_2) = \frac{P(d\cap e_1\cap e_2)}{P(e_1\cap e_2)} = \frac{P(d\cap e1)P(d\cap e2)}{P(e_1)P(e_2)} = P(d|e_1)P(d|e_2)$$
As far as I understand your problem correctly, because one would suppose $e_1$, $e_2$ to be random variables.
A: I understand in this context dependency as linear dependency so $e_1\not=e_2$ and $e_2 = k_1\cdot e_1$ or $e_1 = k_2\cdot e_2$. A third case of dependency that I put apart is $e_1=e_2$.
Independent: $\ p(d\mid e_1,e_2) = p(d\mid e_1) \cdot p(d\mid e_2)$, because
$d$ must divide both $e_1$ and $e_2$.
Dependent: $\ p(d\mid e_1,e_2) = \ p(d\mid e_1)\cdot p(e_1\mid e_2) + \ p(d\mid e_2)\cdot p(e_2\mid e_1)$, because $d$ must divide $e_1$ and $e_1$ must divide $e_2$ or $d$ must divide $e_2$ and $e_2$ must divide $e_1$.
If $e_1 = e_2$, $\ p(d\mid e_1) = p(d\mid e_2) = p(d\mid e_1,e_2)$, they have the same value so the probability is unique, an special case of dependency.
