Conditional Probability Given Multiple Independent Events

Question

I am interested in finding the conditional probability $P(A|E_1,E_2,…,E_n)$ where the $E_i$ are mutually independent events. I know only $P(A)$ and $P(A|E_i)$.

Is this possible? If so, how? If not, what information is missing?

Answer 1

$\begin{align} \mathsf P(A\mid \bigcap_i E_i) & = \frac{\mathsf P(\bigcap_i E_i\mid A)\cdot\mathsf P(A)}{P(\bigcap_i E_i)} \\[1ex] & = \frac{\mathsf P(\bigcap_i E_i\mid A)\cdot \mathsf P(A)}{\prod_i \mathsf P(E_i)} \end{align}$

To proceed further you require a guarantee of conditional independence.   That is: $\mathsf P(\bigcap_i E_i\mid A) = \prod_i\mathsf P(E_i\mid A)$

However, mutually independent events $\{E_1, E_2, \ldots E_n\}$ are not necessarily mutually, conditionally independent given event $A$.