When the joint probability for events $A_1,A_2,A_3,A_4$ is desired, the chain rule is used as follows:$$ P(A_4, A_3, A_2, A_1) = \mathrm P(A_4 \mid A_3, A_2, A_1)\cdot \mathrm P(A_3 \mid A_2, A_1)\cdot \mathrm P(A_2 \mid A_1)\cdot \mathrm P(A_1)$$ However, I have a problem in which I need the probability of the joint event and calculation of the conditional probabilities is not possible. Is there any approach that I could find an upper or lower bound for this probability? or any sort of approximation, no matter how elementary it is. I only have the probabilities of events alone, i.e. $P(A_4), P(A_3), P(A_2), P(A_1)$.
Without more information upperbound is: $$\min(\Pr(A_1),\Pr(A_2),\Pr(A_3),\Pr(A_4))$$
This is based on $A_1\cap A_2\cap A_3\cap A_4\subseteq A_i$ for $i=1,2,3,4$ together with the fact that $=$ instead of $\subseteq$ is not excluded here.
If e.g. $\Pr(A_1)+\Pr(A_2)\leq1$ then it is not excluded that $A_1\cap A_2=\varnothing$ so in such cases $0$ serves as lower bound. Not quite useful of course. For a useful lower bound more information concerning the events is needed.