How to calculate joint probabilities from conditional probabilities in a Bayesian Network?

This question is about Bayesian Networks. I want to calculate the probability for certain events to be in a certain state knowing all conditional probabilities.

Consider that I am totally new to Bayesian Networks.

The problem

Consider a Bayesian Network. For all variables $$A_i,B_i \dots Z_i$$ in the network I know the potentials in the form $$P\{ A_i | par(A_i) \}=P\{A_i | B_1 \dots B_n\}$$ for all possible states $$a_i,b_1 \dots b_n$$. These potentials are calculated using subjective probabilities.

Calculating single probabilities

What I know are just potentials. For example, for variable $$A_i$$ I do not know $$P\{A_i = a_i\}$$ but I know $$P\{A_i = a_i | B_1=b_1 \dots B_n=b_n\}$$. Where $$B_1, B_2 \dots B_n$$ are parents of $$A_i$$.

I want to calculate $$P\{A_i = a_i\}$$. Consider that No restrictions are considered so this variable $$A_i$$ can have parents and children. As well as $$B_1 \dots B_n$$.

It does not seem too complicated. Having potentials I can use the total probability law and:

$$P\{ A_i = a_i \} = \sum_{\forall b_1 \dots b_n} P\{ A_i = a_i | B_1=b_1 \dots B_n=b_n \} \cdot P\{B_1=b_1 \dots B_n=b_n \}$$

In the sum, the first part: $$P\{ A_i = a_i | B_1=b_1 \dots B_n=b_n \}$$ can be calculated using potentials.

What about $$P\{B_1=b_1 \dots B_n=b_n \}$$. That is, How can I calculate the probability that some variables have some state contemporarly? Please note that $$B_1 \dots B_n$$ are not one parent of the other, they are on the same level of the network, they are peers, meaning they have no parent relationship. They are parents of $$A_i$$.

The question

Generally speaking, how to calculate the probability that some variables are in some state contemporarly? When they have no parent relationships? Preconditions are: I know the conditional probabilites (potentials).

Do $A_i$ have no children? Do $B$ have no parents?
In general, you can't know $P(B_i)$ without also evaluating $P(A_i)$ because diagnostic evidence from below $A$ can affect $A$, which affects $B$.
I think the efficient solution is to calculate lambda and pi messages, which are the predictive and diagnostic evidence from the parent and child nodes of $A$.