A BN is a complete model
P(x1,x2,…,xn) is an entry in the JPD.
P(x1,x2,…,xn) = P(xn | x1,…,xn-1) P(x1,…,xn-1)
P(x1,x2,…,xn) = P(xn | x1,…,xn-1) P(xn-1 | x1,…,xn-2) P(x1,…,xn-2)
P(x1,x2,…,xn) = P(xn | x1,…,xn-1) P(xn-1 | x1,…,xn-2) ...P(x2 | x1) P(x1)
In the enumeration x1,x2,…,xn for every j, 1£ j £ n the Parents(Xj) Í {X1,…, Xj-1). Then Parents(Xj) d-separate Xj from {X1,…, Xj-1} - Parents(Xj). Due do the implied conditional independencies we can rewrite the product above as:
P(x1,x2,…,xn) = P(xn | Parents(Xn)) P(xn-1 | Parents(Xn-1)) ...P(x1).