The problem of portfolio credit risk modeling is investigated using background filtrations and an embedding via an \(\mathbb{H}\)-hypothesis. The authors focus on the top-down approach. It is shown that, as opposed to the bottom-up approach, in a top-down approach default contagion is compatible with the background filtration modeling approach. This is because in a top-down approach one can exploit the fact that the event arrival times are ordered. Two \(\mathbb{H}\)-type embedding assumptions are introduced: successive property and one-step property and it is shown that in a top-down portfolio credit risk modeling the two assumptions actually are equivalent. A canonical loss process is constructed such that the successive \(\mathbb{H}\)-property holds. It is proved that under weak regularity conditions the assumption of \(\mathbb{H}\)-property is actually equivalent to the existence of a canonically constructed loss process. The conditional Markov model is treated as a special case in which the complete conditional transition probabilities can be computed in closed form.