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In the first top-down pass, the second quad-tree is swept downward from the root to the leaves to calculate the prior P(cs) recursively. This prior is initialized in each site of the root of the second quad-tree as is the site of the root of the first quad-tree with the same spatial location as s. The partial posterior marginal has been derived in the aforementioned initialization. Then, the top-down pass travels along the other layers until it reaches the leaves

[1.7]

This formulation encourages parent and children sites to share the same class label, although it does not deterministically enforce this condition. It also implies a model for the statistical relations between labels in consecutive layers. Here, the parent–child transition probability is expressed using the parametric model in Bouman (1991):

[1.8]

where is a hyperparameter of the method. Experiments conducted in Hedhli et al. (2016) indicated limited sensitivity of the result of MPM on multiple cascaded quad-trees to the value of this hyperparameter. We also note that equation [1.8] implicitly yields a stationary model for the considered transitions, i.e. the probability depends on the pair of classes, but not on the specific site location

After the first top-down pass, the prior P(cs) is known on every site s of the second quad-tree.