Consistency of the predictions: details and theorem proofs

Consistency of the predictions: details and theorem proofs Additional file 2 provides a simple example to show that we need a per level traversal of the DAG, in which levels are defined in the sense of the maximum distance from the root, to obtain consistent predictions that obey the “true path rule”. Indeed looking at the HTD-DAG scores obtained respectively with the minimum and maximum distance from the root (bottom-left of Additional Figure 2 of the additional file 2), we see that only the maximum distance preserves the consistency of the predictions. For instance, focusing on node 5, by traversing the DAG levels according to the minimum distance from the root, we have that the level of node 5 is 1 (ψ(5) = 1) and in this case by applying the HTD rule (eq. 2 of the main manuscript) the flat score ŷ5 = 0.8 is wrongly modified with the HTD ensemble score ȳ5 = 0.7. If we instead traverse the DAG levels according to the maximum distance from the root, we have ψ(5) = 3 and the HTD ensemble score is correctly set to ȳ5 = 0.3. In other words at the end of the HTD, by traversing the levels according to the minimum distance we have ȳ5 = 0.7 > ȳ4 = 0.3. Therefore a child node has a score larger the score of its parent and the true path rule is not preserved. On the contrary by traversing the levels according to the maximum distance we achieve ȳ5 = 0.3 ≤ ȳ4 = 0.3 and the true path rule consistency is assured. This is due to the fact that by adopting the minimum distance when we visit node 5, node 4 has not just been visited, and hence the value 0.4 has not been transmitted by node 2 to node 4; on the contrary if we visit the DAG according to the maximum distance all the ancestors of node 5 (including node 4) have just been visited and the score 0.4 is correctly transmitted to node 5 along the path 2→ 4→ 5. To prove the consistency of the predictions of the HTD-DAG algorithm, we first introduce a property of the level function ψ. We recall here the definition of ψ, just given in the main paper. If p(r, i) represents a path from the root node r and a node i ∈ V , l (p(r, i)) the length of p(r, i), L = {0, 1, . . . , ξ} the set of observed levels, with ξ the maximum node level, then ψ : V −→ L is a level function which assigns each node i ∈ V to its level ψ(i):