An Upper Bound of the Information Flow From Children to Parent Node on Trees

We consider the transmission of a state from the root of a tree towards its leaves, assuming that each transmission occurs through a noisy channel. The states at the leaves are observed, while at deeper nodes we can compute the likelihood of each state given the observation. In this sense, information flows from child nodes towards the parent node. Here we find an upper bound of this children-to-parent information flow. To do so, first we introduce a new measure of information, the memory vector, whose norm quantifies whether all states have the same likelihood. Then we find conditions such that the norm of the memory vector at the parent node can be linearly bounded by the sum of norms at the child nodes. We also describe the reconstruction problem of estimating the ancestral state at the root given the observation at the leaves. We infer sufficient conditions under which the original state at the root cannot be confidently reconstructed using the observed leaves, assuming that the number of levels from the root to the leaves is large.