论文标题
信息流的上限，从孩子到树上的父节点
An Upper Bound of the Information Flow From Children to Parent Node on Trees
论文作者
论文摘要
假设每个传输都通过嘈杂的通道发生，我们考虑从树的根部向其叶子透射的状态。观察到叶子处的状态，而在更深的节点处，我们可以计算观察到每个状态的可能性。从这个意义上讲，信息从子节点流向父节点。在这里，我们找到了这种与父母的信息流的上限。为此，首先，我们介绍了一种新的信息量度，即记忆向量，其规范量化了所有州是否具有相同的可能性。然后，我们发现条件使得可以通过子节点的规范之和在父节点处的内存向量的规范进行线性界定。我们还描述了鉴于在叶子处进行观察到的观察结果的重建问题的重建问题。我们推断出足够的条件，在该条件下，没有观察到的叶子不能坚信根部的原始状态，假设从根到叶子的水平数量很大。
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.
