Estimating the Directed Information and Testing for Causality
Estimating the Directed Information and Testing for Causality
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DOI：
10.1109/tit.2016.2604842
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发表时间：
2016-11-01
影响因子：
2.5
通讯作者：
Skoularidou, Maria
中科院分区：
文献类型：
--
作者：
Kontoyiannis, Ioannis;Skoularidou, Maria
考虑通过插件（或最大似然）估计器估计两个离散过程{X-n}和{Y-n}之间的有向信息率的问题。当联合过程{(X-n,Y-n)}是给定内存长度的马尔可夫链时，插件估计器表现出渐近高斯分布，并且在适当的条件下以最优速率O(1/v n)收敛；这是第一个被证明可以达到这一速率的估计器。在估计定向信息率的问题和对两个过程之间是否存在因果影响进行假设检验的问题之间建立了重要的联系。在相当一般的条件下，零假设（对应于不存在因果影响）相当于定向信息率等于零的要求。在这种情况下，建立了更精细的结果，表明插件以更快的速率 O(1/n) 收敛，并且它是渐近 chi(2) 分布的。这通过证明该估计量等于上述假设检验的经典似然比统计量（标量倍数）来证明。最后，值得注意的是，这些结果有助于设计是否存在因果影响的实际似然比检验。
The problem of estimating the directed information rate between two discrete processes {X-n} and {Y-n} via the plug-in (or maximum-likelihood) estimator is considered. When the joint process {(X-n, Y-n)} is a Markov chain of a given memory length, the plug-in estimator is shown to be asymptotically Gaussian and to converge at the optimal rate O(1/v n) under appropriate conditions; this is the first estimator that has been shown to achieve this rate. An important connection is drawn between the problem of estimating the directed information rate and that of performing a hypothesis test for the presence of causal influence between the two processes. Under fairly general conditions, the null hypothesis, which corresponds to the absence of causal influence, is equivalent to the requirement that the directed information rate be equal to zero. In that case, a finer result is established, showing that the plug-in converges at the faster rate O(1/n) and that it is asymptotically chi(2)-distributed. This is proved by showing that this estimator is equal to (a scalar multiple of) the classical likelihood ratio statistic for the above hypothesis test. Finally, it is noted that these results facilitate the design of an actual likelihood ratio test for the presence or absence of causal influence.