We present novel data-processing inequalities relating the mutual information and the directed information in systems with feedback. The internal deterministic blocks within such systems are restricted only to be causal mappings, but are allowed to be non-linear and time varying, and randomized by their own external random input, can yield any stochastic mapping. These randomized blocks can for example represent source encoders, decoders, or even communication channels. Moreover, the involved signals can be arbitrarily distributed. Our first main result relates mutual and directed information and can be interpreted as a law of conservation of information flow. Our second main result is a pair of data-processing inequalities (one the conditional version of the other) between nested pairs of random sequences entirely within the closed loop. Our third main result introduces and characterizes the notion of in-the-loop (ITL) transmission rate for channel coding scenarios in which the messages are internal to the loop. Interestingly, in this case the conventional notions of transmission rate associated with the entropy of the messages and of channel capacity based on maximizing the mutual information between the messages and the output turn out to be inadequate. Instead, as we show, the ITL transmission rate is the unique notion of rate for which a channel code attains zero error probability if and only if such an ITL rate does not exceed the corresponding directed information rate from messages to decoded messages. We apply our data-processing inequalities to show that the supremum of achievable (in the usual channel coding sense) ITL transmission rates is upper bounded by the supremum of the directed information rate across the communication channel. Moreover, we present an example in which this upper bound is attained. Finally, we further illustrate the applicability of our results by discussing how they make possible the generalization of two fundamental inequalities known in networked control literature.

中文翻译：

---

有反馈系统的定向数据处理不等式

我们提出了新颖的数据处理不等式，该系统在具有反馈的系统中将相互信息和定向信息联系在一起。此类系统中的内部确定性块仅被限制为因果映射，但被允许为非线性且随时间变化，并通过其自身的外部随机输入进行随机化，可以产生任何随机映射。这些随机块可以例如表示源编码器，解码器或什至通信信道。而且，所涉及的信号可以被任意地分配。我们的第一个主要结果涉及相互和直接的信息，并且可以解释为信息流守恒定律。我们的第二个主要结果是完全在闭环内的嵌套随机序列对之间的一对数据处理不等式（一个是另一个的条件版本）。我们的第三个主要结果介绍并描述了消息在循环内部的信道编码方案的循环中（ITL）传输速率的概念。有趣的是，在这种情况下，与消息的熵有关的传输速率和基于最大化消息与输出之间的互信息的信道容量的传统概念被证明是不适当的。相反，如我们所示，ITL传输速率是唯一的速率概念，当且仅当这样的ITL速率不超过从消息到解码消息的相应定向信息速率时，信道代码的错误概率才为零。我们应用数据处理不等式来表明，可实现的（在通常的信道编码意义上）ITL传输速率的最高上限是整个通信信道上定向信息速率的最高上限。此外，我们提供了一个达到此上限的示例。最后，我们通过讨论结果如何推广网络控制文献中已知的两个基本不等式，进一步说明了我们的结果的适用性。