We investigate the Witsenhausen counterexample in a continuous vector-valued context with a causal encoder and noncausal decoder. Our main result is the optimal single-letter condition that characterizes the set of achievable Witsenhausen power costs and estimation costs, leveraging a modified weak typicality approach. In particular, we accommodate our power analysis to the causal encoder constraint, and provide an improved distortion error analysis for the challenging estimation of the interim state. Interestingly, the idea of dual role of control is explicitly captured by the two auxiliary random variables.

翻译：我们研究了连续向量值场景下具有因果编码器与非因果解码器的Witsenhausen反例。通过改进的弱典型性方法，我们的主要成果是刻画了可达Witsenhausen功率成本与估计成本集合的最优单字母条件。特别地，我们将功率分析适配于因果编码器约束，并针对具有挑战性的中间状态估计问题提出了改进的失真误差分析。有趣的是，控制的双重作用思想被两个辅助随机变量清晰地刻画出来。