Innovation representation of stochastic processes with application to causal inference

Typically, real-world stochastic processes are not easy to analyze. In this paper, we study the representation of different stochastic process as a memoryless innovation process triggering a dynamic system. We show that such a representation is always feasible for innovation processes taking values over a continuous set. However, the problem becomes more challenging when the alphabet size of the innovation is finite. In this case, we introduce both lossless and lossy frameworks, and provide closed-form solutions and practical algorithmic methods. In addition, we discuss the properties and uniqueness of our suggested approach. Finally, we show that the innovation representation problem has many applications. We focus our attention on entropic causal inference, which has recently demonstrated promising performance, compared to alternative methods.