Large-scale signal detection: A unified perspective

There is an overwhelmingly large literature and algorithms already available on "large-scale inference problems" based on different modeling techniques and cultures. Our primary goal in this article is not to add one more new methodology to the existing toolbox but instead (i) to clarify the mystery how these different simultaneous inference methods are connected, (ii) to provide an alternative more intuitive derivation of the formulas that leads to simpler expressions in order (iii) to develop a unified algorithm for practitioners. A detailed discussion on representation, estimation, inference, and model selection is given. Applications to a variety of real and simulated datasets show promise. We end with several future research directions.