A Statistical Perspective on Inverse and Inverse Regression Problems

Inverse problems, where in broad sense the task is to learn from the noisy response about some unknown function, usually represented as the argument of some known functional form, has received wide attention in the general scientific disciplines. How- ever, in mainstream statistics such inverse problem paradigm does not seem to be as popular. In this article we provide a brief overview of such problems from a statistical, particularly Bayesian, perspective. We also compare and contrast the above class of problems with the perhaps more statistically familiar inverse regression problems, arguing that this class of problems contains the traditional class of inverse problems. In course of our review we point out that the statistical literature is very scarce with respect to both the inverse paradigms, and substantial research work is still necessary to develop the fields.