A Statistical Perspective on Inverse and Inverse Regression Problems

TL;DR

This paper reviews inverse and inverse regression problems from a Bayesian statistical perspective, highlighting their differences, similarities, and the need for further research in these areas.

Contribution

It provides a comparative overview of inverse and inverse regression problems within a Bayesian framework, emphasizing the scarcity of statistical literature and future research directions.

Findings

Inverse problems are less common in mainstream statistics.

Inverse regression problems encompass traditional inverse problems.

Significant research gaps exist in the statistical treatment of these problems.

Abstract

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.