Inverse Problem

An inverse problem is the task of inferring causes from observed effects — reconstructing the hidden structure or parameters that produced a measurable outcome. Where a forward problem predicts what a known system will do, an inverse problem asks what system must have done this. The task is mathematically ill-posed: multiple distinct causes can produce the same effect, and small errors in measurement can amplify into catastrophic errors in reconstruction. Inverse problems appear wherever observation must be interpreted: perception, medical imaging, geophysics, and machine learning. The techniques developed to solve them — regularization, Bayesian inference, prior constraints — are themselves theories about what kinds of causes are most probable, and therefore encode assumptions about the structure of the world. An inverse problem without a prior is not unsolved; it is undefined.