Forward and Inverse Problems#

It is not difficult to make an observation — we do it all the time.

Much of scientific work involves contriving reliable observations. Some interesting phenomenon deviates from what is typically observed and this provokes a curiosity that leads us to seek an explanation.

What makes science special is not only its capacity to rigorously observe phenomena but also to work backwards and develop explanations for those phenomena.

Forward Problems#

Mario Bunge used a notation like the following to signify a question of the form "what y does x cause?":

$$x \to \ ?y$$

This is a forward problem.

Bunge gives the example of medical prognosis, which proceeds from an observation of some condition to a forecast of the symptom development:

$$condition \to symptoms$$

If a flesh wound is identified it is relatively trivial for a trained doctor to determine what the consequences will be if the wound is not appropriately treated.

Inverse Problems#

The corresponding inverse problem is "what x is the cause of y?"

$$?x \to \ y$$

Bunge gives medical diagnosis as a concrete example:

$$symptoms \to condition$$

Anyone who has attempted self-diagnosis understands the difficulty here. There can be many symptoms (e.g., cough, sweating, itchiness), of varying degrees of severity, which often overlap as possible symptoms for many diseases.

Forward and Inverse Problems

• Forward: what observation results from a given set of conditions
• Inverse: what are the conditions that result in a given observation

Solving an Inverse Problem

May require its conversion into a tractable set of forward problems.

If we want to work on something important or revolutionary we should seek out new inverse problems. At the same time we need to be wary of metaphorical thinking and other blockers that can needlessly multiply the inverse problems that we need to solve.