Every measurement is the end of a chain of cause and effect. The forward problem walks that chain forwards: from a complete description of the world — the model m — predict what an instrument will record — the data d. We write the forward map as

The inverse problem is the scientist's real task: you have the data d and want the model m that produced it. Reconstruct the body from its X-ray shadows, the seabed from sonar echoes, the sharp image from the blurred one.

Why running the chain backwards is hard

Forward problems are usually well-behaved: plug in m, turn the crank, out comes d. Inverse problems fight back in three ways, all of which we will meet:

• Non-uniqueness. Different models can produce identical data — the data does not pin down a single answer.
• Non-existence. Noisy data may match no model exactly, so we must settle for a best fit.
• Instability. A whisker of noise in d can swing the recovered m wildly — the defining headache of the subject.

The simplest taste of non-uniqueness: a forward map that is not one-to-one. If d = m^2, then a single measured d has two possible causes, m = \pm\sqrt{d} — the data alone cannot choose between them.

One datum, two causes

The curve is a forward map d = G(m) = m^2; the horizontal line is a measured data value d. Slide it: wherever the line crosses the curve is a model consistent with the data — and it almost always crosses twice. That ambiguity, in far subtler forms, is what regularization and priors will later resolve.