To predict the result of a measurement requires (1) a model of the system under investigation, and (2) a physical theory linking the parameters of the model to the parameters being measured. This prediction of observations, given the values of the parameters defining the model constitutes the "normal problem," or, in the jargon of inverse problem theory, the forward problem. The "inverse problem" consists in using the results of actual observations to infer the values of the parameters characterizing the system under investigation.

Inverse problems may be difficult to solve for at least two different reasons: (1) different values of the model parameters may be consistent with the data (knowing the height of the main-mast is not sufficient for calculating the age of the captain), and (2) discovering the values of the model parameters may require the exploration of a huge parameter space (finding a needle in a 100-dimensional haystack is difficult).

Although most of the formulations of inverse problems proceed directly to the setting of an optimization problem, it is actually best to start using a probabilistic formulation, the optimization formulation then appearing as a by-product.

Consider an example in which two planes make two estimations of the geographical coordinates of a shipwrecked man. Let the probabilities be represented by the two volumetric probabilities and . The volumetric probability that combines these two pieces of information is.

The model parameters are coordinates on the model parameter manifold M while the observable parameters are coordinates over the observable parameter manifold O. When the points on M are denoted M, M^', ... and the points on O are denoted O, O^', ..., the relation between the model parameters an the observable parameters is written M|->O=O(M). The three basic elements of a typical inverse problem are: 1. Some a priori information on the model parameters, represented by a volumetric probability rho_(prior)(M) defined over M. 2. Some experimental information obtained on the observable parameters, represented by a volumetric probability sigma_(obs)(O) defined over O.