Statistical model

A statistical model is used in applied statistics. Three basic notions are sufficient to describe all statistical models.
 * 1) We choose a statistical unit which we will observe directly. Multiple observations of the same unit over time is called longitudinal research. Observations of a variety of statistical attributes is a common way of studying relationships among the attributes of a single unit.
 * 2) We may really be interested in a statistical population (or set) of similar units rather than in any statistical unit per se. Survey sampling offers an example of this type of application.
 * 3) Our interest may be focused on a statistical assembly where we examine functional subunits of the statistical unit. Physiology is an example in which we probe the organs which compose the unit. A common model for this type of research is the stimulus-response model.

In mathematical terms, a statistical model is frequently thought of as a parameterized set of probability distributions of the form


 * $$\{ P_\theta | \theta \in \Theta \}.$$

It is assumed that there is a distinct element in the above set from which the observed data are generated. Statistical inference is the art of making statements about which elements of this set are likely to be the true one.

So, for example, Bayes theorem in its raw form may be intractable, but assuming a general model H allows it to become
 * $$P(A | B,H) = \frac{P(B | A,H) P(A | H)}{P(B | H)}$$

which may be easier. Models can also be compared using measures such as Bayes factors or mean square error.