Probabilistic model

A statistical model is a class of mathematical model, which embodies a set of assumptions concerning the generation of some sample data, and similar data from a larger population. A statistical model represents, often in considerably idealized form, the data-generating process.

The assumptions embodied by a statistical model describe a set of probability distributions, some of which are assumed to adequately approximate the distribution from which a particular data set is sampled. The probability distributions inherent in statistical models are what distinguishes statistical models from other, non-statistical, mathematical models.

A statistical model is usually specified by mathematical equations that relate one or more random variables and possibly other non-random variables.

All statistical hypothesis tests and all statistical estimators are derived from statistical models. More generally, statistical models are part of the foundation of statistical inference.

In mathematical terms, a statistical model is usually thought of as a pair (${\displaystyle S,{\mathcal {P}}}$), where ${\displaystyle S}$ is the set of possible observations, i.e. the sample space, and ${\displaystyle {\mathcal {P}}}$ is a set of probability distributions on ${\displaystyle S}$.

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