---
tag: JZ6R7J36K5
kind: definition
name: Statistical model
msc: [62A01]
defines: K3AMMSY8DD
uses: [04WSW8T3SY, 0VGMEBVW1M]
sources:
  - url: https://en.wikipedia.org/wiki/Statistical_model
    locator: Formal definition
---

## Definition

Let $S$ be a set, the sample space, with a [[sigma-algebra|04WSW8T3SY]] $\mathcal{S}$ on $S$.
A *statistical model* on $(S, \mathcal{S})$ is a pair $(S, \mathcal{P})$, where $\mathcal{P}$ is a set of [[probability measures|0VGMEBVW1M]] on $(S, \mathcal{S})$.
The model is often written with an index set $\Theta$ as $\mathcal{P} = \set{P_\theta : \theta \in \Theta}$, and $\Theta$ is the parameter space.

## Remarks

Example: $S = \set{0, 1}^n$ with all subsets measurable, and $P_p(x) = p^{k}(1 - p)^{n - k}$ where $k = x_1 + \dots + x_n$, for $p \in [0, 1]$.
Each $P_p$ is a probability measure, because the binomial theorem gives $\sum_{x} P_p(x) = (p + 1 - p)^n = 1$.
This is the model of $n$ independent tosses of a coin with unknown chance $p$ of heads.
Non-example: the set of measures $\set{c \cdot P_{1/2} : c > 0}$ is not a statistical model, because for $c \ne 1$ the measure $c \cdot P_{1/2}$ has total mass $c$, so it is not a probability measure.
The data are modelled as a value $x$ of a random element $X$ with law in $\mathcal{P}$.