Statistical model

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 $\mathcal{P}$.