= Statistical model
{wiki}
A <statistical model> is a specified family $\mathcal P$ of <probability distributions> for the observations. A <statistical parameter> often indexes it as $\mathcal P=\{P_\theta:\theta\in\Theta\}$, where $\Theta$ is the <parameter space>. The choice states which patterns of variation are considered possible and supplies a basis for <statistical inference>; it does not guarantee that any member exactly describes the data-generating mechanism.
For example, suppose $n$ binary observations are <independent and identically distributed> with a common unknown success <probability> $p$. The <Bernoulli distribution> model has $\Theta=[0,1]$ and <likelihood function>
$$
L(p;x)=p^{\sum_i x_i}(1-p)^{n-\sum_i x_i}.
$$
With $n=10$ and seven successes, this becomes $p^7(1-p)^3$ and is maximized at $p=0.7$. The binary observation type alone does not supply <independence> or a common $p$: those are additional model assumptions. A <regression model> can instead let the response law depend on observed <covariates>, while a <mixed model> represents a specified grouping structure through <random effects>.
Back to article page
