= Statistical inference
{wiki}
<Statistical inference> uses observed data and a <statistical model> to learn about unknown features of the process that generated them. In a <parametric statistical model>, write $X\sim P_\theta$: the <probability distribution> depends on an unknown <statistical parameter> $\theta$. An <estimand> specifies the target, such as a <mean> or a difference of <means>. A <statistic> is a function of the observations; an <estimator> is a <statistic> used to estimate the target. Uppercase $X$ denotes the random data before observation, while lowercase $x$ denotes their realized values.
Three questions lead to different outputs. <Statistical estimation> asks which target value the observations suggest. A <confidence interval> asks for a range produced by a procedure with calibrated <coverage probability>. A <hypothesis test> asks whether the observations depart sufficiently from a specified <null hypothesis> to enter its <rejection region>. These outputs complement one another: an estimated <effect size> describes magnitude, while a <standard error> describes the <sampling distribution> of an <estimator>. A <p-value> describes a tail <probability> under the <null hypothesis>.
The appeal of <statistical inference> is that it turns variable observations into reproducible uncertainty statements. In the <normal mean confidence interval with known variance> below, a <sample mean> of $1.2$ has <standard error> $0.5$: both the estimated shift and its precision matter. The calibration depends on the <statistical model>, including <independence> and the specified <probability distribution>. Increasing <sample size> improves precision under those assumptions; it does not repair a systematically misleading sampling procedure. <Bayesian inference> provides another framework, using a <prior distribution> and a <likelihood function> to obtain a <posterior distribution>.
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