| created |
2024-08-29 |
| lastmod |
2024-09-02 |
Bayesian parametric inference is an approach to [[statistical inference]]. We consider a family of distributions $\Theta = {P_\theta: \theta\in\Theta}$ where $\Theta$ is finite dimensional and we have data $X = (X_1,\dots,X_n)$ generated by some distribution. We call this parametric inference because the distributions are parameterized by $\Theta$. This is contrast to [[Bayesian nonparametrics]] (and nonparametric methods more generally), which replace $\Theta$ by some infinite dimensional parameter space.
In accordance with [[Bayesian statistics]], we put a prior $\pi$ over the parameter space $\Theta$ and then compute our posterior using $X$. That is, we assume that $\theta\sim\pi$ and then compute the posterior using Bayes' theorem: $\Pr(\theta|X) = \Pr(X|\theta)\pi(\theta)/\Pr(X)$. $\Pr(X)$ is called the "evidence" and computing/estimating this integral is a big research area.