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  "body": "\n## Definition\n\nA *Bayesian hierarchical model* (with two levels of prior) consists of the following.\n\n1. A [[parametric model|Q0DXKAQXE1]] for the data: given a parameter $\\theta \\in \\Theta$, the data $x$ have density $f(x \\mid \\theta)$.\n2. A [[prior|00NBBSXRQS]] for $\\theta$ that depends on a hyperparameter $\\varphi \\in \\Phi$: given $\\varphi$, the parameter $\\theta$ has density $\\pi(\\theta \\mid \\varphi)$.\n3. A hyperprior: a probability distribution on $\\Phi$ with density $\\pi(\\varphi)$.\n\nThe joint density of $(x, \\theta, \\varphi)$ is $f(x \\mid \\theta)\\,\\pi(\\theta \\mid \\varphi)\\,\\pi(\\varphi)$, and the [[posterior|0KJ7ZS8AJA]] of $(\\theta, \\varphi)$ given $x$ is proportional to it, by [[Bayes' theorem|0NM6E9VW04]].\n\n## Remarks\n\nModels with more levels add further hyperpriors in the same way.\nExample: data in $J$ groups with $X_{ij} \\mid \\theta_j \\sim \\mathcal{N}(\\theta_j, \\sigma^2)$, group means $\\theta_j \\mid \\mu, \\tau \\sim \\mathcal{N}(\\mu, \\tau^2)$ i.i.d., and a hyperprior on $(\\mu, \\tau)$.\nAll three clauses hold, with $\\theta = (\\theta_1, \\dots, \\theta_J)$ and $\\varphi = (\\mu, \\tau)$.\nGiven $\\mu$ and $\\tau$, the posterior mean of each $\\theta_j$ lies between the sample mean of group $j$ and the overall level $\\mu$.\nNon-example: the same model with $\\mu$ and $\\tau$ fixed numbers has no hyperprior, so clause 3 fails; it is an ordinary one-level Bayesian model.\nIntegrating out $\\theta$ gives the marginal $\\int f(x \\mid \\theta)\\pi(\\theta \\mid \\varphi)\\,d\\theta$; estimating $\\varphi$ from it instead of using a hyperprior is the empirical Bayes approach.\n",
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