Bayesian hierarchical model

A Bayesian hierarchical model is a statistical model used in Bayesian statistics, in which parameters at one level have a prior distribution controlled by parameters at another level. Those shared hyperparameters can themselves receive a hyperprior. For grouped observations $y_j$, group parameters ${\theta }_j$ and shared parameters $\varphi$, one common conditional-independence structure isThe hierarchy specifies how information can be shared; the data are not assumed to become independent merely because they are listed in different groups.^[ref]

A normal distribution example makes partial pooling explicit. Let ${\bar{y}}_j\mid {\theta }_j\sim \mathcal{N}({\theta }_j,{\sigma }^2/n_j)$ and ${\theta }_j\mid \mu ,\tau \sim \mathcal{N}(\mu ,{\tau }^2)$. Conditional on known $\mu ,\tau ,\sigma$, completing the square gives the posterior meanFor $n_j=4$, ${\sigma }^2=4$, ${\tau }^2=1$, $\mu =2$ and ${\bar{y}}_j=6$, this is4, halfway between the group mean and the shared mean. More precise group data increase $w_j$. In a full hierarchy with uncertain hyperparameters, their posterior distribution must also be integrated over; simply fixing fitted values omits that uncertainty.

A measurement application appears in Blaza Toman's2017 calibration study. Shared uncertain calibration inputs induce correlations among results; the Bayesian hierarchical model represents this structure within the model rather than treating all replicates as independent.^[ref]