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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]

A hyperprior is a prior distribution assigned to a hyperparameter that controls another prior distribution. In a hierarchical Bayesian model, this makes uncertainty about the prior's parameters part of the joint probability model, rather than replacing those parameters by fixed numbers.^[ref]

For data $y$, a model parameter $\theta$ and a hyperparameter $\lambda$, a common factorization isHere $p(y\mid \theta )$ supplies the likelihood function, $p(\theta \mid \lambda )$ is the conditional prior distribution, and $p(\lambda )$ is the hyperprior. After observing $y$, Bayesian inference gives a joint posterior distribution. The marginal posterior of $\theta$ averages over the updated hyperparameter distribution:For discrete hyperparameters, the integral is replaced by a sum.

An exact example makes the averaging visible. Let $\theta$ be a success probability and choose a two-point hyperprior,Both conditional beta distributions have expectation $1/2$, but they concentrate differently around it. Now observe two successes that are independent conditional on $\theta$, each with a Bernoulli distribution. Their conditional probability is ${\theta }^2$. Integrating against the two possible priors givesThus the data also update the hyperparameter:The conditional posterior distributions of $\theta$ are respectively $\mathrm{Beta}(3,1)$ and $\mathrm{Beta}(12,10)$. Their posterior mixture has meanThis is an original illustrative probability model, rather than data or a numerical example from the guide.

Fixing $\kappa$ at its prior mean $11$ would instead produce $\mathrm{Beta}(7.5,5.5)$ after the same observations, with posterior mean $7.5/13\approx 0.577$. Averaging the hyperparameter first has changed the model; it does not reproduce averaging the conditional posteriors with their updated probabilities. A hyperprior therefore adds a specified layer of uncertainty, while its own choice remains part of the assumptions of the hierarchical Bayesian model.