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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]

Blaza Toman is a statistician whose work connects Bayesian statistics, design of experiments and measurement uncertainty in metrology. Her institutional biography records a statistics doctorate in1987 and joining the National Institute of Standards and Technology in2000. Her research includes uncertainty assessment and statistical analysis of comparisons between laboratories.^[ref]

Her2017 first-authored paper with colleagues examined uncertainty evaluation in chromatographic measurement calibration. The study compared calculation through measurement equations, including correlations, with a Bayesian hierarchical model. A central point is that several measurements can share uncertain inputs: treating them as independent can underestimate the uncertainty of a combined result.^[ref]

An invented calibration example makes this concrete. Let two detector signals be100 and110 units, with the same estimated response $k=10$ signal units per milligram. The inferred amounts $x_1=100/k$ and $x_2=110/k$ are10 and11 milligrams. If the true $k$ is1% larger, both inferred amounts must decrease together, while their ratio remains1.1. Repeating signal readings can reduce independent reading noise, but it does not by itself eliminate this shared uncertainty in $k$. This illustrates why the structure of an experiment matters to its statistical model.^[ref]