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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]

A spectroscopic response factor relates a specified measured signal in spectroscopy to an amount of material under fixed measurement conditions. In the convention$I$ is the signal after blank correction, $n$ is the amount of substance, and $k$ has units of signal per amount. Specify whether the signal is a peak height or an integrated feature, and retain the same acquisition conditions during measurement calibration and unknown-sample measurement. Some literature instead defines a response factor as amount per signal: it is the reciprocal of $k$ in this simple model.^[ref]

For an original calibration, two standards with amounts $10\mu \mathrm{mol}$ and $20\mu \mathrm{mol}$ give raw integrated signals 25 and 45 in fixed instrument units. Fitting $I_{\mathrm{raw}}=I_0+kn$ givesAn unknown with raw signal 35 therefore hasRatios of uncorrected signal to amount would give 2.5 and 2.25 for the standards, falsely suggesting a changing response. Here the cause is an additive background, not a changing $k$.

Two standards fit this illustrative straight line exactly. Additional standards and repeats are needed to test the useful linear range. The inferred amount inherits measurement uncertainty in the signal, background and response factor. Samples calibrated using the same uncertain factor can have correlated inferred amounts; repeating the unknown measurement does not independently average away that shared calibration contribution.^[ref]