Hierarchical Bayesian model

A hierarchical Bayesian model assigns distributions to local latent variables conditional on shared hyperparameters, then distributions to observations conditional on those latent variables. Its probabilistic graphical model records this conditional factorization, often using a plate for repeated observations. Integrating out latent variables gives the marginal likelihood of the hyperparameters. Priors described as flat on logarithms require a Jacobian determinant when densities are written in the original positive variables.

Partial pooling estimates related local parameters jointly through shared hyperparameters. Each local Bayesian posterior balances its own likelihood function against a group prior distribution, with stronger shrinkage when its data are weak or the group variation is small. It lies between forcing all local parameters to be equal and estimating them independently. Random-effects meta-analysis is an important application.

A Gamma–Poisson hierarchical model uses $Y_i\mid {\theta }_i\sim \mathrm{Poisson}(t_i{\theta }_i)$, ${\theta }_i\mid b\sim \mathrm{Gamma}(a,b)$, and $b\sim \mathrm{Gamma}(c,r)$, with gamma distributions in shape-rate convention. Conditional independence gives full conditional distributions ${\theta }_i\mid b,y\sim \mathrm{Gamma}(a+y_i,b+t_i)$ and $b\mid \theta ,y\sim \mathrm{Gamma}(c+na,r+{\sum }_i{\theta }_i)$.

An observed astronomical colour is the sum of a normal intrinsic colour, a nonnegative dust contribution with an exponential distribution, and normal measurement noise. With latent $C_s\sim N(\mu ,v)$, $E_s\sim \mathrm{Exp}(\text{mean }\tau )$ and $O_s\mid C_s,E_s\sim N(C_s+E_s,r_s)$, the Gibbs sampler conditionals for $C_s$ are normal and those for $E_s$ are truncated normal distributions. A flat prior on $\mu$ and inverse-gamma priors on $v,\tau$ give inverse-gamma conditional updates for the two scales. Log-flat priors on both scales instead make the joint posterior improper when all $r_s>0$, despite formally proper full conditionals at generic latent states: the marginal observed likelihood has a positive limit at either zero-scale boundary. Proper positive-scale priors restore a genuine joint posterior.