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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.

The gamma full conditional distribution has mean $(y_i+1)/(1+\psi )=(y_i+1)\varphi$. The tower property and the beta distribution mean giveWriting $\overline{y}=s/n$ and $w=s/(s+n)=\overline{y}/(1+\overline{y})$ yieldsThus partial pooling pulls every local mean toward the overall sample mean. The common weight arises from the fixed shape-one mixing distribution; it is the Poisson–exponential posterior shrinkage formula.

Model the count relative to exposure, for exampleHere $E_i$ is a measured exposure such as patient-days, $x_i$ contains hospital predictors, and $b_i$ is a random effect. This Poisson regression separates hospital size from rate variation and supplies partial pooling. Use proper, scale-appropriate priors on coefficients and variation; add regional or temporal effects if the data demand them. A gamma mixing law with an estimated shape is another way to allow more flexible overdispersion. Compare the fitted model's counts and dispersion through posterior predictive checks.

The hierarchical Bayesian model separates variability between individuals from variability between repeated measurements on an individual. In WinBUGS, dnorm takes a precision parameter, so inverse variance is the appropriate second argument.

Line (1) gives $\log {\sigma }_j^2\sim N(\varphi ,{\psi }^2)$, hence a positive log-normal distribution for each within-individual variance. Estimating the shared parameters supplies partial pooling of these variances without forcing them equal. Line (2) assigns a proper bounded flat prior to the between-individual standard deviation $\tau$, and line (3) does the same for the standard deviation $\psi$ on the log-variance scale. The broad bounded priors on $\mu$ and $\varphi$ are also proper.

This is a reasonable structure for heterogeneous repeated measurements, but flat does not mean uninformative on all transformed scales. The bounds must be wide enough for plausible values, while such enormous ranges can put substantial mass on implausible predictions. Use prior predictive checks and sensitivity analysis, and check the Markov chain Monte Carlo convergence diagnostics. Serial dependence or systematic trends would require an extension of the independent-measurement likelihood.

Model B replaces independent tree parameters by multivariate normal random effects. It permits correlated asymptote, timing and slope variation and supplies partial pooling, a useful improvement if the five trees can be regarded as exchangeable.

The Wishart distribution prior is on the precision matrix $\Omega$, not the covariance matrix. In the WinBUGS convention, dimension three and degrees of freedom three give a proper but very dispersed prior. The induced inverse-Wishart distribution has no finite mean at those degrees of freedom. Thus this is not a neutral choice of covariance prior, and estimating six covariance components from only five trees is weakly informed. The broad mean priors and possible negative growth slopes also remain. Check prior-predictive curves and sensitivity, and consider separate, interpretable scale and correlation priors.

The resulting posterior means of the second and third parameters are quite similar across trees, motivating a simpler shared-shape model; their uncertainty means this similarity is evidence to explore, rather than a proof of exact equality.

A common second parameter means the trees reach the same fraction of their asymptotic sizes at the centered day; a common third parameter means they have the same standardized growth rate. The trees may differ mainly in final size while sharing a growth shape. Similar environment and the overlapping posterior estimates make this biologically plausible. Equality is a stronger modelling assumption than partial pooling, and should be checked predictively.

The common normal distribution for the child-specific random intercepts produces partial pooling. Children with little information borrow strength from the common population, while well-observed children retain more of their individual estimate. For fixed slopes and hyperparameters, let ${\overline{r}}_i$ average the three responses after subtracting their slope terms. Multiplying the normal likelihood by the normal intercept prior distribution giveswith conditional variance $(3/{\sigma }^2+1/{\tau }^2{)}^{-1}$. Thus small between-child variation produces stronger shrinkage towards $\delta$. The common mean and variation are themselves learned from all children in this hierarchical Bayesian model.

Model 2 is an errors-in-variables model: the follow-up regression depends on a latent true baseline rather than substituting its noisy observation. A common normal distribution for the true baselines provides partial pooling and allows uncertainty about each baseline to propagate to the regression coefficients and predictions. Conditional on the shared parameters, the child-specific factorization isThe Directed acyclic graph below includes the shared parameters, the two local latent variables, both observed measurements, and the deterministic regression mean. The outer plate repeats over children and the inner plate over follow-up visits. Filled nodes are observed; rectangular mean nodes are deterministic.

One complete adapted BUGS specification uses the non-centered form of the random intercept, retaining the original priors and adding proper broad priors for $\theta$ and $\psi$:

for (i in 1:N) {
  z[i] ~ dnorm(0, 1)
  alpha[i] <- delta + tau*z[i]
  truebase[i] ~ dnorm(theta, baseprec)
  baseline[i] ~ dnorm(truebase[i], noiseprec)
  for (j in 1:3) {
    followmean[i,j] <- alpha[i] + beta*log(time[i,j]) + gamma*truebase[i]
    followup[i,j] ~ dnorm(followmean[i,j], noiseprec)
  }
}
noiseprec ~ dgamma(0.001, 0.001)
beta ~ dunif(-100, 100)
gamma ~ dunif(-100, 100)
delta ~ dunif(-100, 100)
tau ~ dunif(0, 100)
theta ~ dunif(-100, 100)
psi ~ dunif(0, 100)
baseprec <- 1/(psi*psi)

Here $N$ is the number of children, baseline and followup contain the observed data, and dnorm uses precision rather than variance. The priors on the newly introduced population baseline mean and standard deviation are explicit choices, not consequences of the likelihood; sensible alternatives can be checked for sensitivity. The code uses the noisy baseline only in its own measurement likelihood, and uses truebase in every follow-up mean.

The reported deviance information criterion differs byModel 3 has a mean Bayesian deviance worse by only $0.2$, but its effective parameter count in DIC is lower by $2.1$. The small fit cost is more than offset by the reduced complexity penalty, so the criterion gives a mild preference to Model 3. A difference this small should not be presented as decisive evidence, especially without the Monte Carlo error of the criterion difference.

The counts $143.6$ and $141.5$ are effective rather than literal parameter numbers. The likelihood includes child-specific intercepts and uncertain true baselines as well as global coefficients. With 106 of each local quantity and seven global parameters, Model 2 has 219 stochastic unknowns in a natural full representation, but partial pooling and other prior information reduce their effective contribution. Fixing two slopes lowers the effective count by about two, with a further small change from the fitted posterior. There is no reason for $p_D$ to be an integer, or to equal just the number of population-level regression coefficients.

For illustration, the deviances evaluated at posterior means implied by the table are $\overline{D}-p_D=984.5$ for Model 2 and $986.8$ for Model 3. These differ from the average deviances because averaging a nonlinear deviance is not the same as evaluating it at an averaged parameter. This interpretation presumes the same observed-data likelihood convention in both fits; integrating local variables out before forming the deviance would define a different comparison. The DIC difference is not a Bayes factor.

Calling the effects exchangeable random variables means that their joint prior distribution is unchanged by permuting study labels. In a hierarchical Bayesian model, conditional independent draws ${\beta }_j\mid \mu ,\tau \sim N(\mu ,{\tau }^2)$ achieve this, and integrating shared hyperparameters induces dependence between studies. This permits partial pooling without asserting that all effects are identical.

The assumption is reasonable when the studies concern comparable treatments, populations, outcomes and follow-up, and no known study characteristic gives one effect a systematically different prior center. Relevant differences can instead enter a linear regression for study effects, after which the residual effects may be exchangeable. The numerical table counts deaths, so its event probabilities are mortality probabilities and ${\beta }_j<0$ indicates a lower mortality odds ratio. Interpreting those counts as beneficial responses would reverse the clinical meaning.

For the common Bayesian deviance convention $D=-2\log L$ used in all three models,Here Dhat is $D(\overline{\theta })$, an at-posterior-mean fit measure, and $p_D$ is an effective parameter count. The independent model's Dhat of 53.1 is almost identical to the exchangeable model's 53.2; both improve on the common model's 57.8. The common model's $p_D=7$ corresponds to six intercepts plus one shared effect. Independence uses roughly twelve effective parameters. Partial pooling reduces the exchangeable model's effective complexity to about 8.7 while retaining nearly the same fitted likelihood function as independence.

The exchangeable model has the lowest reported DIC, but the common model is competitive. Their difference is only about 1.3, whereas independence is worse by about 6.3. The deviance information criterion measures penalized fit for a predictive comparison, not model posterior probabilities, and these numbers do not establish overwhelming evidence for heterogeneity. The displayed exchangeable $\overline{D}+p_D$ is $61.9+8.7=70.6$, rather than the printed 70.5; rounding of the underlying values can account for a tenth and does not change this interpretation.

For $Y_i\mid {\theta }_i\sim \mathrm{Poisson}({\theta }_i)$ and ${\theta }_i\mid \psi \sim \mathrm{Exp}(\psi )$, a scale-invariant prior $d\psi /\psi$ induces the Haldane prior on $\varphi =(1+\psi {)}^{-1}$. Integrating out each local mean gives $p(y_i\mid \varphi )=(1-\varphi ){\varphi }^{y_i}$. Thus $\varphi \mid y\sim \mathrm{Beta}(s,n)$ for $s={\sum }_i{y}_i>0$. The gamma full conditional distribution has mean $(y_i+1)\varphi$, so the tower property gives $(y_i+1)s/(s+n)$, equal to the displayed partial pooling formula. For $s=0$, the posterior is improper and this expectation is undefined.

A Student t random-effect model assigns Student's t-distributions to exchangeable study effects, allowing heavier tails than a normal distribution hierarchy. An equivalent Gaussian scale mixture is ${\lambda }_j\sim {\chi }_{\nu }^2$, ${\beta }_j\mid {\lambda }_j,\mu ,\psi \sim N(\mu ,\nu {\psi }^2/{\lambda }_j)$, with independent latent draws. For $\nu >2$ the variance is $\nu {\psi }^2/(\nu -2)$, so $\psi$ is a scale rather than a standard deviation. Small latent precisions weaken shrinkage for atypical studies while retaining partial pooling for the rest.