Sun-Tsutakawa-Speckman (1999) Posterior Distribution of Hierarchical Models Using CAR(1) Distributions

Summary

Sun, Tsutakawa, and Speckman (1999) examine when the joint posterior is proper for Bayesian hierarchical models whose spatial random effects follow a conditional autoregressive (CAR(1)) distribution. The key result (Theorem 2) gives a rank condition on the data design matrix and the CAR precision matrix that is both necessary and sufficient for the posterior to be proper. This matters because the intrinsic CAR prior of Besag-York-Mollié (1991) — the default spatial random effect in disease mapping — has a singular precision matrix, making its joint distribution improper. The paper extends the result to generalized linear mixed models (GLMMs), covering the Poisson log-linear disease-mapping model as a special case, and warns that Gibbs sampling runs without error even when the posterior is improper.

Key Claims

Three CAR models: Model 1A ( $B = D - \rho C$ , $|\rho| < 1$ , proper joint); Model 1 (Besag-York-Mollié, $B = D - C$ at $\rho=1$ , row/column sums all zero, rank $q-1$ , improper); Model 2 (Clayton-Kaldor 1987, $B = I_q - \rho C$ , proper iff $\lambda_1^{-1} < \rho < \lambda_q^{-1}$ by Theorem 1, where $\lambda_1 < 0 < \lambda_q$ are the extreme eigenvalues of the adjacency matrix $C$ ).
Partially informative normal: When $B$ is singular nonnegative-definite of rank $r < q$ , the joint distribution of spatial effects $Z$ is improper — $f(Z) \propto \delta_1^{-q/2}\exp\!\bigl(-\tfrac{1}{2\delta_1}Z'BZ\bigr)$ — but the conditionals remain proper and "functionally compatible" (Hobert-Casella 1998). This subtlety is why Gibbs sampling appears to work but is exploring an improper distribution.
Theorem 2 — sufficient conditions: For the linear mixed model $V = X_1\theta + X_2 Z + e$ with flat prior on $\theta$ and inverse-gamma $(a_i,b_i)$ priors on $(\delta_0,\delta_1)$ , the joint posterior of $(\theta, Z, \delta_0, \delta_1)$ is proper iff: (a) $\mathrm{rank}(X_2'R_1X_2 + B) = q$ where $R_1 = I_n - X_1(X_1'X_1)^{-1}X_1'$ ; (b) $a_1 > 0$ and $b_1 > 0$ ; (c) $n - p - q + 2a_0 > 0$ and $\mathrm{SSE} + 2b_0 > 0$ .
Theorem 3 — necessity: If $\mathrm{rank}(X_2'R_1X_2 + B) < q$ , the posterior fails to exist for any prior on the variance components.
One-way Analysis of Variance (ANOVA) illustration: Balanced $Y_{ij} = \theta + Z_i + e_{ij}$ ; Case 1 ( $B = I_q$ ): posterior proper; Case 2 ( $B = D - C$ , intrinsic CAR, rank $q-1$ ): posterior does NOT exist; Case 3 ( $B = I_q - \rho C$ , limiting eigenvalues): depends on graph — a path-graph with $q=3$ gives posterior-proper at $\rho = \pm 1/\sqrt{2}$ .
Theorem 4 — GLMM extension: If at least $n$ observations have integrable conditional likelihoods and the reduced design matrices satisfy the rank condition, the posterior of the GLMM is proper. The Poisson log-linear disease-mapping model is a direct special case.
Practical hazard: Gibbs sampling on an improper posterior runs without error and produces output that looks like convergence; the rank condition $\mathrm{rank}(X_2'R_1X_2 + B) = q$ is a necessary pre-flight diagnostic.
Generalizes Hobert-Casella (1996): That paper required $B$ positive definite; Sun-Tsutakawa-Speckman extend to the singular $B$ case that arises with all standard CAR models used in disease mapping.

Concepts Introduced or Extended

Entities Mentioned

Quotes

"The use of the CAR(1) model to represent spatial effects may be illustrated by a log-linear model in mortality analysis." (p. 341)

"One implication of our results is that, among the assumptions of Theorem 2, $\text{rank}(X_2'R_1X_2 + B) = q$ is both necessary and sufficient for the posterior distribution of $(\theta, Z, \delta_0, \delta_1)$ given $V$ to be proper." (p. 347)

My Take

A compact and technically precise 10-page paper that closes the gap left by Hobert-Casella (1996), who required $B$ positive definite — a condition the most widely used CAR model (intrinsic CAR, $\rho=1$ ) fails to satisfy. The rank condition $\mathrm{rank}(X_2'R_1X_2 + B) = q$ is the key diagnostic: if the data design does not span the null space of $B$ , the posterior does not exist regardless of how informative the variance hyperpriors are. The practical implication for disease mapping is that area-level random effects with a single fixed intercept and the intrinsic CAR prior ( $B = D-C$ , rank $q-1$ ) will only identify the null space through $X_2$ ; practitioners must verify this condition, not assume it. The GLMM extension to Theorem 4 also directly extends the companion Tsutakawa (1988) geographic mortality model — see Tsutakawa (1988) — to the spatially correlated case.