Conditional Autoregressive Model

Definition

A conditional autoregressive (CAR) model specifies spatial random effects $Z = (Z_1, \ldots, Z_q)'$ for $q$ geographic regions through their full conditional distributions: each $Z_i | Z_{-i}$ follows a normal distribution whose mean is a weighted average of neighboring values. When the implied $q \times q$ precision matrix $B$ is positive definite the joint distribution of $Z$ is proper; when $B$ is only nonnegative definite (singular), the joint distribution is improper — a "partially informative normal" — requiring additional identification from the data for the posterior to be well-defined.

Key Ideas

• Conditional specification (Besag 1974): $f(Z_i | Z_{-i}) = N\!\bigl(\sum_{j \in \Lambda_i} \beta_{ij} Z_j,\; \delta_1/a_i\bigr)$, where $\Lambda_i$ is the set of regions adjacent to $i$. The precision matrix $B$ has diagonal elements $a_i$ and off-diagonal element $-a_i\beta_{ij}$ for $j \in \Lambda_i$.
• Proper joint distribution: If $B$ is symmetric positive definite then $Z \sim N(0, \delta_1 B^{-1})$ — a proper multivariate Normal. Besag (1974) shows this is the unique proper joint consistent with the conditionals.
• Partially informative normal: When $B$ is singular of rank $r < q$, $f(Z) \propto \delta_1^{-q/2}\exp\!\bigl(-\tfrac{1}{2\delta_1}Z'BZ\bigr)$ is improper; the $(q-r)$-dimensional null space of $B$ is completely unidentified from the prior alone.
• Three model variants (Sun-Tsutakawa-Speckman 1999): (i) Model 1A — $B = D - \rho C$, $|\rho| < 1$, where $D$ = diagonal degree matrix and $C$ = adjacency matrix; proper joint. (ii) Intrinsic CAR (Besag-York-Mollié 1991) — $B = D - C$ (i.e., $\rho=1$), all row/column sums zero, rank $q-1$, improper; standard in disease mapping. (iii) Clayton-Kaldor (1987) — $B = I_q - \rho C$; proper iff $\lambda_1^{-1} < \rho < \lambda_q^{-1}$ where $\lambda_1 < 0 < \lambda_q$ are the extreme eigenvalues of $C$.
• Posterior propriety rank condition: For the hierarchical model $V = X_1\theta + X_2 Z + e$ with flat prior on $\theta$ and inverse-gamma priors on variance components, the posterior is proper if and only if $\mathrm{rank}(X_2'R_1X_2 + B) = q$ where $R_1 = I_n - X_1(X_1'X_1)^{-1}X_1'$. This condition is both necessary and sufficient (Theorems 2–3 of Sun-Tsutakawa-Speckman 1999).
• Functional compatibility: The improper intrinsic CAR prior has full conditionals that are "functionally compatible" with the improper joint (Hobert-Casella 1998) — the conditionals are proper and consistent, but the joint they formally imply has infinite mass. Gibbs sampling on such conditionals runs without error but explores an improper distribution if the rank condition fails.

How It Works

1. Construct adjacency structure: Build adjacency matrix $C$ (entries $c_{ij}=1$ if regions $i$ and $j$ share a boundary, 0 otherwise) and degree matrix $D = \mathrm{diag}(d_1, \ldots, d_q)$ with $d_i = \#\{j: c_{ij}=1\}$.
2. Choose CAR variant: Intrinsic CAR ($B = D - C$, $\rho=1$) encodes "no overall level" — only local contrasts are identified from the prior. Clayton-Kaldor ($B = I - \rho C$) has a proper joint for $|\rho| < 1/\lambda_q$.
3. Check rank condition: Before fitting, verify $\mathrm{rank}(X_2'R_1X_2 + B) = q$. For intrinsic CAR with grand mean in $X_1$: the rank condition is equivalent to checking that the regression design $X_2$ identifies the null space of $D - C$ (the constant vector $\mathbf{1}_q$).
4. Gibbs sampling: The full conditional $Z_i | Z_{-i}, \theta, \delta_0, \delta_1$ is Normal; the full conditional for $\delta_1 | Z$ is inverse-gamma — both conjugate. The sampler is straightforward to implement but will appear to converge even when the posterior is improper.
5. Generalized Linear Mixed Model (GLMM) extension: Replace the Gaussian observation layer with a Poisson log-linear response ($Y_i \sim \mathrm{Poisson}(m_i e^{V_i})$) for disease mapping. The sufficient condition for a proper posterior requires at least $n$ observations with integrable likelihoods and the rank condition on the design matrices (Theorem 4).

Why It Matters

• Disease mapping: The Besag-York-Mollié intrinsic CAR is the default spatial random effect in WinBUGS, R-INLA, and CARBayes. Its improper joint prior means the posterior's existence depends entirely on the data design satisfying the rank condition — a check that most applied users do not perform.
• Geographic mortality: Directly extends Tsutakawa (1988)'s gamma-Poisson model (see Tsutakawa (1988)) to the spatial correlation setting; spatial dependence across counties is now encoded in $B$ rather than treated as independent regional effects.
• Gibbs sampling failure mode: The functionally compatible but improper CAR prior is a canonical example of the "proper conditionals, improper joint" failure mode described in the Gibbs Sampler literature (Casella-George 1992; Hobert-Casella 1996).
• Spatial econometrics: CAR models are the discrete-space analogue of Gaussian random fields; the same conditional-specification idea underlies spatial lag and spatial error models in regional econometrics.

Open Questions

• Identifiability with intrinsic CAR: The intrinsic CAR prior ($\rho=1$) is only identified up to its null space through the data; practitioners typically impose a sum-to-zero or corner constraint on $Z$ to resolve the intercept-level non-identifiability.
• Multivariate CAR: Jointly modeling spatially correlated effects for multiple outcomes (e.g., multi-disease mapping) with a block-structured $B$ raises additional rank conditions and partial-identifiability issues.
• Hyperprior sensitivity: Even when the rank condition passes, near-improper inverse-gamma hyperpriors on $\delta_1$ (with very small shape parameter $a_1$) can produce near-improper posteriors in finite samples.

Related

• Bayesian Hierarchical Model
• Generalized Linear Mixed Model
• Gibbs Sampler
• Tsutakawa (1988) Mixed Model for Analyzing Geographic Variability in Mortality Rates
• Sun-Tsutakawa-Speckman (1999) Posterior Distribution of Hierarchical Models Using CAR(1) Distributions
• Dongchu Sun
• Robert K. Tsutakawa
• Space-Varying Regression Model
• Gamerman-Moreira-Rue (2003)
• Dani Gamerman