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Assuming a spatially structured prior distribution for r means to take into account that geograph- ically closed areas tend to have similar relative risks. To express in mathematical terms a local spatial variation of risks, nearest neighbour Markov Random Fields (MRF) can be useful:

[xi|xj, j 6= i] = [xi|xj, j ∈ δi] (3.5)

Assuming the conditionally specification (3.5) for parameters x = log r means assuming that the conditional distribution of the log relative risk in area i, given values for the log relative risks in all other areas j 6= i, depends only on the log relative risks in the neighbouring areas (denoted as δi)

of area i. The joint distribution of the log relative risks can be determined, up to a normalizing constant, from the knowledge of each conditional distribution (3.5) by applying Brook’s Lemma (Besag, 1974). Moreover, the Hammersley-Clifford Theorem shows that if we have a MRF, i.e. if a set of full conditionals defines a unique joint distribution, then this joint distribution is a Gibbs distribution. Informally, [x1, ..., xN], is a Gibbs distribution if it is a function of the xi

only through a function of those xj which belong to the set of the neighbouring areas of area i

(j ∈ δi). Specifying the prior model for r by a set of full conditional distributions such that the

joint distribution is uniquely determined as a Gibbs distribution allows to make posterior inference by implementing Gibbs sampler algorithm. It is thus possible to simulate realizations from the joint posterior distribution of the log relative risks by simulating from each full conditional separately, still being sure that there is a unique equilibrium distribution for this sampler; see Banerjee et al. (2004) and references therein for more theoretical details.

A very useful prior specification for x = log(r) is the intrinsic Autoregressive model (IAR or intrinsic CAR). [xi|x−i, σ2] = N ormal( ¯xi, σ2 wi+ ) (3.6) where ¯xi =P_{j∈δi w}xi

i+ denotes the mean of the xj in areas adjacent to area i, and x−i denotes the

log relative risks in all the areas j 6= i.

Therefore, the conditional prior distribution of xi, given all the other log relative risks in the

map, is assumed normal with mean the average of the xj in the neighbouring areas and variance

inversely proportional to the number of neighbouring areas (denoted as wi+). This model differs

from the proper conditional autoregressive model (CAR) where the conditional variance for xigiven

all the other log relative risks is constant. CAR is suitable for regular maps, whereas IAR is more appropriate for irregular maps, i.e. where the number of neighbors varies. Model (3.6) identify the following joint prior distribution for x given the hyperparameter γ (here γ = σ2 since it is the

variance of a normal distribution): [x|γ] = [x|σ2] ∝ 1 σnexp{− 1 2σ2 n X i=1 X j<i wij(xi− xj)2} (3.7)

This is a Gaussian MRF where the mean is zero and its precision matrix has diagonal elements wi+

σ2

and off-diagonal elements −wij

σ2 , wij are pre-chosen non-negative weights, with (in the simplest case)

wij = 1 if i and j are neighbouring areas, wij = 0 for the remaining areas and wi+ = PNj=1wij.

A prior of the form (3.7) is a pairwise difference distribution and it is not proper, i.e. its integral is not finite, hence the mean and other moments of such a distribution cannot be determined. The impropriety is also evident since we can add any constant to all of the log relative risks xi

and (3.7) is unaffected. Constraining the set of the xi’s to sum to zero can solve the problem.

Thus, such a model can never be taken as a model for describing data, since data could not arise under an improper density function and yet we could not impose a centering constraint on random realizations. Model (3.7) can however be assumed as a prior for parameters in the model that play the role of random effects. In our case, we chose this model for specifying the distribution of area-specific log relative risks x = (x1, ..., xN). This choice is appropriate and

yields a posterior distribution [x|y] that is proper (Molli´e, 1996). However, for the identification of posterior log relative risks the impropriety of (3.7) cause troubles; such parameters can be identified only up to an additive constant. Thus, it is convenient to introduce an intercept α such that x = α + u can be identified by imposing the constraint PN

i=1ui= 0. Indeed, constraining the

random effects to sum to zero and specifying a separate intercept term with a uniform prior on the whole real line is equivalent to the unconstrained parameterisation with no separate intercept (Besag and Kooperberg, 1995). Note, in implementing Gibbs sampler this constraint can be imposed numerically by recentring each sampled u vector around its own mean in each MCMC iteration. OpenBugs free software can automatically impose such a sum-to-zero constraint. To sum up, a specification of the spatially structured prior above as a CAR (or IAR) is usually proposed and it is fruitful since model (3.7) is also a Gibbs distribution, precisely a distribution for xi which

depends only on neighbouring areas log relative risks. Working with the N full conditionals (3.6) is better than seeking to write down the joint distribution for several reasons. First, the possibly large number of areal units, second, and most important in practice, it has the advantage of developing MCMC computation by implementing Gibbs sampler algorithm to sample realizations of each separate log-relative risk from its full conditional distribution. Furthermore, a local specification where the risk of area i is dependent on risk of its neighbors is a natural prior belief in many applications.