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Autoregressive Parameter Priors

All of the autoregressive parameter priors we describe here appear to allow the autoregressive parameters to range from $ -\infty$ to $ -\infty$. However, in practice we effectively truncate these priors to only allow values between $ -1$ to $ 1$ by rejecting proposed values outside this range when we perform the MCMC sampling. If we are using the spatially stationary spatial AR model (equation 10) then we use a noninformative prior of a disperse Gaussian:
$\displaystyle p(\beta_d\vert\phi_\beta) \sim N(0,1/\phi_\beta)$     (12)

with $ \phi_\beta$ set to a small number ($ 0.001$). For both the non-stationary spatial ( $ \beta _{ij}$) and temporal autoregressive parameters ( $ \alpha _{pi}$) we consider three different possible priors. These are now described for $ \alpha_p$; equivalent priors are used for $ \beta _{ij}$.

Subsections