A four-parameter Beta specification as the prior for the spatial autoregressive parameter \(\rho\), as proposed by LeSage and Parent (2007) .
betapdf(rho, a = 1, b = 1, rmin = 0, rmax = 1)Density value evaluated at rho.
The scalar value for \(\rho\)
The first shape parameter of the Beta distribution
The second shape parameter of the Beta distribution
Scalar \(\underline{\rho}_{min}\): the minimum value of \(\rho\)
Scalar \(\underline{\rho}_{max}\): the maximum value of \(\rho\)
The prior density is given by:
$$ p(\rho) \sim \frac{1}{Beta(a,b)} \frac{(\rho - \underline{\rho}_{min})^{(a-1)} (\underline{\rho}_{max} - \rho)^{(b-1)} }{2^{a + b - 1}} $$
where \(Beta(a, b)\) (\(a,b > 0\)) represents the Beta function, \(Beta(a, b)= \int_{0}^{1} t^{a-1} (1-t)^{b-1} dt\).
LeSage, J. P., and Parent, O. (2007) Bayesian model averaging for spatial econometric models. Geographical Analysis, 39(3), 241-267.