| bbinompdf {estimateW} | R Documentation |
Probability density for a hierarchical prior setup for the elements of the adjacency matrix based on the beta binomial distribution
Description
A hierarchical prior setup can be used in W_priors to anchor the prior
number of expected neighbors. Assuming a fixed prior inclusion probability \underline{p}=1/2
for the off-diagonal entries in the binary n by n adjacency matrix \Omega implies
that the number of neighbors (i.e. the row sums of \Omega) follows a Binomial distribution
with a prior expected number of neighbors for the n spatial observations of (n-1)\underline{p}.
However, such a prior structure has the potential undesirable effect of promoting a relatively large
number of neighbors. To put more prior weight on parsimonious neighborhood structures and promote sparsity
in \Omega, the beta binomial prior accounts for the number of neighbors in each row of \Omega.
Usage
bbinompdf(x, nsize, a, b, min_k = 0, max_k = nsize)
Arguments
x |
Number of neighbors (scalar) |
nsize |
Number of potential neighbors: |
a |
Scalar prior parameter |
b |
Scalar prior parameter |
min_k |
Minimum prior number of neighbors (defaults to 0) |
max_k |
Maximum prior number of neighbors (defaults to |
Details
The beta-binomial distribution is the result of treating the prior inclusion probability \underline{p}
as random (rather than being fixed) by placing a hierarchical beta prior on it.
For the number of neighbors x, the resulting prior on the elements of \Omega, \omega_{ij},
can be written as:
p(\omega_{ij} = 1 | x)\propto \Gamma\left(a+ x \right)\Gamma\left(b+(n-1)-x\right),
where \Gamma(\cdot ) is the Gamma function, and a and
b are hyperparameters from the beta prior. In the case of a = b = 1, the prior takes the
form of a discrete uniform distribution over the number of neighbors. By fixing a = 1
the prior can be anchored around the expected number of neighbors m through
b=[(n-1)-m]/m (see Ley and Steel, 2009).
The prior can be truncated by setting a minimum (min_k) and/or a maximum number of
neighbors (max_k). Values outside this range have zero prior support.
Value
Prior density evaluated at x.
References
Ley, E., & Steel, M. F. (2009). On the effect of prior assumptions in Bayesian model averaging with applications to growth regression. Journal of Applied Econometrics, 24(4). doi:10.1002/jae.1057.