Prior distributions

uniform(lower, upper, variable = NULL)

normal(location = 0, scale, variable = NULL)

student_t(df = 10, location = 0, scale, variable = NULL)

gamma2(alpha, beta, variable = NULL)

hs(global_scale = 1, slab_df = 10, slab_scale, variable = "beta_ev")

Source

Donegan, C., Y. Chun and A. E. Hughes (2020). Bayesian estimation of spatial filters with Moran’s Eigenvectors and hierarchical shrinkage priors. Spatial Statistics. doi:10.1016/j.spasta.2020.100450 (open access: doi:10.31219/osf.io/fah3z ).

Polson, N.G. and J.G. Scott (2010). Shrink globally, act locally: Sparse Bayesian regularization and prediction. Bayesian Statistics 9, 501-538.

Piironen, J and A. Vehtari (2017). Sparsity information and regularization in the horseshoe and other shrinkage priors. In Electronic Journal of Statistics, 11(2):5018-5051.

Arguments

lower, upper: lower and upper bounds of the distribution
variable: A reserved slot for the variable name; if provided by the user, this may be ignored by geostan.
location: Location parameter(s), numeric value(s)
scale: Scale parameter(s), positive numeric value(s)
df: Degrees of freedom, positive numeric value(s)
alpha: shape parameter, positive numeric value(s)
beta: inverse scale parameter, positive numeric value(s)
global_scale: Control the (prior) degree of sparsity in the horseshoe model (0 < global_scale < 1).
slab_df: Degrees of freedom for the Student's t model for large coefficients in the horseshoe model (slab_df > 0).
slab_scale: Scale parameter for the Student's t model for large coefficients in the horseshoe model (slab_scale > 0).

Value

An object of class prior which will be used internally by geostan to set parameters of prior distributions.

Student's t

Return value for student_t depends on the input; if no arguments are provided (specifically, if the scale parameter is missing), this will return an object of class 'family'; if at least the scale parameter is provided, student_t will return an object of class prior containing parameter values for the Student's t distribution.

Details

The prior distribution functions are used to set the values of prior parameters.

Users can control the values of the parameters, but the distribution (model) itself is fixed. The intercept and regression coefficients are given Gaussian prior distributions and scale parameters are assigned Student's t prior distributions. Degrees of freedom parameters are assigned gamma priors, and the spatial autocorrelation parameter in the CAR model, rho, is assigned a uniform prior. The horseshoe (hs) model is used by stan_esf.

Note that the variable argument is used internally by geostan, and any user provided values will be ignored.

Parameterizations

For details on how any distribution is parameterized, see the Stan Language Functions Reference document: https://mc-stan.org/users/documentation/.

The horseshoe prior

The horseshoe prior is used by stan_esf as a prior for the eigenvector coefficients. The horseshoe model encodes a prior state of knowledge that effectively states, 'I believe a small number of these variables may be important, but I don't know which of them is important.' The horseshoe is a normal distribution with unknown scale (Polson and Scott 2010):

       beta_j ~ Normal(0, tau^2 * lambda_j^2)

The scale parameter for this prior is the product of two terms: lambda_j^2 is specific to the variable beta_j, and tau^2 is known as the global shrinkage parameter.

The global shrinkage parameter is assigned a half-Cauchy prior:

       tau ~ Cauchy(0, global_scale * sigma)

where global_scale is provided by the user and sigma is the scale parameter for the outcome variable; for Poisson and binomial models, sigma is fixed at one. Use global_scale to control the overall sparsity of the model.

The second part of the model is a Student's t prior for lambda_j. Most lambda_j will be small, since the model is half-Cauchy:

       lambda_j ~ Cauchy(0, 1)

This model results in most lambda_j being small, but due to the long tails of the Cauchy distribution, strong evidence in the data can force any particular lambda_j to be large. Piironen and Vehtari (2017) adjust the model so that those large lambda_j are effectively assigned a Student's t model:

       Big_lambda_j ~ Student_t(slab_df, 0, slab_scale)

This is a schematic representation of the model; see Piironen and Vehtari (2017) or Donegan et al. (2020) for details.

Examples

# \donttest{
# normal priors for k=2 covariates
data(georgia)
prior <- list()
k <- 2
prior$beta <- normal(location = rep(0, times = k),
                     scale = rep(2, times = k))
prior$intercept <- normal(-5, 3)
print(prior)
fit <- stan_glm(deaths.male ~ offset(log(pop.at.risk.male)) + log(income) + college,
                re = ~ GEOID,
                centerx = TRUE,
                data = georgia,
                family = poisson(),
                prior = prior,
                chains = 2, iter = 600) # for speed only
plot(fit)

# setting (hyper-) priors in ME models
se <- data.frame(insurance = georgia$insurance.se)
prior <- list()
prior$df <- gamma2(3, 0.2)
prior$location <- normal(50, 50)
prior$scale <- student_t(12, 10, 20)
print(prior)
ME <- prep_me_data(se = se, prior = prior)
fit <- stan_glm(log(rate.male) ~ insurance, 
                data = georgia,
                centerx = TRUE,
                ME = ME,
               chains = 2, iter = 600) # for speed only
# }