Compute direct Monte Carlo samples from the posterior and predictive distributions of a STAR linear regression model with a g-prior.
blm_star_exact(
y,
X,
X_test = X,
transformation = "np",
y_max = Inf,
psi = NULL,
method_sigma = "mle",
approx_Fz = FALSE,
approx_Fy = FALSE,
nsave = 5000,
compute_marg = FALSE
)a list with the following elements:
coefficients the posterior mean of the regression coefficients
post.beta: nsave x p samples from the posterior distribution
of the regression coefficients
post.pred: draws from the posterior predictive distribution of y
post.pred.test: nsave x n0 samples
from the posterior predictive distribution at test points X_test
(if given, otherwise NULL)
sigma: The estimated latent data standard deviation
post.g: nsave posterior samples of the transformation
evaluated at the unique y values (only applies for 'bnp' transformations)
marg.like: the marginal likelihood (if requested; otherwise NULL)
n x 1 vector of observed counts
n x p matrix of predictors
n0 x p matrix of predictors for test data
transformation to use for the latent data; must be one of
"identity" (identity transformation)
"log" (log transformation)
"sqrt" (square root transformation)
"bnp" (Bayesian nonparametric transformation using the Bayesian bootstrap)
"np" (nonparametric transformation estimated from empirical CDF)
"pois" (transformation for moment-matched marginal Poisson CDF)
"neg-bin" (transformation for moment-matched marginal Negative Binomial CDF)
a fixed and known upper bound for all observations; default is Inf
prior variance (g-prior)
method to estimate the latent data standard deviation; must be one of
"mle" use the MLE from the STAR EM algorithm
"mmle" use the marginal MLE (Note: slower!)
logical; in BNP transformation, apply a (fast and stable) normal approximation for the marginal CDF of the latent data
logical; in BNP transformation, approximate
the marginal CDF of y using the empirical CDF
number of Monte Carlo simulations
logical; if TRUE, compute and return the marginal likelihood
STAR defines a count-valued probability model by (1) specifying a Gaussian model for continuous *latent* data and (2) connecting the latent data to the observed data via a *transformation and rounding* operation. Here, the continuous latent data model is a linear regression.
There are several options for the transformation. First, the transformation
can belong to the *Box-Cox* family, which includes the known transformations
'identity', 'log', and 'sqrt'. Second, the transformation
can be estimated (before model fitting) using the empirical distribution of the
data y. Options in this case include the empirical cumulative
distribution function (CDF), which is fully nonparametric ('np'), or the parametric
alternatives based on Poisson ('pois') or Negative-Binomial ('neg-bin')
distributions. For the parametric distributions, the parameters of the distribution
are estimated using moments (means and variances) of y. The distribution-based
transformations approximately preserve the mean and variance of the count data y
on the latent data scale, which lends interpretability to the model parameters.
Lastly, the transformation can be modeled using the Bayesian bootstrap ('bnp'),
which is a Bayesian nonparametric model and incorporates the uncertainty
about the transformation into posterior and predictive inference.
The Monte Carlo sampler produces direct, discrete, and joint draws from the posterior distribution and the posterior predictive distribution of the linear regression model with a g-prior.