Posterior inference for STAR linear model
blm_star(
y,
X,
X_test = NULL,
transformation = "np",
y_max = Inf,
prior = "gprior",
use_MCMC = TRUE,
nsave = 5000,
nburn = 5000,
nskip = 0,
method_sigma = "mle",
approx_Fz = FALSE,
approx_Fy = FALSE,
psi = NULL,
compute_marg = FALSE
)a list with at least the following elements:
coefficients: the posterior mean of the regression coefficients
post.beta: posterior draws of the regression coefficients
post.pred: draws from the posterior predictive distribution of y
post.log.like.point: draws of the log-likelihood for each of the n observations
WAIC: Widely-Applicable/Watanabe-Akaike Information Criterion
p_waic: Effective number of parameters based on WAIC
If test points are passed in, then the list will also have post.predtest,
which contains draws from the posterior predictive distribution at test points.
Other elements may be present depending on the choice of prior, transformation, and sampling approach.
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 process; must be one of
"identity" (identity transformation)
"log" (log transformation)
"sqrt" (square root transformation)
"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)
"box-cox" (box-cox transformation with learned parameter)
"ispline" (transformation is modeled as unknown, monotone function using I-splines)
"bnp" (Bayesian nonparametric transformation using the Bayesian bootstrap)
a fixed and known upper bound for all observations; default is Inf
prior to use for the latent linear regression; currently implemented options are "gprior", "horseshoe", and "ridge". Not all modeling options and transformations are available with the latter two priors.
= TRUE,
number of MCMC iterations to save (or MC samples to draw if use_MCMC=FALSE)
number of MCMC iterations to discard
number of MCMC iterations to skip between saving iterations, i.e., save every (nskip + 1)th draw
method to estimate the latent data standard deviation in exact sampler; 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
prior variance (g-prior)
logical; if TRUE, compute and return the marginal likelihood (only available when using exact sampler, i.e. use_MCMC=FALSE)
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', as well as a version in which the Box-Cox parameter
is inferred within the MCMC sampler ('box-cox'). 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 (use_MCMC=FALSE) produces direct, discrete, and joint draws
from the posterior distribution and the posterior predictive distribution
of the linear regression model with a g-prior.