STAR_gprior_gibbs: Gibbs sampler for STAR linear regression with a g-prior

In drkowal/rSTAR: MCMC and EM algorithms for Simultaneous Transformation and Rounding (STAR) Models

Gibbs sampler for STAR linear regression with a g-prior

Description

Compute MCMC samples from the posterior and predictive distributions of a STAR linear regression model with a g-prior. Compared to the Monte Carlo sampler STAR_gprior, this function incorporates a prior (and sampling step) for the latent data standard deviation.

Usage

STAR_gprior_gibbs(
  y,
  X,
  X_test = X,
  transformation = "np",
  y_max = Inf,
  psi = 1000,
  approx_Fz = FALSE,
  approx_Fy = FALSE,
  nsave = 1000,
  nburn = 1000,
  nskip = 0,
  verbose = TRUE
)

Arguments

y	n x 1 vector of observed counts
X	n x p matrix of predictors
X_test	n0 x p matrix of predictors for test data; default is the observed covariates X
transformation	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)
y_max	a fixed and known upper bound for all observations; default is Inf
psi	prior variance (g-prior)
approx_Fz	logical; in BNP transformation, apply a (fast and stable) normal approximation for the marginal CDF of the latent data
approx_Fy	logical; in BNP transformation, approximate the marginal CDF of y using the empirical CDF
nsave	number of MCMC iterations to save
nburn	number of MCMC iterations to discard
nskip	number of MCMC iterations to skip between saving iterations, i.e., save every (nskip + 1)th draw
verbose	logical; if TRUE, print time remaining

Details

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.

Value

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_sigma: nsave samples from the posterior distribution of the latent data standard deviation

• post_ytilde: nsave x n0 samples from the posterior predictive distribution at test points X_test

• post_g: nsave posterior samples of the transformation evaluated at the unique y values (only applies for 'bnp' transformations)

Note

The 'bnp' transformation simply calls STAR_gprior, since the latent data SD is not identified anyway.

Examples

# Simulate some data:
sim_dat = simulate_nb_lm(n = 100, p = 10)
y = sim_dat$y; X = sim_dat$X

# Fit a linear model:
fit = STAR_gprior_gibbs(y, X)
names(fit)

# Check the efficiency of the MCMC samples:
getEffSize(fit$post_beta)

drkowal/rSTAR documentation built on July 5, 2023, 2:18 p.m.