This function generates a sample from the posterior distribution of a probit regression model using the data augmentation approach of Albert and Chib (1993). The user supplies data and priors, and a sample from the posterior distribution is returned as an mcmc object, which can be subsequently analyzed with functions provided in the coda package.
MCMCprobit(
formula,
data = NULL,
burnin = 1000,
mcmc = 10000,
thin = 1,
verbose = 0,
seed = NA,
beta.start = NA,
b0 = 0,
B0 = 0,
bayes.resid = FALSE,
marginal.likelihood = c("none", "Laplace", "Chib95"),
...
)An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.
Model formula.
Data frame.
The number of burn-in iterations for the sampler.
The number of Gibbs iterations for the sampler.
The thinning interval used in the simulation. The number of Gibbs iterations must be divisible by this value.
A switch which determines whether or not the progress of the
sampler is printed to the screen. If verbose is greater than 0 the
iteration number and the betas are printed to the screen every
verboseth iteration.
The seed for the random number generator. If NA, the Mersenne
Twister generator is used with default seed 12345; if an integer is passed
it is used to seed the Mersenne twister. The user can also pass a list of
length two to use the L'Ecuyer random number generator, which is suitable
for parallel computation. The first element of the list is the L'Ecuyer
seed, which is a vector of length six or NA (if NA a default seed of
rep(12345,6) is used). The second element of list is a positive
substream number. See the MCMCpack specification for more details.
The starting value for the \(\beta\) vector. This can either be a scalar or a column vector with dimension equal to the number of betas. If this takes a scalar value, then that value will serve as the starting value for all of the betas. The default value of NA will use the maximum likelihood estimate of \(\beta\) as the starting value.
The prior mean of \(\beta\). This can either be a scalar or a column vector with dimension equal to the number of betas. If this takes a scalar value, then that value will serve as the prior mean for all of the betas.
The prior precision of \(\beta\). This can either be a scalar or a square matrix with dimensions equal to the number of betas. If this takes a scalar value, then that value times an identity matrix serves as the prior precision of \(\beta\). Default value of 0 is equivalent to an improper uniform prior on \(\beta\).
Should latent Bayesian residuals (Albert and Chib, 1995) be returned? Default is FALSE meaning no residuals should be returned. Alternatively, the user can specify an array of integers giving the observation numbers for which latent residuals should be calculated and returned. TRUE will return draws of latent residuals for all observations.
How should the marginal likelihood be calculated?
Options are: none in which case the marginal likelihood will not be
calculated, Laplace in which case the Laplace approximation (see Kass
and Raftery, 1995) is used, or Chib95 in which case Chib (1995)
method is used.
further arguments to be passed
MCMCprobit simulates from the posterior distribution of a probit
regression model using data augmentation. The simulation proper is done in
compiled C++ code to maximize efficiency. Please consult the coda
documentation for a comprehensive list of functions that can be used to
analyze the posterior sample.
The model takes the following form:
$$y_i \sim \mathcal{B}ernoulli(\pi_i)$$
Where the inverse link function:
$$\pi_i = \Phi(x_i'\beta)$$
We assume a multivariate Normal prior on \(\beta\):
$$\beta \sim \mathcal{N}(b_0,B_0^{-1})$$
See Albert and Chib (1993) for estimation details.
Albert, J. H. and S. Chib. 1993. ``Bayesian Analysis of Binary and Polychotomous Response Data.'' J. Amer. Statist. Assoc. 88, 669-679
Albert, J. H. and S. Chib. 1995. ``Bayesian Residual Analysis for Binary Response Regression Models.'' Biometrika. 82, 747-759.
Andrew D. Martin, Kevin M. Quinn, and Jong Hee Park. 2011. ``MCMCpack: Markov Chain Monte Carlo in R.'', Journal of Statistical Software. 42(9): 1-21. tools:::Rd_expr_doi("10.18637/jss.v042.i09").
Siddhartha Chib. 1995. ``Marginal Likelihood from the Gibbs Output.'' Journal of the American Statistical Association. 90: 1313-1321. <doi: 10.1080/01621459.1995.10476635>
Daniel Pemstein, Kevin M. Quinn, and Andrew D. Martin. 2007. Scythe Statistical Library 1.0. http://scythe.wustl.edu.s3-website-us-east-1.amazonaws.com/.
Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2006. ``Output Analysis and Diagnostics for MCMC (CODA)'', R News. 6(1): 7-11. https://CRAN.R-project.org/doc/Rnews/Rnews_2006-1.pdf.
plot.mcmc,summary.mcmc,
glm
if (FALSE) {
data(birthwt)
out1 <- MCMCprobit(low~as.factor(race)+smoke, data=birthwt,
b0 = 0, B0 = 10, marginal.likelihood="Chib95")
out2 <- MCMCprobit(low~age+as.factor(race), data=birthwt,
b0 = 0, B0 = 10, marginal.likelihood="Chib95")
out3 <- MCMCprobit(low~age+as.factor(race)+smoke, data=birthwt,
b0 = 0, B0 = 10, marginal.likelihood="Chib95")
BayesFactor(out1, out2, out3)
plot(out3)
summary(out3)
}
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