MCMCregress {MCMCpack} | R Documentation |
This function generates a sample from the posterior distribution of a linear regression model with Gaussian errors using Gibbs sampling (with a multivariate Gaussian prior on the beta vector, and an inverse Gamma prior on the conditional error variance). 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.
MCMCregress(formula, data = parent.frame(), burnin = 1000, mcmc = 10000, thin = 1, verbose = 0, seed = NA, beta.start = NA, b0 = 0, B0 = 0, c0 = 0.001, d0 = 0.001, marginal.likelihood = c("none", "Laplace", "Chib95"), ...)
formula |
Model formula. |
data |
Data frame. |
burnin |
The number of burn-in iterations for the sampler. |
mcmc |
The number of MCMC iterations after burnin. |
thin |
The thinning interval used in the simulation. The number of MCMC iterations must be divisible by this value. |
verbose |
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, the
beta vector, and the error variance are printed to
the screen every verbose th iteration. |
seed |
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. |
beta.start |
The starting values for the beta vector. This can either be a scalar or a column vector with dimension equal to the number of betas. The default value of of NA will use the OLS estimate of beta as the starting value. If this is a scalar, that value will serve as the starting value mean for all of the betas. |
b0 |
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. |
B0 |
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 for beta. |
c0 |
c0/2 is the shape parameter for the inverse Gamma prior on sigma^2 (the variance of the disturbances). The amount of information in the inverse Gamma prior is something like that from c0 pseudo-observations. |
d0 |
d0/2 is the scale parameter for the inverse Gamma prior on sigma^2 (the variance of the disturbances). In constructing the inverse Gamma prior, d0 acts like the sum of squared errors from the c0 pseudo-observations. |
marginal.likelihood |
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, and
Chib95 in which case the method of Chib (1995) is used. |
... |
further arguments to be passed |
MCMCregress
simulates from the posterior distribution using
standard Gibbs sampling (a multivariate Normal draw for the betas, and an
inverse Gamma draw for the conditional error variance). 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 = x_i'beta + epsilon_i
Where the errors are assumed to be Gaussian:
epsilon_i ~ N(0, sigma^2)
We assume standard, semi-conjugate priors:
beta ~ N(b0,B0^(-1))
And:
sigma^(-2) ~ Gamma(c0/2, d0/2)
Where beta and sigma^(-2) are assumed a priori independent. Note that only starting values for beta are allowed because simulation is done using Gibbs sampling with the conditional error variance as the first block in the sampler.
An mcmc object that contains the posterior sample. This object can be summarized by functions provided by the coda package.
Siddhartha Chib. 1995. "Marginal Likelihood from the Gibbs Output." Journal of the American Statistical Association. 90: 1313-1321.
Robert E. Kass and Adrian E. Raftery. 1995. "Bayes Factors." Journal of the American Statistical Association. 90: 773-795.
Andrew D. Martin, Kevin M. Quinn, and Daniel Pemstein. 2004. Scythe Statistical Library 1.0. http://scythe.wustl.edu.
Martyn Plummer, Nicky Best, Kate Cowles, and Karen Vines. 2002. Output Analysis and Diagnostics for MCMC (CODA). http://www-fis.iarc.fr/coda/.
## Not run: line <- list(X = c(-2,-1,0,1,2), Y = c(1,3,3,3,5)) posterior <- MCMCregress(Y~X, data=line, verbose=1000) plot(posterior) raftery.diag(posterior) summary(posterior) ## End(Not run)