MCMCregress: Markov chain Monte Carlo for Gaussian Linear Regression

Description

This function generates a posterior density sample from 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 density is returned as an mcmc object, which can be subsequently analyzed with functions provided in the coda package.

Usage

MCMCregress(formula, data = list(), burnin = 1000, mcmc = 10000,
   thin = 5, verbose = FALSE, seed = 0, beta.start = NA, sigma2.start = NA,
   b0 = 0, B0 = 0, nu = 0.001, delta = 0.001, ...)

Arguments

formula

Model formula.

data

Data frame.

burnin

The number of burn-in iterations for the sampler.

mcmc

The number of Gibbs iterations for the sampler.

thin

The thinning interval used in the simulation. The number of Gibbs 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 TRUE, the iteration number, the $\beta$ vector, and the conditional error variance is printed to the screen every 500 iterations.

seed

The seed for the random number generator. The code uses the Mersenne Twister, which requires an integer as an input. If nothing is provided, the Scythe default seed is used.

beta.start

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

sigma2.start

The starting value for the conditional error variance. The default value of of NA will use the maximum likelihood estimate of $\sigma^2$ 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.

$\nu/2$ is the shape parameter for inverse-Gamma prior on the conditional error variance.

delta

$\delta/2$ is the scale parameter for inverse-Gamma prior on the conditional error variance.

...

further arguments to be passed

Value

An mcmc object that contains the posterior density sample. This object can be summarized by functions provided by the coda package.

Details

MCMCregress simulates from the posterior density 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 density sample. The model takes the following form: $$y_i = x_i ' \beta + \varepsilon_{i}$$ Where the errors are assumed to be Gaussian: $$\varepsilon_{i} \sim \mathcal{N}(0, \sigma^2)$$ We assume standard, conjugate priors: $$\beta \sim \mathcal{N}(b_0,B_0^{-1})$$ And: $$\sigma^{-2} \sim \mathcal{G}amma(\nu/2, \delta/2)$$ Where $\beta$ and $\sigma^{-2}$ are assumed a priori independent.

References

Andrew D. Martin, Kevin M. Quinn, and Daniel Pemstein. 2002. Scythe Statistical Library 0.3. 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/.

Examples

Run this code

line   <- list(X = c(1,2,3,4,5), Y = c(1,3,3,3,5))
line$X <- line$X - mean(line$X)
post2  <- MCMCregress(Y~X, data=line)

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