This function generates a sample from the posterior distribution of a Poisson regression model using a random walk Metropolis algorithm. 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.
MCMCpoisson(
formula,
data = NULL,
burnin = 1000,
mcmc = 10000,
thin = 1,
tune = 1.1,
verbose = 0,
seed = NA,
beta.start = NA,
b0 = 0,
B0 = 0,
marginal.likelihood = c("none", "Laplace"),
...
)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 Metropolis iterations for the sampler.
The thinning interval used in the simulation. The number of mcmc iterations must be divisible by this value.
Metropolis tuning parameter. Can be either a positive scalar or a \(k\)-vector, where \(k\) is the length of \(\beta\).Make sure that the acceptance rate is satisfactory (typically between 0.20 and 0.5) before using the posterior sample for inference.
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 current beta vector, and the Metropolis acceptance
rate 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 for beta.
How should the marginal likelihood be calculated?
Options are: none in which case the marginal likelihood will not be
calculated or Laplace in which case the Laplace approximation (see
Kass and Raftery, 1995) is used.
further arguments to be passed.
MCMCpoisson simulates from the posterior distribution of a Poisson
regression model using a random walk Metropolis algorithm. 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:
$$y_i \sim \mathcal{P}oisson(\mu_i)$$
Where the inverse link function:
$$\mu_i = \exp(x_i'\beta)$$
We assume a multivariate Normal prior on \(\beta\):
$$\beta \sim \mathcal{N}(b_0,B_0^{-1})$$
The Metropois proposal distribution is centered at the current value of
\(\theta\) and has variance-covariance \(V = T (B_0 + C^{-1})^{-1} T \)
where \(T\) is a the diagonal positive definite matrix formed from the
tune, \(B_0\) is the prior precision, and \(C\) is the
large sample variance-covariance matrix of the MLEs. This last calculation
is done via an initial call to glm.
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").
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) {
counts <- c(18,17,15,20,10,20,25,13,12)
outcome <- gl(3,1,9)
treatment <- gl(3,3)
posterior <- MCMCpoisson(counts ~ outcome + treatment)
plot(posterior)
summary(posterior)
}
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