Laplace.sampling.lr {PrevMap}

Langevin-Hastings MCMC for conditional simulation (low-rank approximation)

Description

This function simulates from the conditional distribution of the random effects of binomial and Poisson models.

Usage

Laplace.sampling.lr(
  mu,
  sigma2,
  K,
  y,
  units.m,
  control.mcmc,
  messages = TRUE,
  plot.correlogram = TRUE,
  poisson.llik = FALSE
)

Arguments

mu

mean vector of the linear predictor.

sigma2

variance of the random effect.

K

random effect design matrix, or kernel matrix for the low-rank approximation.

y

vector of binomial/Poisson observations.

units.m

vector of binomial denominators, or offset if the Poisson model is used.

control.mcmc

output from control.mcmc.MCML.

messages

logical; if messages=TRUE then status messages are printed on the screen (or output device) while the function is running. Default is messages=TRUE.

plot.correlogram

logical; if plot.correlogram=TRUE the autocorrelation plot of the conditional simulations is displayed.

poisson.llik

logical; if poisson.llik=TRUE a Poisson model is used or, if poisson.llik=FALSE, a binomial model is used.

Details

Binomial model. Conditionally on Z, the data y follow a binomial distribution with probability p and binomial denominators units.m. Let K denote the random effects design matrix; a logistic link function is used, thus the linear predictor assumes the form

\log(p/(1-p))=\mu + KZ

where \mu is the mean vector component defined through mu. Poisson model. Conditionally on Z, the data y follow a Poisson distribution with mean m\lambda, where m is an offset set through the argument units.m. Let K denote the random effects design matrix; a log link function is used, thus the linear predictor assumes the form

\log(\lambda)=\mu + KZ

where \mu is the mean vector component defined through mu. The random effect Z has iid components distributed as zero-mean Gaussian variables with variance sigma2.

Laplace sampling. This function generates samples from the distribution of Z given the data y. Specifically, a Langevin-Hastings algorithm is used to update \tilde{Z} = \tilde{\Sigma}^{-1/2}(Z-\tilde{z}) where \tilde{\Sigma} and \tilde{z} are the inverse of the negative Hessian and the mode of the distribution of Z given y, respectively. At each iteration a new value \tilde{z}_{prop} for \tilde{Z} is proposed from a multivariate Gaussian distribution with mean

\tilde{z}_{curr}+(h/2)\nabla \log f(\tilde{Z} | y),

where \tilde{z}_{curr} is the current value for \tilde{Z}, h is a tuning parameter and \nabla \log f(\tilde{Z} | y) is the the gradient of the log-density of the distribution of \tilde{Z} given y. The tuning parameter h is updated according to the following adaptive scheme: the value of h at the i-th iteration, say h_{i}, is given by

h_{i} = h_{i-1}+c_{1}i^{-c_{2}}(\alpha_{i}-0.547),

where c_{1} > 0 and 0 < c_{2} < 1 are pre-defined constants, and \alpha_{i} is the acceptance rate at the i-th iteration (0.547 is the optimal acceptance rate for a multivariate standard Gaussian distribution). The starting value for h, and the values for c_{1} and c_{2} can be set through the function control.mcmc.MCML.

Value

A list with the following components

samples: a matrix, each row of which corresponds to a sample from the predictive distribution.

h: vector of the values of the tuning parameter at each iteration of the Langevin-Hastings MCMC algorithm.

Author(s)

Emanuele Giorgi [e-mail]

Peter J. Diggle [e-mail]

See Also

control.mcmc.MCML.