Laplace.sampling.lr {PrevMap}R Documentation

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 ZZ, the data y follow a binomial distribution with probability pp and binomial denominators units.m. Let KK denote the random effects design matrix; a logistic link function is used, thus the linear predictor assumes the form

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

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

log(λ)=μ+KZ\log(\lambda)=\mu + KZ

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

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

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

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

hi=hi1+c1ic2(αi0.547),h_{i} = h_{i-1}+c_{1}i^{-c_{2}}(\alpha_{i}-0.547),

where c1>0c_{1} > 0 and 0<c2<10 < c_{2} < 1 are pre-defined constants, and αi\alpha_{i} is the acceptance rate at the ii-th iteration (0.5470.547 is the optimal acceptance rate for a multivariate standard Gaussian distribution). The starting value for hh, and the values for c1c_{1} and c2c_{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.giorgi@lancaster.ac.uk

Peter J. Diggle p.diggle@lancaster.ac.uk

See Also

control.mcmc.MCML.


[Package PrevMap version 1.5.4 Index]