R: Langevin-Hastings MCMC for conditional simulation

Laplace.sampling {PrevMap}

R Documentation

Langevin-Hastings MCMC for conditional simulation

Description

This function simulates from the conditional distribution of a Gaussian random effect, given binomial or Poisson observations y.

Usage

Laplace.sampling(
  mu,
  Sigma,
  y,
  units.m,
  control.mcmc,
  ID.coords = NULL,
  messages = TRUE,
  plot.correlogram = TRUE,
  poisson.llik = FALSE
)

Arguments

`mu`	mean vector of the marginal distribution of the random effect.
`Sigma`	covariance matrix of the marginal distribution of the random effect.
`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`.
`ID.coords`	vector of ID values for the unique set of spatial coordinates obtained from `create.ID.coords`. These must be provided if, for example, spatial random effects are defined at household level but some of the covariates are at individual level. Warning: the household coordinates must all be distinct otherwise see `jitterDupCoords`. Default is `NULL`.
`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 the random effect S, the data y follow a binomial distribution with probability p and binomial denominators units.m. The logistic link function is used for the linear predictor, which assumes the form

\log(p/(1-p))=S.

Poisson model. Conditionally on the random effect S, the data y follow a Poisson distribution with mean m\lambda, where m is an offset set through the argument units.m. The log link function is used for the linear predictor, which assumes the form

\log(\lambda)=S.

The random effect S has a multivariate Gaussian distribution with mean mu and covariance matrix Sigma.

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

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

where \tilde{s}_{curr} is the current value for \tilde{S}, h is a tuning parameter and \nabla \log f(\tilde{S} | y) is the the gradient of the log-density of the distribution of \tilde{S} 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.

Random effects at household-level. When the data consist of two nested levels, such as households and individuals within households, the argument ID.coords must be used to define the household IDs for each individual. Let i and j denote the i-th household and the j-th person within that household; the logistic link function then assumes the form

\log(p_{ij}/(1-p_{ij}))=\mu_{ij}+S_{i}

where the random effects S_{i} are now defined at household level and have mean zero. Warning: this modelling option is available only for the binomial model.

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