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 |
ID.coords |
vector of ID values for the unique set of spatial coordinates obtained from |
messages |
logical; if |
plot.correlogram |
logical; if |
poisson.llik |
logical; if |
Details
Binomial model. Conditionally on the random effect , the data
y
follow a binomial distribution with probability and binomial denominators
units.m
. The logistic link function is used for the linear predictor, which assumes the form
Poisson model. Conditionally on the random effect , the data
y
follow a Poisson distribution with mean , where
is an offset set through the argument
units.m
. The log link function is used for the linear predictor, which assumes the form
The random effect has a multivariate Gaussian distribution with mean
mu
and covariance matrix Sigma
.
Laplace sampling. This function generates samples from the distribution of given the data
y
. Specifically a Langevin-Hastings algorithm is used to update where
and
are the inverse of the negative Hessian and the mode of the distribution of
given
y
, respectively. At each iteration a new value for
is proposed from a multivariate Gaussian distribution with mean
where is the current value for
,
is a tuning parameter and
is the the gradient of the log-density of the distribution of
given
y
. The tuning parameter is updated according to the following adaptive scheme: the value of
at the
-th iteration, say
, is given by
where and
are pre-defined constants, and
is the acceptance rate at the
-th iteration (
is the optimal acceptance rate for a multivariate standard Gaussian distribution).
The starting value for
, and the values for
and
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 and
denote the
-th household and the
-th person within that household; the logistic link function then assumes the form
where the random effects 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
See Also
control.mcmc.MCML
, create.ID.coords
.