Description Usage Arguments Details Value Author(s) See Also
This function simulates from the conditional distribution of the random effects of binomial and Poisson models.
1 2 3 4 5 6 7 8 9 10 11 | Laplace.sampling.lr(
mu,
sigma2,
K,
y,
units.m,
control.mcmc,
messages = TRUE,
plot.correlogram = TRUE,
poisson.llik = FALSE
)
|
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 |
messages |
logical; if |
plot.correlogram |
logical; if |
poisson.llik |
logical; if |
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))=μ + KZ
where μ is the mean vector component defined through mu
.
Poisson model. Conditionally on Z, the data y
follow a Poisson distribution with mean mλ, 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(λ)=μ + KZ
where μ 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{Σ}^{-1/2}(Z-\tilde{z}) where \tilde{Σ} 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}}(α_{i}-0.547),
where c_{1} > 0 and 0 < c_{2} < 1 are pre-defined constants, and α_{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
.
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.
Emanuele Giorgi e.giorgi@lancaster.ac.uk
Peter J. Diggle p.diggle@lancaster.ac.uk
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