RejectionSampling {LaplacesDemon}  R Documentation 
Rejection Sampling
Description
The RejectionSampling
function implements rejection sampling
of a target density given a proposal density.
Usage
RejectionSampling(Model, Data, mu, S, df=Inf, logc, n=1000, CPUs=1, Type="PSOCK")
Arguments
Model 
This is a model specification function. For more
information, see 
Data 
This is a list of data. For more information, see

mu 
This is a mean vector 
S 
This is a convariance matrix 
df 
This is a scalar degrees of freedom parameter

logc 
This is the logarithm of the rejection sampling constant. 
n 
This is the number of independent draws to be simulated from the proposal density. 
CPUs 
This argument accepts an integer that specifies the number
of central processing units (CPUs) of the multicore computer or
computer cluster. This argument defaults to 
Type 
This argument specifies the type of parallel processing to
perform, accepting either 
Details
Rejection sampling (von Neumann, 1951) is a Monte Carlo method for
drawing independent samples from a distribution that is proportional
to the target distribution, f(x)
, given a sampling distribution,
g(x)
, from which samples can readily be drawn, and for which
there is a finite constant c
.
Here, the target distribution, f(x)
, is the result of the
Model
function. The sampling distribution, g(x)
, is
either a multivariate normal or multivariate tdistribution. The
parameters of g(x)
(mu
, S
, and df
) are used
to create random draws, \theta
, of the sampling
distribution, g(x)
. These draws, \theta
, are used
to evaluate the target distribution, f(x)
, via the Model
specification function. The evaluations of the target distribution,
sampling distribution, and the constant are used to create a
probability of acceptance for each draw, by comparing to a vector of
n
uniform draws, u
. Each draw, \theta
is
accepted if
u \le \frac{f(\theta\textbf{y})}{cg(\theta)}
Before beginning rejection sampling, a goal of the user is to find the
bounding constant, c
, such that f(\theta\textbf{y}) \le
cg(\theta)
for all
\theta
. These are all expressed in logarithms, so the
goal is to find \log f(\theta\textbf{y})  \log g(\theta) \le
\log(c)
for all
\theta
. This is done by maximizing \log
f(\theta\textbf{y})  \log g(\theta)
over all \theta
. By using, say,
LaplaceApproximation
to find the modes of the
parameters of interest, and using the resultant LP
, the mode
of the logarithm of the joint posterior distribution, as
\log(c)
.
The RejectionSampling
function performs one iteration of
rejection sampling. Rejection sampling is often iterated, then called
the rejection sampling algorithm, until a sufficient number or
proportion of \theta
is accepted. An efficient
rejection sampling algorithm has a high acceptance rate. However,
rejection sampling becomes less efficient as the model dimension (the
number of parameters) increases.
Extensions of rejection sampling include Adaptive Rejection Sampling (ARS) (either derivativebased or derivativefree) and Adaptive Rejection Metropolis Sampling (ARMS), as in Gilks et al. (1995). The randomwalk Metropolis algorithm (Metropolis et al., 1953) combined the rejection sampling (a method of Monte Carlo simulation) of von Neumann (1951) with Markov chains.
Parallel processing may be performed when the user specifies
CPUs
to be greater than one, implying that the specified number
of CPUs exists and is available. Parallelization may be performed on a
multicore computer or a computer cluster. Either a Simple Network of
Workstations (SNOW) or Message Passing Interface (MPI) is used. With
small data sets and few samples, parallel processing may be slower,
due to computer network communication. With larger data sets and more
samples, the user should experience a faster runtime.
This function is similar to the rejectsampling
function in the
LearnBayes
package.
Value
The RejectionSampling
function returns an object of class
rejection
, which is a matrix of accepted, independent,
simulated draws from the target distribution.
Author(s)
Statisticat, LLC. software@bayesianinference.com
References
Gilks, W.R., Best, N.G., Tan, K.K.C. (1995). "Adaptive Rejection Metropolis Sampling within Gibbs Sampling". Journal of the Royal Statistical Society. Series C (Applied Statistics), Vol. 44, No. 4, p. 455–472.
Metropolis, N., Rosenbluth, A.W., Rosenbluth, M.N., and Teller, E. (1953). "Equation of State Calculations by Fast Computing Machines". Journal of Chemical Physics, 21, p. 10871092.
von Neumann, J. (1951). "Various Techniques Used in Connection with Random Digits. Monte Carlo Methods". National Bureau Standards, 12, p. 36–38.
See Also
dmvn
,
dmvt
,
IterativeQuadrature
,
LaplaceApproximation
,
LaplacesDemon
, and
VariationalBayes
.
Examples
library(LaplacesDemon)
### Suppose an output object of class laplace is called Fit:
#rs < RejectionSampling(Model, MyData, mu=Fit$Summary1[,1],
# S=Fit$Covar, df=Inf, logc=Fit$LP.Final, n=1000)
#rs