The `RejectionSampling`

function implements rejection sampling
of a target density given a proposal density.

`RejectionSampling(Model, Data, mu, S, df=Inf, logc, n=1000, CPUs=1, Type="PSOCK")`

Model

This is a model specification function. For more
information, see `LaplaceApproximation`

.

Data

This is a list of data. For more information, see
`LaplaceApproximation`

.

mu

This is a mean vector \(\mu\) for the multivariate normal or multivariate t proposal density.

S

This is a convariance matrix \(\Sigma\) for the multivariate normal or multivariate t proposal density.

df

This is a scalar degrees of freedom parameter \(\nu\). It defaults to infinity, in which case the multivariate normal density is used.

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 `CPUs=1`

, in which
parallel processing does not occur.

Type

This argument specifies the type of parallel processing to
perform, accepting either `Type="PSOCK"`

or
`Type="MPI"`

.

The `RejectionSampling`

function returns an object of class
`rejection`

, which is a matrix of accepted, independent,
simulated draws from the target distribution.

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 t-distribution. 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 derivative-based or derivative-free) and Adaptive Rejection Metropolis Sampling (ARMS), as in Gilks et al. (1995). The random-walk 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 run-time.

This function is similar to the `rejectsampling`

function in the
`LearnBayes`

package.

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. 1087-1092.

von Neumann, J. (1951). "Various Techniques Used in Connection with Random Digits. Monte Carlo Methods". National Bureau Standards, 12, p. 36--38.

`dmvn`

,
`dmvt`

,
`IterativeQuadrature`

,
`LaplaceApproximation`

,
`LaplacesDemon`

, and
`VariationalBayes`

.

```
# NOT RUN {
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
# }
```

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