# Independence Chain Metropolis-Hastings Algorithm using an Adaptive Mixture of Student-t Distributions as the Candidate Density

### Description

Performs independence chain Metropolis-Hastings (M-H) sampling using an adaptive mixture of Student-t distributions as the candidate density

### Usage

1 |

### Arguments

`N` |
number of draws generated by the independence chain M-H algorithm (positive
integer number). Default: |

`KERNEL` |
kernel function of the target density on which the adaptive mixture is fitted. This
function should be vectorized for speed purposes (i.e., its first
argument should be a matrix and its output a vector). Moreover, the function must contain
the logical argument |

`mit` |
list containing information on the mixture approximation (see *Details*). |

`...` |
further arguments to be passed to |

### Details

The argument `mit`

is a list containing information on the
adaptive mixture of Student-t distributions. The following components must
be provided:

`p`

vector (of length

*H*) of mixing probabilities.`mu`

matrix (of size

*Hxd*) containing the vectors of modes (in row) of the mixture components.`Sigma`

matrix (of size

*Hxd*d*) containing the scale matrices (in row) of the mixture components.`df`

degrees of freedom parameter of the Student-t components (real number not smaller than one).

where *H (>=1)* is the number of components and
*d (>=1)* is the dimension of the first argument in `KERNEL`

. Typically,
`mit`

is estimated by the function `AdMit`

.

### Value

A list with the following components:

`draws`

: matrix (of size `N`

*xd*) of draws
generated by the independence chain M-H algorithm,
where `N`

*(>=1)* is the number of draws
and *d (>=1)* is the
dimension of the first argument in `KERNEL`

.

`accept`

: acceptance rate of the independence chain M-H algorithm.

### Note

Further details and examples of the **R** package `AdMit`

can be found in Ardia, Hoogerheide and van Dijk (2009a,b). See also
the package vignette by typing `vignette("AdMit")`

and the
files ‘AdMitJSS.txt’ and ‘AdMitRnews.txt’ in the ‘/doc’ package's folder.

Further information on the Metropolis-Hastings algorithm can be found in Chib and Greenberg (1995) and Koop (2003).

Please cite the package in publications. Use `citation("AdMit")`

.

### Author(s)

David Ardia

### References

Ardia, D., Hoogerheide, L.F., van Dijk, H.K. (2009a).
AdMit: Adaptive Mixture of Student-t Distributions.
*The R Journal* **1**(1), pp.25–30.
http://journal.r-project.org/2009-1/

Ardia, D., Hoogerheide, L.F., van Dijk, H.K. (2009b).
Adaptive Mixture of Student-t Distributions as a Flexible Candidate
Distribution for Efficient Simulation: The R Package AdMit.
*Journal of Statistical Software* **29**(3), pp.1–32.
http://www.jstatsoft.org/v29/i03/

Chib, S., Greenberg, E. (1995).
Understanding the Metropolis-Hasting Algorithm.
*The American Statistician* **49**(4), pp.327–335.

Koop, G. (2003).
*Bayesian Econometrics*.
Wiley-Interscience (London, UK).
ISBN: 0470845678.

### See Also

`AdMitIS`

for importance sampling using an adaptive
mixture of Student-t distributions as the importance density,
`AdMit`

for fitting
an adaptive mixture of Student-t distributions to a target density
through its `KERNEL`

function; the package coda for MCMC output
analysis.

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 | ```
## NB : Low number of draws for speedup. Consider using more draws!
## Gelman and Meng (1991) kernel function
GelmanMeng <- function(x, A = 1, B = 0, C1 = 3, C2 = 3, log = TRUE)
{
if (is.vector(x))
x <- matrix(x, nrow = 1)
r <- -.5 * (A * x[,1]^2 * x[,2]^2 + x[,1]^2 + x[,2]^2
- 2 * B * x[,1] * x[,2] - 2 * C1 * x[,1] - 2 * C2 * x[,2])
if (!log)
r <- exp(r)
as.vector(r)
}
## Run the AdMit function to fit the mixture approximation
set.seed(1234)
outAdMit <- AdMit(KERNEL = GelmanMeng,
mu0 = c(0.0, 0.1), control = list(Ns = 1e4))
## Run M-H using the mixture approximation as the candidate density
outAdMitMH <- AdMitMH(N = 1e4, KERNEL = GelmanMeng, mit = outAdMit$mit)
options(digits = 4, max.print = 40)
print(outAdMitMH)
## Use functions provided by the package coda to obtain
## quantities of interest for the density whose kernel is 'GelmanMeng'
library("coda")
draws <- as.mcmc(outAdMitMH$draws)
draws <- window(draws, start = 1001)
colnames(draws) <- c("X1", "X2")
summary(draws)
summary(draws)$stat[,3]^2 / summary(draws)$stat[,4]^2 ## RNE
plot(draws)
``` |