## 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

```AdMitMH(N = 1e5, KERNEL, mit = list(), ...)
```

### Arguments

 `N` number of draws generated by the independence chain M-H algorithm (positive integer number). Default: `N = 1e5`. `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 `log`. If `log = TRUE`, the function returns (natural) logarithm values of kernel function. `NA` and `NaN` values are not allowed. (See the function `AdMit` for examples of `KERNEL` implementation.) `mit` list containing information on the mixture approximation (see *Details*). `...` further arguments to be passed to `KERNEL`.

### 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")`.

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")`.

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. doi: 10.32614/RJ-2009-003

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. doi: 10.18637/jss.v029.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.

`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

```  ## 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)
mu0 = c(0.0, 0.1), control = list(Ns = 1e4))

## Run M-H using the mixture approximation as the candidate density
options(digits = 4, max.print = 40)