## Adaptive Mixture of Student-t Distributions

### Description

Function which performs the fitting of an adaptive mixture of Student-t distributions to approximate a target density through its kernel function

### Usage

```AdMit(KERNEL, mu0, Sigma0 = NULL, control = list(), ...)
```

### Arguments

 `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 the kernel function. `NA` and `NaN` values are not allowed. (See *Details* for examples of `KERNEL` implementation.) `mu0` initial value in the first stage optimization (for the location of the first Student-t component) in the adaptive mixture, or location of the first Student-t component if `Sigma0` is not `NULL`. `Sigma0` scale matrix of the first Student-t component (square, symmetric and positive definite). Default: `Sigma0 = NULL`, i.e., the scale matrix of the first Student-t component is estimated by the function `AdMit`. `control` control parameters (see *Details*). `...` further arguments to be passed to `KERNEL`.

### Details

The argument `KERNEL` is the kernel function of the target density, and it should be vectorized for speed purposes.

As a first example, consider the kernel function proposed by Gelman-Meng (1991):

k(x1,x2) = exp( 0.5*[A*x1^2*x2^2 + x1^2 + x2^2 - 2*B*x1*x2 - 2*C1*x1 - 2*C2*x2] )

where commonly used values are A=1, B=0, C1=3 and C2=3.

A vectorized implementation of this function might be:

```    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)
}
```

This way, we may supply a point (x1,x2) for `x` and the function will output a single value (i.e., the kernel estimated at this point). But the function is vectorized, in the sense that we may supply a Nx2 matrix of values for `x`, where rows of `x` are points (x1,x2) and the output will be a vector of length N, containing the kernel values for these points. Since the `AdMit` procedure evaluates `KERNEL` for a large number of points, a vectorized implementation is important. Note also the additional argument `log = TRUE` which is used for numerical stability.

As a second example, consider the following (simple) econometric model:

y_t ~ i.i.d. N(mu,sigma^2) t=1,...,T

where mu is the mean value and sigma is the standard deviation. Our purpose is to estimate theta=(mu,sigma) within a Bayesian framework, based on a vector y of T observations; the kernel thus consists of the product of the prior and the likelihood function. As previously mentioned, the kernel function should be vectorized, i.e., treat a (Nx2) matrix of points theta for which the kernel will be evaluated. Using the common (Jeffreys) prior p(theta)=1/sigma for sigma>0, a vectorized implementation of the kernel function might be:

```     KERNEL <- function(theta, y, log = TRUE)
{
if (is.vector(theta))
theta <- matrix(theta, nrow = 1)

## sub function which returns the log-kernel for a given
## thetai value (i.e., a given row of theta)
KERNEL_sub <- function(thetai)
{
if (thetai[2] > 0) ## check if sigma>0
{ ## if yes, compute the log-kernel at thetai
r <- - log(thetai[2])
+ sum(dnorm(y, thetai[1], thetai[2], TRUE))
}
else
{ ## if no, returns -Infinity
r <- -Inf
}
as.numeric(r)
}

## 'apply' on the rows of theta (faster than a for loop)
r <- apply(theta, 1, KERNEL_sub)
if (!log)
r <- exp(r)
as.numeric(r)
}
```

Since this kernel function also depends on the vector y, it must be passed to `KERNEL` in the `AdMit` function. This is achieved via the argument ..., i.e., `AdMit(KERNEL, mu = c(0, 1), y = y)`.

To gain even more speed, implementation of `KERNEL` might rely on C or Fortran code using the functions `.C` and `.Fortran`. An example is provided in the file ‘AdMitJSS.R’ in the package's folder.

The argument `control` is a list that can supply any of the following components:

`Ns`

number of draws used in the evaluation of the importance sampling weights (integer number not smaller than 100). Default: `Ns = 1e5`.

`Np`

number of draws used in the optimization of the mixing probabilities (integer number not smaller than 100 and not larger than `Ns`). Default: `Np = 1e3`.

`Hmax`

maximum number of Student-t components in the adaptive mixture (integer number not smaller than one). Default: `Hmax = 10`.

`df`

degrees of freedom parameter of the Student-t components (real number not smaller than one). Default: `df = 1`.

`CVtol`

tolerance for the relative change of the coefficient of variation (real number in [0,1]). The adaptive algorithm stops if the new component leads to a relative change in the coefficient of variation that is smaller or equal than `CVtol`. Default: `CVtol = 0.1`, i.e., 10%.

`weightNC`

weight assigned to the new Student-t component of the adaptive mixture as a starting value in the optimization of the mixing probabilities (real number in [0,1]). Default: `weightNC = 0.1`, i.e., 10%.

`trace`

tracing information on the adaptive fitting procedure (logical). Default: `trace = FALSE`, i.e., no tracing information.

`IS`

should importance sampling be used to estimate the mode and the scale matrix of the Student-t components (logical). Default: `IS = FALSE`, i.e., use numerical optimization instead.

`ISpercent`

vector of percentage(s) of largest weights used to estimate the mode and the scale matrix of the Student-t components of the adaptive mixture by importance sampling (real number(s) in [0,1]). Default: `ISpercent = c(0.05, 0.15, 0.3)`, i.e., 5%, 15% and 30%.

`ISscale`

vector of scaling factor(s) used to rescale the scale matrix of the mixture components (real positive numbers). Default: `ISscale = c(1, 0.25, 4)`.

`trace.mu`

Tracing information on the progress in the optimization of the mode of the mixture components (non-negative integer number). Higher values may produce more tracing information (see the source code of the function `optim` for further details). Default: `trace.mu = 0`, i.e., no tracing information.

`maxit.mu`

maximum number of iterations used in the optimization of the modes of the mixture components (positive integer). Default: `maxit.mu = 500`.

`reltol.mu`

relative convergence tolerance used in the optimization of the modes of the mixture components (real number in [0,1]). Default: `reltol.mu = 1e-8`.

`trace.p`, `maxit.p`, `reltol.p`

the same as for the arguments above, but for the optimization of the mixing probabilities of the mixture components.

### Value

A list with the following components:

`CV`: vector (of length H) of coefficients of variation of the importance sampling weights.

`mit`: list (of length 4) containing information on the fitted mixture of Student-t distributions, with the following components:

`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.

where H (>=1) is the number of components in the adaptive mixture of Student-t distributions and d (>=1) is the dimension of the first argument in `KERNEL`.

`summary`: data frame containing information on the optimization procedures. It returns for each component of the adaptive mixture of Student-t distribution: 1. the method used to estimate the mode and the scale matrix of the Student-t component (‘USER’ if `Sigma0` is provided by the user; numerical optimization: ‘BFGS’, ‘Nelder-Mead’; importance sampling: ‘IS’, with percentage(s) of importance weights used and scaling factor(s)); 2. the time required for this optimization; 3. the method used to estimate the mixing probabilities (‘NLMINB’, ‘BFGS’, ‘Nelder-Mead’, ‘NONE’); 4. the time required for this optimization; 5. the coefficient of variation of the importance sampling weights.

### Note

By using `AdMit` you agree to the following rules:

• You must cite Ardia et al. (2009a,b) in working papers and published papers that use `AdMit`. Use `citation("AdMit")`.

• You must place the following URL in a footnote to help others find `AdMit`: https://CRAN.R-project.org/package=AdMit.

• You assume all risk for the use of `AdMit`.

Further details and examples of the R package `AdMit` can be found in Ardia et al. (2009a,b).

Further details on the core algorithm are given in Hoogerheide (2006), Hoogerheide, Kaashoek, van Dijk (2007) and Hoogerheide, van Dijk (2008).

The adaptive mixture `mit` returned by the function `AdMit` is used by the function `AdMitIS` to perform importance sampling using `mit` as the importance density or by the function `AdMitMH` to perform independence chain Metropolis-Hastings sampling using `mit` as the candidate density.

### Author(s)

David Ardia for the R port, Lennart F. Hoogerheide and Herman K. van Dijk for the `AdMit` algorithm.

### References

Ardia, D., Hoogerheide, L.F., van Dijk, H.K. (2009a). AdMit: Adaptive Mixture of Student-t Distributions. 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

Gelman, A., Meng, X.-L. (1991). A Note on Bivariate Distributions That Are Conditionally Normal. The American Statistician 45(2), pp.125-126.

Hoogerheide, L.F. (2006). Essays on Neural Network Sampling Methods and Instrumental Variables. PhD thesis, Tinbergen Institute, Erasmus University Rotterdam (NL). ISBN: 9051708261. (Book nr. 379 of the Tinbergen Institute Research Series.)

Hoogerheide, L.F., Kaashoek, J.F., van Dijk, H.K. (2007). On the Shape of Posterior Densities and Credible Sets in Instrumental Variable Regression Models with Reduced Rank: An Application of Flexible Sampling Methods using Neural Networks. Journal of Econometrics 139(1), pp.154-180.

Hoogerheide, L.F., van Dijk, H.K. (2008). Possibly Ill-Behaved Posteriors in Econometric Models: On the Connection between Model Structures, Non-elliptical Credible Sets and Neural Network Simulation Techniques. Tinbergen Institute discussion paper 2008-036/4.

`AdMitIS` for importance sampling using an adaptive mixture of Student-t distributions as the importance density, `AdMitMH` for the independence chain Metropolis-Hastings algorithm using an adaptive mixture of Student-t distributions as the candidate density.

### 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 AdMit (with default values)
set.seed(1234)
outAdMit <- AdMit(GelmanMeng, mu0 = c(0.0, 0.1), control = list(Ns = 1e4))