Blurb::
Calculate the Kernel Density Estimate of the posterior distribution

Description::
A kernel density estimate (KDE) is a non-parametric, smooth approximation of 
the probability density function of a random variable. It is calculated using
a set of samples of the random variable. If \f$X\f$ is a univariate random 
variable with unknown density \f$f\f$ and independent and identically 
distributed samples \f$x_{1}, x_{2}, \ldots, x_{n}\f$, the KDE is given by
\f[
\hat{f} = \frac{1}{nh} \sum_{i = 1}^{n} K \left( \frac{x - x_{i}}{h} \right).
\f]
The kernel \f$K\f$ is a non-negative function which integrates to one. Although
the kernel can take many forms, such as uniform or triangular, Dakota uses a
normal kernel. The bandwidth \f$h\f$ is a smoothing parameter that should be
optimized. Choosing a large value of \f$h\f$ yields a wide KDE with large 
variance, while choosing a small value of \f$h\f$ yields a choppy KDE with 
large bias. Dakota approximates the bandwidth using Silverman's rule of thumb,
\f[
h = \hat{\sigma} \left( \frac{4}{3n} \right)^{1/5},
\f]
where \f$\hat{\sigma}\f$ is the standard deviation of the sample set 
\f$\left\{ x_{i} \right\}\f$.

For multivariate cases, the random variables are treated as independent, and a
separate KDE is calculated for each.

<b> Expected Output </b>

If \c kde is specified, calculated values of \f$\hat{f}\f$ will be output to 
the file \c kde_posterior.dat. Example output is given below.

Examples::
Below is a \c method block of a Dakota input file that indicates the 
calculation of the KDE

\verbatim
method,
    bayes_calibration queso
      dram
      seed = 34785
      chain_samples = 1000
      posterior_stats kde
\endverbatim

The calculated KDE values are output to the file \c kde_posterior.dat, as
shown below

\verbatim
uuv_1  KDE PDF estimate  
0.406479    61.2326
0.406338    64.0245
0.402613    114.468
0.402613    114.468
0.40249    114.409
0.40282    114.162
0.398899    65.2361
0.400093    84.9285
0.401264    104.105
0.400917    98.7803
\endverbatim

