kde

Calculate the Kernel Density Estimate of the posterior distribution

Specification

• Alias: None

• Arguments: None

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 $X$ is a univariate random variable with unknown density $f$ and independent and identically distributed samples $x_1,x_2,\dots ,x_n$ , the KDE is given by

$$\hat{f}=\frac{1}{nh}\sum _{i=1}^{n}K\left(\frac{x-x_i}{h}\right).$$

The kernel $K$ 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 $h$ is a smoothing parameter that should be optimized. Choosing a large value of $h$ yields a wide KDE with large variance, while choosing a small value of $h$ yields a choppy KDE with large bias. Dakota approximates the bandwidth using Silverman’s rule of thumb,

$$h=\hat{\sigma }{\left(\frac{4}{3n}\right)}^{1/5},$$

where $\hat{\sigma }$ is the standard deviation of the sample set $\left\{x_i\right\}$ .

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

Expected Output

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