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# 15.1 Approximating a Density From a Set of Monte Carlo Samples

## 15.1.1 Generating Samples from a Two-Dimensional Gaussian Mixture Density

In this section, we discuss several methods for estimating the form of a density based on only a set of samples drawn from that density. It is often the case that we must make inferences about an unknown density from a set of samples (data) from that density, so we must adopt a nonparametric approach. Such nonparametric density estimation methods can provide visual cues that reveal skewness in distributions or the presence of multiple modes in the data and may provide the basis for important physical interpretations of the observations [2]. By way of example, consider the dual mode Gaussian mixture density

(15.1)

where Samples can be easily drawn from such a mixture density. If the total number of desired samples is N, then draw samples from and samples from . These samples for N = 1000 and are shown in Figure 15.1. We have also set

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