use the given discrete data y with a discrete Dirac means

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Figure 1 shows a visualization of eq. (3) for the setting of image denoising, where noise is added to a specific instance Y = y. Moreover, the process of denoising a noisy image X = x can be described by the posterior

P(YX = x).

(4) Since we don't know the complete distribution over the target space p(y) we need to make some assumptions given our finite training dataset y = {y1,..., YN}. We want to compare two options to estimate our prior distribution p(y), see also Figure 2:

1. use the given discrete data y with a discrete Dirac means

use kernel density estimation (KDE), a non-parametric approach for density estimation with a kernel k and a bandwidth 

where any kernel function k can be used, however, a smooth kernel, such as the Gauss kernel is desirable to avoid discontinuities in the estimated probability density function and yields the following=density estimation with KDE

Figure 2: Given a finite numer of 2D samples (red markers) we can either use the data itself (Dirac measure) or a kernel density estimation (KDE). Note that the latter allows for a smooth prior distribution and for h→0 it approaches the Dirac measure.

Having a model for the likelihood and the prior distribution, we now have two possibilities to use the posterior eq. (4) for making predictions, as shown in the lecture notes and slides. The first possibility is the so called conditional mean, also know as regression function, and is given by

 

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