Python Automation and Machine Learning for EM and ICs

Density estimation is a general term in statistics and machine learning that refers to the process of estimating the probability density function (PDF) of a random variable. It is a way of modeling the distribution of data in order to understand its underlying structure. There are various techniques for density estimation, and Kernel Density Estimation (KDE) is one of them.

Density Estimation:

Figure 4981 shows density estimation with KDE in 2D (with clusters).

Figure 4981. Density estimation with KDE in 2D (with clusters) (code).

In a density estimation problem, assuming we have a model with a joint probability distribution, p(x, z; θ) over observed variables x and latent variables z, parameterized by θ. Then, the log-likelihood function, which is the sum of the log probabilities of observing the given data under the model, is given by,

          ------------------------------- [4981a]

                      ------------------------------- [4981b]

where,

          x⁽ⁱ⁾ is the observed data for the i-th instance.

When we only observe x and not z, it implies that we are dealing with a latent variable model where the latent variables are not directly observed but are inferred based on the observed data. Our goal is to find the parameters θ that maximize the likelihood of observing the given data x⁽¹⁾, x⁽²⁾, ..., x^(m). This is done by maximizing the log-likelihood function L(θ). Note that maximizing this function with respect to θ is a common approach in maximum likelihood estimation (MLE). The latent variable z is not observed, so we would typically use techniques like the Expectation-Maximization (EM) algorithm to iteratively estimate the latent variables and update the model parameters until convergence.

The formulation in Equation 4981b accounts for the latent variable z, and the sum over z(i) represents the marginalization of the latent variable. It reflects the idea that we are summing over all possible values of the latent variable for each instance i, which is necessary when the latent variables are not directly observed.