Gaussian Mixture Models and Expectation-Maximization A Gaussian mixture model (GMM) is a probabilistic model that assumes that a data sample is generate...
Gaussian Mixture Models and Expectation-Maximization
A Gaussian mixture model (GMM) is a probabilistic model that assumes that a data sample is generated from a mixture of underlying distributions. These underlying distributions, called components or modes, are typically Gaussian, with each component having its own mean and covariance.
The probability density of a data point in a GMM is the sum of the probabilities of being generated by each component. The weights of each component determine how much their contribution to the overall density.
The expectation-maximization (EM) algorithm is a widely used technique for estimating the parameters of a GMM. EM involves iteratively updating the means and covariances of the components based on the observed data.
Here's how EM works:
Start with initial estimates for the means and covariances of the components.
These estimates can be based on prior knowledge, external information, or even random sampling.
For each iteration:
For each data point, find the component with the highest probability of generating that data point.
Update the mean and covariance of that component based on the observed data.
Update the weights of all components based on the new means and covariances.
Example:
Suppose we have three components, each with a different mean and covariance. We observe a data point that is more likely to be generated by the third component. EM would iteratively update the means and covariances of these components to better match the observed data point