mmd and density

combination of MMD and density: my understanding is that we can obtain MMD, density, prior weights from GMM, when GMM can be calculated from ground-trurh label of

• MMD: first-order information

• Density: second-order information, the larger variance, the sparser, the small variance, the denser

• Gaussian Mixed Model: A Gaussian mixture model is a probabilistic model that assumes all the data points are generated from a mixture of a finite number of Gaussian distributions with unknown parameters. One can think of mixture models as generalizing k-means clustering to incorporate information about the covariance structure of the data as well as the centers of the latent Gaussians.

• Differentce between k-means and GMM: k-means notes that each point is assigned to different clusters, while GMM can calculate the probabiliy of each point belong to each clusters.

• Import parameters of GMM: K Gaussion models, also K clusters, $pi$, $mu$, $Sigma$
  begin{aligned}
  p(x) & = sum_{k=1}^K p(k)p(x|k) = sum_{k=1}^K pi_k mathcal{N}(x|mu_k, Sigma_k)
  end{aligned}