KDE is a non-parametric method: it can be used without specifying any assumptions regarding the density your are trying to estimate. … Maximum Likelihood Estimation (MLE, which I guess is what you are referring to with "Max LLE") refers to a family of methods for performing parametric density estimation. Let us assume that you know that your observations {𝑥𝑖}1≤𝑖≤𝑁 have been generated from a Gaussian distribution, but that you don't know the parameters 𝜃 (e.g. the mean 𝜇 and variance 𝜎2) of this distribution.