höchstwahrscheinlichkeitsdichte

Höchstwahrscheinlichkeitsdichte, often translated as maximum likelihood estimate (MLE), is a method for estimating the parameters of a statistical model. Given a dataset and a probabilistic model, the goal of MLE is to find the parameter values that make the observed data most probable. It involves defining a likelihood function, which quantifies the probability of observing the given data for different values of the model's parameters.

The procedure is to find the parameters that maximize this likelihood function. This is typically done by taking the derivative of the log-likelihood function with respect to each parameter, setting these derivatives to zero, and solving the resulting equations. The log-likelihood is often used because it simplifies calculations, especially when dealing with independent observations, as the logarithm of a product becomes a sum.

MLE is a widely used estimation technique due to its desirable statistical properties. Under certain regularity conditions, MLEs are consistent, meaning they converge to the true parameter values as the sample size increases. They are also asymptotically efficient, achieving the lowest possible variance among unbiased estimators as the sample size grows large. However, MLEs can sometimes be biased for finite sample sizes. The method is fundamental in various fields, including machine learning, econometrics, and signal processing.