# The Azimuth Project Maximum likelihood estimator (changes)

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# Maximum likelihood estimator

## Idea

The maximum likelihood estimator (MLE) is one preferred method for esimating estimating the parameters of a partially probabilistic model.

## Details

Given a model with parameters $\theta$ – including a model of the kinds of deviations due to noise – along with a set of data $\{x_i\}$ believed to be generated from the model, the likelihood $L(\theta;\{x_i\})$ is the probability of model generating the data. The MLE estimate $\hat{\theta}$ of $\theta$ is then

$\hat{\theta}=\argmax_\theta L(\theta;\{x_i\})$

For some models $L$ has a sufficiently nice form that it can be analytically differentiated and set to zero to determine $\hat{\theta}$. For more difficult models other techniques are required.