# Maximum Likelihood Estimation

## Contents

# Maximum Likelihood Estimation#

Maximum likelihood estimation (MLE) is a widely used method in statistics and machine learning for parameter estimation. The idea behind maximum likelihood estimation is to find the parameters that maximize the likelihood of the observed data given the parameters. In other words, the goal is to find the values of the parameters that are most likely to have generated the observed data.

Maximum likelihood estimation is used in a wide range of applications, including regression analysis, logistic regression, and Gaussian mixture models. The method is based on the concept of maximum likelihood, which is the probability of observing the data given the parameters. The maximum likelihood estimate is the value of the parameters that maximizes this probability.

Sometimes, a closed-form solution to the MLE does not exist. In these cases, the process is to start with an initial guess for the parameters, and then iteratively adjust the parameters until the maximum likelihood is achieved. This can be done using numerical optimization methods such as gradient descent, expectation-maximization (EM), or Newton’s method.

One advantage of maximum likelihood estimation is that it is a consistent estimator, meaning that as the sample size increases, the estimates will converge to the true values of the parameters. It is also asymptotically efficient, meaning that the estimates are as good as the best possible estimates based on the data.

However, (iterative) maximum likelihood estimation can be sensitive to the choice of initial conditions, and the method can become trapped in a local maximum instead of finding the global maximum. To mitigate this issue, multiple starting points can be used and the results can be averaged to obtain a more robust estimate.

Furthermore, maximum likelihood estimation can be sensitive to outliers, especially when the number of data points is limited. In such cases, alternative methods such as MAP estimation can be used.

In summary, maximum likelihood estimation is a widely used and effective method for parameter estimation in machine learning and statistics. It provides a way of finding the values of the parameters that are most likely to have generated the observed data, and has the advantages of consistency and asymptotic efficiency.

## Where to Learn More#

I’ve covered maximum likelihood estimation in-depth in the following course:

Classical Statistical Inference and A/B Testing in Python

And we apply MLE in the following courses:

Bayesian Machine Learning in Python: A/B Testing

Deep Learning Prerequisites: Linear Regression in Python