Maximum Likelihood Estimation
"And when is there time to remember, to sift, to weigh, to estimate, to total?"
- Tillie Olson
Maximum likelihood estimation (MLE) stands as a cornerstone in the realm of statistical inference, offering a powerful method for estimating the parameters of a probability distribution. Rooted in the principle of finding the most probable values for the parameters given observed data, MLE provides a systematic framework for making inferences about unknown quantities. Whether in fields like economics, biology, or engineering, where uncertainty reigns, understanding and applying MLE empowers researchers and practitioners to glean insights from data and make informed decisions.
In this blog post, we will explore the concept of MLE, its mathematical underpinnings, and consider a few examples.

What is Maximum Likelihood Estimation?
Maximum likelihood estimation is an estimation method used to find the parameter values of a probability distribution that maximize the likelihood of the observed data. In other words, MLE seeks to find the most probable values for the parameters of a distribution given the observed data. Consider the following definition.
Let
With those definitions in mind, let's do some examples to illustrate the concept.
MLE Examples
For our first example, let's consider a simple case where we have a random sample from a Bernoulli distribution. Recall that the p.m.f.
for the Bernoulli distribution is given by
Now, one thing to note is that the math does not always work out this nice. However, there is a way around that!
Log-Likelihood!
The idea of log-likelihood is to take the natural logarithm of the likelihood function and solve for the maximum likelihood estimate of our
parameter. It is important to note that this will still give us the same answer. Since the logarithm is a monotonicly increasing function (always increasing)
and the likelihood function is nonnegative, the log-likelihood
Let's see this in action, first with the Bernoulli distribution. We have
Why is MLE Useful?
Earlier I mentioned that MLE is a powerful method for estimating the parameters of a probability distribution. But truthfully this is an understatement. MLE is the best method for estimating the parameters of a probability distribution. The principle of MLE is used as a foundation for many machine learning algorithms and statistical methods.
Truthfully, the question should not be when is MLE useful, but when is it not useful.
Conclusion
Maximum likelihood estimation is a powerful method for estimating the parameters of a probability distribution. It provides a systematic framework for making inferences about unknown quantities, and is used as a foundation for many machine learning algorithms and statistical methods. In this blog post, we explored the concept of MLE, its mathematical underpinnings, and considered an example with the Bernoulli distribution. Stay tuned as we talk about different important ideas that use MLE!