An Intuitive Explanation to Maximum Likelihood Estimation (MLE) in Machine Learning

MLE: Find the most likely explanation.

When using statistical models, modeling labeled data (X, y) is about finding the specific set of model parameters (θ) that maximize the likelihood of observing the given data.

More formally, a model attempts to find the specific set of parameters (θ), which maximizes the likelihood function (L):

In simple words, the above expression says that:

• maximize the likelihood of observing y

• given X

• when the prediction is parameterized by some parameters θ

When we begin modeling:

• We know X.

• We also know y.

• The only unknown is θ, which we are trying to determine.

This is called maximum likelihood estimation (MLE).

MLE is a method for estimating the parameters of a statistical model by maximizing the likelihood of the observed data.

MLE in one line: Find the parameter values that maximize the likelihood of data

While MLE is quite popular, I have often seen many struggling to intuitively understand this critical concept.

Let’s understand this today with a real-life analogy!

Imagine you are a teacher and you conducted an exam.

Surprisingly, every student received a perfect score (20/20, say).

Here’s a question: “Which of these three events would have most likely happened?”

1. Everyone studied thoroughly for the exam.

2. The question paper was leaked before the exam.

3. The invigilator allowed the students to cheat.

Which of the above events was most likely to have happened?

In this context, one may guess that the likelihood of everyone getting a perfect score given:

• “They studied thoroughly” is LOW.

• “They knew the questions beforehand” is HIGH.

• “The invigilator wasn’t keeping an eye on the students” is MEDIUM.

  

Based on this, we proceed with the event with the highest likelihood, and that’s “The question paper was leaked.”

So what did we do here?

Essentially, we maximized the conditional probability of the event (perfect score) given an explanation.

• The event is what we observed — “Everyone getting a perfect score.”

• An “explanation” represents a possible cause we came up with.

We formulated our problem this way because there’s a probability of:

• “Perfect score” given “Everyone studied thoroughly.”

• “Perfect score” given “The paper was leaked.”

• “Perfect score” given “The invigilator allowed them to cheat.”

After estimating the probability, we picked the explanation with the highest conditional probability P(event|explanation).

Simply put, we tried to find the scenario that most likely led to the observed event.

That’s it.

This is the core idea behind maximum likelihood estimation (MLE).

And this is precisely what we do in machine learning at times.

We have a bunch of observed data and several models that could have possibly generated that data.

Find the conditional probability of the data given parameter

“Several models” can be thought as the same type of model but with different parameter values.

First, we estimate the probability of seeing the data given “Parameter 1”, “Parameter 2”, and so on.

Finally, we pick those parameters that most likely produced the data.

Select the highest conditional probability

In other words, we maximize the probability of the data given parameter.

And this is called maximum likelihood estimation (MLE).

As a departing note, always remember that MLE is used to explain events that have already happened. The “event”, in this case, being the observed data.

Hope that helped :)

If you wish to learn more practical/hands-on usage of MLE, we discussed this in the following articles:

• Generalized Linear Models (GLMs): The Supercharged Linear Regression

• The Probabilistic Origin of Regularization

• Where Did The Assumptions of Linear Regression Originate From?

👉 Over to you: What other concepts would you like to learn more intuitively?

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