Maximum Likelihood Estimation Explained

1. The Core Idea: Finding the Best-Fitting Distribution

Imagine you have a set of data points, and you suspect they come from a certain type of probability distribution (e.g., a Normal distribution), but you don't know its specific parameters (e.g., the mean and standard deviation). How would you pick the "best" parameters?

The principle of Maximum Likelihood Estimation gives us a clear answer: Choose the parameters that make your observed data most probable.

Let's watch a video from StatQuest that illustrates this idea beautifully. It uses an example of mouse weights to show how we can visually "slide" a Normal distribution around and change its width until it provides the best possible fit for the data we actually saw.

As the video demonstrates, MLE is an optimization problem. We are searching through all possible values of our model's parameters ($\theta$) to find the one specific set of values ($\hat{\theta }$) that maximizes the likelihood function, $L(\theta |\text{data})$.

2. Why We Need a Model: Constraints and Assumptions

You might wonder: to make the data "most likely," why don't we just create a "magic" model that perfectly reproduces the exact data points we saw and gives zero probability to everything else?

This is a great question, and it highlights a key aspect of MLE: we operate under the constraints of a pre-defined probability model. We don't just find any function that maximizes the data's probability; we find the best parameters for a specific family of distributions (like Normal, Bernoulli, etc.) that we believe is a good representation of the underlying data-generating process.

The following video explains this concept very well.

By assuming a model, we are making a powerful generalization. Instead of just describing our sample, we are creating a model that can describe the broader population from which the sample was drawn.

3. The Maximization Process: From Calculus to Loss Functions

In the previous lesson, we established that maximizing the likelihood is the same as maximizing the log-likelihood, and that this is mathematically and computationally much easier.

So, the principle of MLE is formally stated as finding the parameters $\hat{\theta }$ that solve this problem:

${\hat{\theta }}_{\text{MLE}}=\arg {\max }_{\theta }\ell (\theta |X)=\arg {\max }_{\theta }{\sum }_{i=1}^n\log P(x_i|\theta )$

How do we solve this? This is where your calculus knowledge becomes very practical. To find the maximum of a function, we can:

1. Take the derivative of the function with respect to the parameter(s).
2. Set the derivative equal to zero.
3. Solve for the parameter(s).

This process finds the "peak" of the log-likelihood curve, where the slope is zero.

The following reading provides a formal, step-by-step summary of this principle.

The Bridge to Machine Learning: Minimizing Loss

In machine learning, we usually frame problems in terms of minimizing a loss function (or cost function), which measures how "bad" a model's predictions are. How does this connect to maximizing likelihood?

The connection is simple but powerful: Maximizing the log-likelihood is equivalent to minimizing the Negative Log-Likelihood (NLL).

$\text{Loss}(\theta )=-\ell (\theta |X)=-{\sum }_{i=1}^n\log P(x_i|\theta )$

By finding the $\theta$ that minimizes this NLL loss, we are, by definition, finding the Maximum Likelihood Estimate. This reframing is incredibly important because it means that many common ML models (like logistic regression) are trained simply by using an optimization algorithm (like gradient descent) to find the parameters that minimize the NLL of the training data.

This is the "why" you've been looking for: when you train a model by minimizing a cross-entropy or NLL loss, you are implicitly finding the MLE for that model's parameters.

Conclusion

Excellent work! You have now grasped one of the most fundamental principles in both statistics and machine learning.

Key Takeaways:

• The Principle of MLE: To estimate the parameters of a model, choose the parameter values that maximize the likelihood (the probability) of observing the data you have collected.
• A Constrained Search: MLE finds the best parameters within a chosen probability model (e.g., finding the best mean for a Normal distribution).
• The Process: We maximize the log-likelihood function. This is often done analytically by using calculus (taking the derivative, setting to zero, and solving) or numerically using optimization algorithms.
• Connection to Machine Learning: Maximizing the log-likelihood is equivalent to minimizing the Negative Log-Likelihood (NLL). The NLL is a very common loss function used to train classification and other probabilistic models.

Preview of the Next Lesson:

We have now covered the "what" (formulating the function) and the "why" (the principle of maximizing it). In the next lesson, "Apply MLE to find the parameters of a simple model," we will do the "how." You'll apply your calculus skills to take the log-likelihood functions we've discussed, find their derivatives, and solve for the maximum likelihood estimators of common distributions like the Bernoulli, Poisson, and Normal.