Maximum Likelihood Estimation in Machine Learning

Maximum Likelihood Estimation (MLE) is a statistical technique used to estimate the parameters of a probability distribution by maximizing the likelihood function. It is widely applied in machine learning, statistics, and AI to optimize models for tasks such as classification, regression, and generative modeling.

MLE is commonly used in logistic regression, Gaussian Mixture Models (GMMs), Hidden Markov Models (HMMs), and Natural Language Processing (NLP). In AI-driven applications, it helps in predictive modeling, speech recognition, and anomaly detection. By finding parameter values that make the observed data most probable, MLE ensures that models generalize well and make reliable predictions.

What is Likelihood in Statistics?

Likelihood is a fundamental concept in statistics and machine learning that measures how well a set of parameters explains a given dataset. Unlike probability, which measures the chance of an event occurring, likelihood quantifies how probable the observed data is under a specific model. The difference between Likelihood and Probability:

• Probability: Given a known model and parameters, probability predicts future outcomes.
• Likelihood: Given observed data, likelihood estimates the best parameters for a model.

For example, in a coin toss experiment, if we assume a fair coin (P(Heads) = 0.5), we use probability to predict future flips. However, if we observe 70 heads in 100 flips, likelihood helps us estimate the coin’s bias (e.g., P(Heads) ≈ 0.7) by adjusting model parameters to maximize the chance of observing our dataset.

Example in Statistical Modeling:

In linear regression, likelihood evaluates how well a line fits given data points. The model with parameters that maximize this likelihood is chosen. Similarly, in logistic regression, likelihood helps estimate the probability of classification labels based on input features.

What is Maximum Likelihood Estimation (MLE)?

Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a probability distribution by maximizing the likelihood function. It helps determine the set of parameters that makes the observed data most probable under the assumed model.

For a dataset X = {x₁, x₂, …, xₙ}, MLE estimates parameters θ by maximizing the likelihood function:

$$L(\theta | X) = P(X | \theta)$$

Since probabilities are often very small, it is computationally convenient to take the log-likelihood function:

$$\log L(\theta | X) = \sum_{i=1}^{n} \log P(x_i | \theta)$$

The MLE estimates θ by maximizing this log-likelihood function.

This is how MLE selects optimal parameters:

1. Assume a probability distribution (e.g., Normal, Bernoulli, or Poisson).
2. Define the likelihood function based on observed data.
3. Take the logarithm to simplify calculations.
4. Compute the derivative of the log-likelihood and set it to zero to solve for optimal parameters.

Relationship between MLE and Bayesian Inference

MLE provides point estimates by maximizing likelihood, whereas Bayesian inference considers prior distributions and computes a posterior probability:

$$P(\theta | X) \propto P(X | \theta) P(\theta)$$

Unlike MLE, Bayesian inference incorporates prior knowledge, making it useful when data is limited or uncertain. However, MLE remains a widely used technique for parameter estimation due to its simplicity and efficiency.

Steps in Maximum Likelihood Estimation

MLE follows a structured approach to estimate parameters that maximize the likelihood of observed data. Below are the key steps involved:

1. Defining the Model and Collecting the Sample

The first step in MLE is to choose an appropriate probability distribution that best represents the given dataset. The choice depends on the nature of the data:

• Normal Distribution – Used for continuous, symmetric data (e.g., height, weight).
• Bernoulli Distribution – Used for binary classification (e.g., success/failure).
• Poisson Distribution – Used for count data (e.g., number of website visits per day).

Once the model is selected, data is collected and preprocessed to remove noise and handle missing values before estimation.

2. Constructing the Likelihood Function

The likelihood function expresses the probability of observing the given data under the assumed distribution. If the dataset X = {x₁, x₂, …, xₙ} follows a normal distribution with unknown parameters μ (mean) and σ² (variance), the likelihood function is:

$$L(\mu, \sigma^2 | X) = \prod_{i=1}^{n} \frac{1}{\sqrt{2\pi\sigma^2}} \exp \left( -\frac{(x_i – \mu)^2}{2\sigma^2} \right)$$

Taking the log-likelihood simplifies computations:

$$\log L(\mu, \sigma^2) = -\frac{n}{2} \log(2\pi\sigma^2) – \sum_{i=1}^{n} \frac{(x_i – \mu)^2}{2\sigma^2}$$

3. Maximizing the Likelihood Function

MLE finds parameter estimates by maximizing L(θ | X). This is done by:

• Taking the derivative of the log-likelihood with respect to the parameters.
• Setting derivatives to zero and solving for the optimal parameters.
• Using numerical optimization techniques (e.g., Gradient Ascent, Newton’s Method) when solving analytically is difficult.

These steps ensure that MLE produces optimal parameter estimates that maximize the likelihood of the observed data.

Python Implementation of Maximum Likelihood Estimation

MLE can be implemented in Python using numerical optimization techniques. The scipy.optimize library is commonly used to find parameter estimates that maximize the likelihood function. Here’s the step-by-step implementation:

1. Import Required Libraries

import numpy as np

from scipy.optimize import minimize

from scipy.stats import norm

2. Generate Sample Data (Assuming a Normal Distribution)

np.random.seed(42)

data = np.random.normal(loc=5, scale=2, size=100)  # Mean=5, Std Dev=2

3. Define the Negative Log-Likelihood Function

Since we maximize the likelihood function, we minimize the negative log-likelihood:

def negative_log_likelihood(params, data):

    mu, sigma = params

    likelihood = -np.sum(norm.logpdf(data, mu, sigma))  # Log-likelihood

    return likelihood

4. Optimize the Parameters Using Scipy

initial_guess = [np.mean(data), np.std(data)]  # Initial estimates

result = minimize(negative_log_likelihood, initial_guess, args=(data,), method='L-BFGS-B')

mu_mle, sigma_mle = result.x

5. Evaluate and Interpret the Results

print(f"MLE Estimated Mean: {mu_mle:.4f}")

print(f"MLE Estimated Standard Deviation: {sigma_mle:.4f}")

Key Takeaways and Common Challenges

Maximum Likelihood Estimation (MLE) is a powerful method for parameter estimation, widely used in machine learning and statistics. Its key advantages include:

• Consistency – MLE provides accurate estimates with sufficient data.
• Efficiency – It achieves minimum variance among unbiased estimators.
• Wide Applicability – Used in logistic regression, Gaussian Mixture Models (GMMs), and Hidden Markov Models (HMMs).

However, MLE has limitations:

• Overfitting – Small datasets can lead to biased estimates.
• Sample Size Dependency – Requires large datasets for reliable results.
• Computational Complexity – Optimization methods can be slow for high-dimensional problems.

Alternative Estimation Methods:

• Bayesian Estimation – Incorporates prior knowledge to refine estimates.
• Method of Moments (MoM) – Estimates parameters by equating sample moments to theoretical moments, often simpler but less efficient.

Conclusion

Maximum Likelihood Estimation (MLE) is a fundamental technique in machine learning and statistics for estimating parameters that best fit observed data. It is widely used in probabilistic models, regression, classification, and deep learning applications. Despite challenges like sample size dependency and computational complexity, MLE remains a powerful and efficient method for parameter estimation.

Future advancements in likelihood-based estimation will focus on Bayesian inference, variational methods, and neural network-based optimization to enhance scalability and robustness. As AI evolves, MLE will continue to play a key role in statistical modeling, decision-making, and real-world machine learning applications.

References:

• Maximum likelihood estimation – Wikipedia