Maximum Likelihood Estimation (MLE): Method to Estimate Parameters by Maximizing the Likelihood Function

A comprehensive look at Maximum Likelihood Estimation (MLE), a method used to estimate the parameters of a statistical model by maximizing the likelihood function. This article covers its historical context, applications, mathematical foundation, key events, comparisons, and examples.

Maximum Likelihood Estimation (MLE) is a widely used method in statistics to estimate the parameters of a statistical model. By maximizing the likelihood function, MLE identifies the parameter values that make the observed data most probable.

Historical Context

The concept of likelihood was introduced by Ronald Fisher in the early 20th century. Fisher’s pioneering work laid the groundwork for modern statistical inference, and MLE has since become a fundamental technique in both theoretical and applied statistics.

Mathematical Foundation

MLE involves selecting the parameter values that maximize the likelihood function, denoted by \(L(\theta; x)\), where \(\theta\) represents the parameters and \(x\) represents the observed data.

Likelihood Function

For a given set of independent and identically distributed data points \(x_1, x_2, \ldots, x_n\) with probability density function \(f(x; \theta)\), the likelihood function \(L(\theta; x)\) is given by:

$$ L(\theta; x) = \prod_{i=1}^n f(x_i; \theta) $$

Log-Likelihood

To simplify the computation, the logarithm of the likelihood function, known as the log-likelihood, is often used:

$$ \log L(\theta; x) = \sum_{i=1}^n \log f(x_i; \theta) $$

Maximization

The goal of MLE is to find the parameter values \(\hat{\theta}\) that maximize the log-likelihood:

$$ \hat{\theta} = \arg \max_{\theta} \log L(\theta; x) $$

Key Events

• 1921: Ronald Fisher formally introduced the method of Maximum Likelihood Estimation.
• 1960s-1970s: MLE became widely adopted in various fields such as economics, biology, and engineering.
• 1990s-Present: Advancements in computational methods have facilitated the application of MLE in complex models.

Applications

MLE is extensively used in different domains, including:

• Economics: Estimating parameters of economic models.
• Biology: Inferring evolutionary trees.
• Machine Learning: Training models like logistic regression and neural networks.

Examples

Example 1: Bernoulli Distribution

For a Bernoulli distribution with parameter \(p\), the likelihood function for a dataset \(x\) consisting of \(n\) trials with \(k\) successes is:

$$ L(p; x) = p^k (1 - p)^{n - k} $$

The log-likelihood function is:

$$ \log L(p; x) = k \log p + (n - k) \log (1 - p) $$

Maximizing this function gives the MLE for \(p\):

$$ \hat{p} = \frac{k}{n} $$

Considerations

• Assumptions: MLE assumes that the data is independent and identically distributed (i.i.d.).
• Bias: In small samples, MLE estimates can be biased.
• Complexity: Involves solving optimization problems, which can be computationally intensive.

Related Terms

• Parameter Estimation: The process of using data to infer the values of parameters.
• Bayesian Estimation: An alternative method that incorporates prior distributions of the parameters.
• Frequentist Approach: The framework under which MLE operates, focusing on long-term frequency properties of estimators.

Comparisons

• MLE vs. Bayesian Estimation: MLE relies solely on the observed data, while Bayesian estimation incorporates prior beliefs.
• MLE vs. Method of Moments: The method of moments estimates parameters by equating sample moments to theoretical moments.

Inspirational Story

Ronald Fisher’s development of the MLE method revolutionized the field of statistics and showcased the power of innovative thinking in solving complex problems. Fisher’s legacy continues to inspire statisticians and data scientists worldwide.

Famous Quotes

• “To consult the rules of grammar with scrupulous nicety and in a spirit of anxiety is not particularly likely to cause eventual publication.” — Ronald Fisher

FAQs

Q: What is MLE used for?

A: MLE is used to estimate the parameters of a statistical model by finding the values that maximize the likelihood function based on the observed data.

Q: Is MLE always unbiased?

A: No, MLE estimates can be biased, especially in small samples. However, they are asymptotically unbiased, meaning they become unbiased as the sample size increases.

Q: How does MLE compare to Bayesian estimation?

A: MLE uses only the observed data to estimate parameters, while Bayesian estimation incorporates prior information along with the observed data.

References

1. Fisher, R.A. (1922). “On the Mathematical Foundations of Theoretical Statistics.” Philosophical Transactions of the Royal Society A.
2. Casella, G., & Berger, R. L. (2002). “Statistical Inference.” Duxbury.
3. Hastie, T., Tibshirani, R., & Friedman, J. (2009). “The Elements of Statistical Learning.” Springer.

Summary

Maximum Likelihood Estimation (MLE) is a fundamental statistical method for parameter estimation. By maximizing the likelihood function, MLE provides a way to derive estimates that are consistent and efficient. Its widespread applications in various fields and its theoretical significance make MLE an essential tool in the arsenal of statisticians and data analysts.