Maximum Likelihood Estimation in Machine Learning.

Introduction:

Maximum Likelihood Estimation (MLE) is a powerful statistical method used to estimate the parameters of a statistical model. It is widely applied in various fields, including finance, biology, and machine learning. MLE aims to find the values of the parameters that maximize the likelihood function, representing the probability of the observed data given the model.

The Likelihood Function:

The likelihood function, denoted as L(θ∣x), expresses the probability of observing the given data (x) under a specific set of parameter values (θ). The goal of MLE is to find the values of θ that make the observed data most probable.

Mathematically, the likelihood function is defined as the joint probability density function (pdf) or probability mass function (pmf) of the observed data:

Here, x1, x2, ...,,xn are the observed data points, and θ is the parameter vector.

Log-Likelihood Function:

To simplify calculations and make the optimization process more manageable, it is common to work with the log-likelihood function, denoted as l(θ∣x), which is the natural logarithm of the likelihood function:

Taking the logarithm does not alter the location of the maximum likelihood because the natural logarithm is a monotonic function.

Finding Maximum Likelihood Estimators:

To find the maximum likelihood estimators (θ), we differentiate the log-likelihood function with respect to each parameter and set the resulting equations to zero:

Solving this system of equations provides the values of the parameters that maximize the likelihood of the observed data.

Example:

Let's consider a simple example with a normal distribution. If we have a set of independent and identically distributed observations x1 ,x2 ,..., xn from a normal distribution with unknown mean (μ) and variance (σ^2), the likelihood function would be:

Taking the natural logarithm and differentiating, we can find the MLEs for μ and σ^2.

Maximum Likelihood Estimation is a powerful tool for estimating unknown parameters in statistical models. By maximizing the likelihood function, MLE provides a robust and principled way to make inferences about the underlying parameters of a given dataset.

MLE is a fundamental concept in statistics, and its applications are widespread in various domains. Understanding the likelihood function, log-likelihood, and the optimization process is crucial for harnessing the full potential of MLE in statistical modeling.