Maximum Likelihood Estimation

What is Maximum Likelihood Estimation? Maximum Likelihood Estimation Explained

Maximum Likelihood Estimation (MLE) is a statistical method used to estimate the parameters of a statistical model based on observed data. It seeks to find the parameter values that maximize the likelihood function, which measures the probability of obtaining the observed data for different values of the parameters.

Here are the key points about Maximum Likelihood Estimation:

Likelihood function: The likelihood function represents the probability of obtaining the observed data given the parameter values. It is derived from the statistical model that describes the relationship between the parameters and the data. The likelihood function is typically denoted as L(θ|X), where θ represents the parameters and X represents the observed data.

Maximum likelihood estimation: The goal of MLE is to find the parameter values that maximize the likelihood function. In other words, it seeks to find the parameter values that make the observed data most likely. Mathematically, the MLE estimates are obtained by maximizing the log-likelihood function, which simplifies calculations and has the same optimal solutions as the likelihood function.

Log-likelihood function: The log-likelihood function is the natural logarithm of the likelihood function. Taking the logarithm allows the likelihood to be expressed as a sum of log-probabilities, which simplifies calculations and avoids potential numerical issues. Maximizing the log-likelihood is equivalent to maximizing the likelihood function, as the logarithm is a monotonic function.

Estimating parameters: MLE estimates the parameters that maximize the likelihood function by solving optimization problems. Depending on the complexity of the model and the availability of closed-form solutions, optimization algorithms such as gradient descent, Newton's method, or Expectation-Maximization (EM) algorithm may be employed.

Properties of MLE: Under certain regularity conditions, MLE has desirable properties, such as consistency, asymptotic normality, and efficiency. Consistency implies that as the sample size increases, the MLE estimates converge to the true parameter values. Asymptotic normality indicates that for large sample sizes, the distribution of the MLE estimates approaches a normal distribution. Efficiency suggests that among all consistent estimators, MLE achieves the smallest asymptotic variance, making it the most efficient estimator.

Inference: Once the MLE estimates are obtained, they can be used for various statistical inference tasks, such as hypothesis testing, confidence interval construction, and model selection. Likelihood ratio tests and Wald tests are commonly used for hypothesis testing based on MLE estimates.

MLE is a widely used method for parameter estimation in statistical modeling. It provides a principled approach to finding the most likely values of the parameters given the observed data. By maximizing the likelihood function, MLE produces estimates that are consistent, asymptotically normal, and efficient, making them suitable for making statistical inferences and predictions.