Likelihood estimation is a statistical technique used to infer the parameters of a model by maximizing the likelihood function, which measures how well the model explains the observed data. The likelihood function provides a way to quantify the probability of the observed data given specific values of the parameters. This method is widely used in various fields, including machine learning, economics, and biological sciences.
In mathematical terms, if we have a model with parameters θ and observed data y, the likelihood function L(θ | y) is defined as the probability of observing the data y given the parameters θ. The goal of likelihood estimation is to find the parameter values that maximize this likelihood function, often referred to as the maximum likelihood estimates (MLE).
To perform likelihood estimation, one typically follows these steps: first, specify the statistical model that describes the data-generating process; next, formulate the likelihood function based on this model; and finally, use optimization techniques to find the parameter values that maximize the likelihood function. This can involve analytical methods or numerical optimization, depending on the complexity of the model.
Likelihood estimation has several desirable properties, such as consistency and asymptotic normality, which means that as the sample size increases, the estimates converge to the true parameter values and the distribution of the estimates approaches a normal distribution. However, it is important to note that likelihood estimation can be sensitive to model assumptions and the choice of the likelihood function.