Maximum likelihood equation helps in estimation of most probable values of the estimator’s predictor variable coefficients which produces results which are the most likely or most probable and are quite close to the truth values.
In logistic regression, the maximum likelihood estimation (MLE) is used to determine the optimal values for the model parameters. The objective is to find the parameter values that maximize the likelihood of observing the given set of outcomes in the training data.
Here’s a more detailed explanation:
- Likelihood Function:
- Logistic regression models the probability of a binary outcome (0 or 1) given a set of input features. The likelihood function represents the probability of observing the actual outcomes in the training data, given the input features and the model parameters.
- The likelihood function for logistic regression is derived from the logistic distribution, and it provides the probability of the observed outcomes.
- Maximizing Likelihood:
- The goal is to find the values of the model parameters that maximize the likelihood function. This is achieved through an optimization process.
- Since working directly with probabilities can lead to very small values (and potential numerical instability), the optimization often involves working with the logarithm of the likelihood, referred to as the log-likelihood.
- Log-Likelihood and Cost Function:
- The log-likelihood is used as the objective function to be maximized. Alternatively, in machine learning, the negative log-likelihood is often used as a cost function to be minimized. This is equivalent to minimizing the cross-entropy loss.
- Minimizing the negative log-likelihood is equivalent to maximizing the likelihood of the observed outcomes.
- Optimization Techniques:
- Various optimization algorithms, such as gradient descent or its variants, are employed to find the parameter values that maximize the likelihood.
- The optimization process involves iteratively updating the parameters in the direction that increases the likelihood until convergence is reached.
In summary, the role of maximum likelihood in logistic regression is to determine the parameter values that make the observed outcomes in the training data most probable according to the logistic regression model.