A Good Mind

In statistics, a likelihood function is a function that measures the goodness of fit of a statistical model to a set of data. It is defined as the probability of observing the given data, assuming a specific model and parameter values. The likelihood function is a crucial component of maximum likelihood estimation, a widely used method for estimating the parameters of statistical models.

The likelihood function is based on the assumption that the data are independent and identically distributed, meaning that each data point is unrelated to the others and has the same underlying probability distribution. The function takes as input the parameter values of the model and outputs the probability of observing the given data for those parameter values.

The likelihood function can be used to compare different statistical models and choose the best one based on how well it fits the data. The maximum likelihood estimator is the set of parameter values that maximises the likelihood function, and it provides a point estimate of the true parameter values of the model.

In practise, the likelihood function is often difficult to calculate analytically, and numerical methods such as gradient descent and Newton-Raphson algorithms are used to find the maximum likelihood estimator. The likelihood function can also be used to perform hypothesis testing and calculate confidence intervals for the estimated parameters.