Likelihood function

In the likelihood function (of English. Likelihood, likelihood ") is a mathematical function that is used in the maximum likelihood method to estimate the parameters of a probability density or function. The log-likelihood function is the log-likelihood function.

Definition

Denote a random variable with an associated density or probability function. Here is a (possibly multidimensional ) of unknown parameters. Furthermore, are various realizations of these random variables. The likelihood function of this sample is defined as the function that each parameter value the value

So the joint density or probability function assigns.

Log- likelihood function

The log-likelihood function is the log-likelihood function. It is usually used in connection with the determination of a maximum likelihood estimation. For determining the maximum of the likelihood function, the first and second derivatives need to be calculated according to, and the zeros of the first derivative is determined. To facilitate the calculation of these derivatives, using the log-likelihood function,

Since, in many cases, the derivative of the log-likelihood to determine by simple, according as the derivative of the likelihood. In general: If a maximum of the log-likelihood function, it is also a maximum of the likelihood function, and vice versa.

A significant simplification in the use of the log-likelihood, for example, when independent realizations of the random variables. Then the likelihood function is obtained as

That is, as the product of one-dimensional density or probability functions. The log-likelihood is obtained in this case by the sum of

Example

The normal distribution with mean and variance has the probability density

Now, are independent realizations of a normal distribution with unknown mean and variance, we obtain the corresponding likelihood function with

And the log - likelihood function to

Pseudo- likelihood function

For solving the maximum likelihood problem, only to find the maximum of the likelihood function of concern. This is one of the reasons why the maximum likelihood method often works even though the conditions are not met. In the following cases it is called a pseudo - likelihood function:

• Distribution conditions for the maximum likelihood method is not satisfied: then it is called the likelihood function, a pseudo - likelihood function, and
• The actual likelihood function or log-likelihood function is to difficult to maximize and is replaced by a smoothed version, for example, and this pseudo- likelihood function is maximized.