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The log-likelihood function being plotted is used in the computation of the score (the gradient of the log-likelihood) and Fisher information (the curvature of the log-likelihood). Thus, the graph has a direct interpretation in the context of maximum likelihood estimation and likelihood-ratio tests.
For logistic regression, the measure of goodness-of-fit is the likelihood function L, or its logarithm, the log-likelihood ℓ. The likelihood function L is analogous to the ε 2 {\displaystyle \varepsilon ^{2}} in the linear regression case, except that the likelihood is maximized rather than minimized.
In statistics, maximum likelihood estimation (MLE) is a method of estimating the parameters of an assumed probability distribution, given some observed data.This is achieved by maximizing a likelihood function so that, under the assumed statistical model, the observed data is most probable.
Maximum likelihood estimation (MLE) is a standard statistical tool for finding parameter values (e.g. the unmixing matrix ) that provide the best fit of some data (e.g., the extracted signals ) to a given a model (e.g., the assumed joint probability density function (pdf) of source signals).
[16]: 308 By finding the zero of the derivative of the log-likelihood function when the distribution is defined over , the maximum likelihood estimator can be found to be ^ = ¯, where ¯ is the sample mean. [18]
The use of log probabilities improves numerical stability, when the probabilities are very small, because of the way in which computers approximate real numbers. [1] Simplicity. Many probability distributions have an exponential form. Taking the log of these distributions eliminates the exponential function, unwrapping the exponent.
Specifically, the likelihood function is maximized subject to the constraints imposed by the model, and expressed in terms of the additional variables that describe the model's structure. This approach requires powerful optimization software such as Artelys Knitro because of the high dimensionality of the optimization problem.
[1] [6] In a slightly different formulation suited to the use of log-likelihoods (see Wilks' theorem), the test statistic is twice the difference in log-likelihoods and the probability distribution of the test statistic is approximately a chi-squared distribution with degrees-of-freedom (df) equal to the difference in df-s between the two ...