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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.
In statistics, the likelihood-ratio test is a hypothesis test that involves comparing the goodness of fit of two competing statistical models, typically one found by maximization over the entire parameter space and another found after imposing some constraint, based on the ratio of their likelihoods.
Similarly, likelihoods are often transformed to the log scale, and the corresponding log-likelihood can be interpreted as the degree to which an event supports a statistical model. The log probability is widely used in implementations of computations with probability, and is studied as a concept in its own right in some applications of ...
In statistics, the score (or informant [1]) is the gradient of the log-likelihood function with respect to the parameter vector. Evaluated at a particular value of the parameter vector, the score indicates the steepness of the log-likelihood function and thereby the sensitivity to infinitesimal changes to the parameter
Recall that for the multinomial model, the MLE of ^ given some data is defined by ^ = Furthermore, we may represent each null hypothesis parameter ~ as ~ = Thus, by substituting the representations of ~ and ^ in the log-likelihood ratio, the equation simplifies to ((~ |) (^ |)) = = = = Relabel the variables with and with .
The log-likelihood that a particular set of K measurements or data points will be generated by the above probabilities can now be calculated. Indexing each measurement by k , let the k -th set of measured explanatory variables be denoted by x k {\displaystyle {\boldsymbol {x}}_{k}} and their categorical outcomes be denoted by y k {\displaystyle ...
The log-likelihood of a normal variable is simply the log of its probability density function: = (). Since this is a scaled and shifted square of a standard normal variable, it is distributed as a scaled and shifted chi-squared variable.
Maximum likelihood estimation is a generic technique for estimating the unknown parameters in a statistical model by constructing a log-likelihood function corresponding to the joint distribution of the data, then maximizing this function over all possible parameter values. In order to apply this method, we have to make an assumption about the ...