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What is Log-likelihood?
An example would be great.
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The log-likelihood value of a regression model is a way to measure the goodness of fit for a model. The higher the value of the log-likelihood, the better a model fits a dataset. The log-likelihood value for a given model can range from negative infinity to positive infinity.
It's a cost function that is used as loss for machine learning models, telling us how bad it's performing, the lower the better.
Application & Interpretation: Log Likelihood value is a measure of goodness of fit for any model. Higher the value, better is the model. We should remember that Log Likelihood can lie between -Inf to +Inf. Hence, the absolute look at the value cannot give any indication.
For logistic regression, there is no longer a closed-form solution, due to the nonlinearity of the logistic sigmoid function. The departure from a quadratic form is not substantial. The error function is convex and hence has a unique minimum.
The only reason to use the log-likelihood instead of the plain old likelihood is mathematical convenience, because it lets you turn multiplication into addition. The plain old likelihood is P(parameters | data), i.e. assuming your data is fixed and you vary the parameters of your model. Maximizing this is one way to do parameter estimation and is known as maximum likelihood.
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Log-likelihood ratio
A likelihood-ratio test is a statistical test relying on a test statistic computed by taking the ratio of the maximum value of the likelihood function under the constraint of the null hypothesis to the maximum with that constraint relaxed. If that ratio is Λ and the null hypothesis holds, then for commonly occurring families of probability distributions, −2 log Λ has a particularly handy asymptotic distribution. Many common test statistics such as the Z-test, the F-test and Pearson's chi-square test can be phrased as log-likelihood ratios or approximations thereof.
http://www.knowledgerush.com/kr/encyclopedia/Log-likelihood_ratio/
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