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When we use AIC to select the best model, we choose the one with the lowest AIC. What is the rule in case of log likelihood?

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Likelihood will tend to be better for larger models (ones with more degrees of freedom to fit the data). In particular, for nested models with continuous parameters over the whole of the reals, adding parameters must almost surely improve the likelihood.

AIC can be seen as an attempt to adjust the bias in the expected likelihood for the fact that we maximized it over a set of parameters. As such you can regard it quite directly as an approach to comparing likelihood across models of different sizes.

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Your question is not clear. You cannot use the log likelihood to select between models because you will always get a better value of the log likelihood for bigger models. The rule is to use a penalized log likelihood: for example AIC or BIC.

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