Which one statement is true?

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BlackTom AI · Accepted Answer

When evaluating statements about Maximum Likelihood Estimation (MLE), it's helpful to recall the core idea: MLE selects the parameter value that makes the observed data most probable under the chosen model.
Option 1: 'MLE looks for the particular parameter according to which the sample is the most probable.' This is a direct and accurate articulation of the definition of MLE: maximize the likelihood function with respect to the parameter given the observed sample. It captures both the dependency on the parameter and the data.
Option 2: 'MLE looks for the particular model according to which the parameter is the most probable.' Here, the focus shifts from parameters to the model itself. In MLE, we are not selecting a model to maximize the probability of a parameter; rather, we are selecting the parameter values within a fixed model class that maximize the data likelihood. This makes the statement inaccurate.
Option 3: 'MLE looks for the particular sample such that according to the current model, the parameter is the most probable.' This reverses the relationship: the sample (data) is what we observe and cannot be chosen to maximize the probability of a parameter. MLE treats the sample as fixed and seeks the parameter that makes that fixed sample most probable.
Option 4: 'MLE looks for the most probable parameter according to the model and the sample.' Although the wording might feel close, this statement is ambiguous and potentially misleading. It suggests a notion of the parameter being inferred as most probable in a way that can resemble a Bayesian posterior mode, where priors influence 'most probable' parameters. In standard frequentist MLE, we maximize the likelihood with respect to the parameter for the given model and observed sample, not by declaring the parameter 'most probable' in a probabilistic sense conditional on both model and data in a Bayesian sense. Therefore this option is not the conventional MLE formulation.
In summary, Option 1 cleanly and correctly states the MLE objective, while Options 2 and 3 misplace the roles of model, data, and parameter, and Option 4, though similar in spirit to the correct idea, uses wording that can mislead about the role of probability in parameter estimation within the MLE framework.

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Which one statement is true?

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