Which one statement is true?

Question

BlackTom AI · Accepted Answer

To tackle this question, I will evaluate each statement in turn and connect it to the core idea of maximum likelihood estimation (MLE).
Option 1: 'MLE looks for the particular parameter according to which the sample is the most probable.' This aligns with the fundamental definition of MLE: choose the parameter value that maximizes the likelihood of observing the given sample. In other words, we select the parameter θ that makes the observed data most probable under the assumed statistical model. This option is correct.
Option 2: 'MLE looks for the particular model according to which the parameter is the most probable.' This mixes up the roles of model and parameter. MLE does not seek the model that makes a parameter most probable; rather, it seeks the parameter value within a specified model that makes the data most probable. The wording incorrectly elevates the model choice over the parameter estimation step, so this is incorrect.
Option 3: 'MLE looks for the particular sample such that according to the current model, the parameter is the most probable.' This reverses the perspective: MLE does not search for a sample; the sample is given (observed). The goal is to infer the parameter, not to select a sample that makes a parameter appear probable. This statement confuses the direction of inference and is false.
Option 4: 'MLE looks for the most probable parameter according to the model and the sample.' While this statement is close in spirit, it is somewhat ambiguous. The standard definition emphasizes choosing the parameter that maximizes the likelihood function given the observed sample and the model. Saying 'according to the model and the sample' is not incorrect in substance, but it is not as precise as Option 1. Depending on interpretation, it can be considered less precise, and in strict testing contexts, Option 1 is the clearest and most accurate phrasing.
In summary, the statements differ in precision and alignment with the core MLE idea. The first option directly states that the parameter yielding the most probable sample under the model is chosen, which captures the essence of maximum likelihood estimation, while the other options introduce conceptual mix-ups or vagueness.

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Which one statement is true?

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