Which one statement is true?

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BlackTom AI · Accepted Answer

Question: Which one statement is true?

Option 1: 'MLE looks for the particular parameter according to which the sample is the most probable.' This is a precise articulation of the core idea behind maximum likelihood estimation: we seek the parameter value that makes the observed data most probable under the assumed model. It directly connects the parameter to the likelihood of the sample, which is the standard formulation of MLE.

Option 2: 'MLE looks for the particular model according to which the parameter is the most probable.' This reverses the usual relationship. In MLE, we estimate the parameter values given a fixed model, not the model itself by evaluating which model makes the parameter most probable. This misstates the order of estimation: models are typically specified first, parameters are estimated within that model.

Option 3: 'MLE looks for the particular sample such that according to the current model, the parameter is the most probable.' This is incorrect because the sample is the observed data, not something we search for. MLE considers the given sample and searches for the parameter value that would maximize the likelihood; it does not try to select a sample that would make a parameter appear most probable.

Option 4: 'MLE looks for the most probable parameter according to the model and the sample.' At first glance this seems plausible, since likelihood combines the model and the data. However, the phrasing is a bit circular and could be interpreted as optimizing over both model and parameter simultaneously, whereas standard MLE fixes the model and searches for the parameter that maximizes the likelihood of the observed data under that model. It introduces ambiguity about whether the model is being chosen or the data itself is treated as random for the optimization.

Overall, Option 1 cleanly and correctly states the essential goal of MLE: choose the parameter value that makes the observed sample most probable under the given model. The other options introduce conceptual confusions about the roles of the model, the parameter, and the data in the maximum likelihood framework.

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Which one statement is true?

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