Which one statement is true?

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BlackTom AI · Accepted Answer

Let's begin by identifying what the question is asking: which statement correctly describes the Maximum Likelihood Estimation (MLE) principle.
Option 1: 'MLE looks for the particular parameter according to which the sample is the most probable.' This is a precise restatement of the core idea of MLE: choose the parameter value that maximizes the likelihood of the observed data. It directly ties the parameter to the probability of the sample under the model.
Option 2: 'MLE looks for the particular model according to which the parameter is the most probable.' This misplaces the focus: MLE searches over parameter values, not over entire models, and it doesn't seek a model for which the parameter is most probable in that sense. The phrasing swaps the roles of model and parameter, causing conceptual confusion.
Option 3: 'MLE looks for the particular sample such that according to the current model, the parameter is the most probable.' This reverses the causal order: MLE does not search for a sample; we already have a sample (data) and we seek the parameter that makes this sample most probable under the model. The wording incorrectly makes the sample the variable to be chosen.
Option 4: 'MLE looks for the most probable parameter according to the model and the sample.' While it hints at maximizing a probability given data and model, its wording is less precise and potentially misleading: it suggests a two-sided consideration without explicitly stating maximizing the likelihood p(x|theta) over theta. The standard definition is more clearly captured in option 1.
In summary, option 1 accurately captures the essence of MLE by identifying the parameter that makes the observed data most probable under the model. The other options either misplace the roles of parameters and models, invert the order of inference, or provide vaguer language that does not precisely convey the MLE objective.

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Which one statement is true?

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