maximum-likelihood-estimation

MLE is fundamentally defined as argmax_θ L(θ|x) — the likelihood function is its constitutive input. Remove likelihood and MLE has no mathematical object to maximize. Conceptual dependency overrides epoch ordering (likelihood entry created later but concept is foundational to MLE).

MLE finds θ̂ by optimizing the likelihood function L(θ|x) or equivalently its log l(θ|x). Remove likelihood-function and MLE has no mathematical object to optimize — the entire estimation procedure ceases to exist. Direction correct: MLE (epoch 0.03) depends on likelihood-function (epoch 0.01), the foundational object. The likelihood function is the constitutive mathematical ingredient of MLE.

MLE is estimation in a specific, named sense: determining an unknown parameter by maximizing the likelihood of the observed data. It passes the nearest-kind test: 'estimation' is the general procedure for determining approximate values from partial information, and no accepted intermediate kind (parameter-estimation, likelihood-based method, ...) sits between. A competent practitioner says 'MLE is a method of estimation' without qualification. INSTANCE_OF against the nearest kind (Laws 9, 11e).

MLE estimates parameters of a statistical model — the model defines the likelihood family p(x|θ). Remove statistical models and MLE has no parameter space, no likelihood family, nothing to estimate. Epoch test confirms: MLE (0.01) newer than statistical-model (0), arrow MLE → statistical-model.

Law 8d 'for whose sake?': EM is built to find maximum-likelihood (and MAP) parameter estimates in models with unobserved latent variables — that estimation task is its designed purpose, and its two steps (E-step posterior, M-step maximization of the expected complete-data log-likelihood) exist only to produce the MLE. Servant (the iterative algorithm) points at master (the estimation method it was built for). Service need not be exclusive (it also serves MAP estimation); the pinned sense is MLE as the primary designed target.

Law 8d servant points at master. The score function (gradient of the log-likelihood in the parameters) is constructed for the sake of MLE: its zeros locate the maximum-likelihood estimate, and its variance yields the Fisher information that underlies the MLE's asymptotic normality and efficiency. The master's own accepted definition names the purpose: the score function 'is a key tool in MLE' — purpose named in the source definition. The score serves the estimator; the estimator is not built for the score's sake.