log-likelihood

Log-likelihood is only defined relative to a statistical model: the likelihood function L(θ|data) = P(data|model, θ). Remove the statistical model and log-likelihood has no meaning — the probability function requires a model specification. Constitutive dependency (Law 8).

Log-likelihood is built for the sake of statistical inference: inference methods (MLE, likelihood ratio tests) use the likelihood function as their operative tool. For whose sake? Inference. Servant (log-likelihood) points at master (statistical-inference). Law 8d.

The likelihood function was formalized by R.A. Fisher in the 1920s, treating the probability of the data as a function of the unknown parameters. The log-likelihood is the later computational re-parameterization: the natural logarithm of that function, adopted because it turns products into sums and makes maximum-likelihood optimization numerically stable — and its own accepted definition states it is 'the natural logarithm of the likelihood function'. The likelihood came first and gives rise to the log-likelihood; hence log-likelihood DERIVED_FROM likelihood (Law 7: which came first).

log-likelihood is the pointwise logarithm of the likelihood function: l(θ|x) = log L(θ|x). The log-likelihood was derived from the likelihood function (likelihood came first at epoch 0.01 vs log-likelihood at 0.03). Direction correct: newer (log-likelihood) → older (likelihood-function). The log-likelihood is a monotone transformation used for numerical stability and additive decomposition; it is fully derived from the likelihood.

Removal test in the pinned sense: the score function U(theta;x) is defined as the gradient of the log-likelihood, U = d/dtheta log L(theta;x) (its accepted definition says so explicitly). Remove log-likelihood and U has no mathematical object to differentiate -- the score is undefined. Fisher information is then its negative expected Hessian, so the whole MLE curvature machinery runs on the log-likelihood. Direction matches the note; functional->input pattern.

Fisher information I(θ) = E[(∂/∂θ log L(θ;X))²] = -E[∂²/∂θ² log L(θ;X)] operates directly on log-likelihood. Remove log-likelihood and Fisher information has no mathematical object to differentiate or evaluate — it ceases to exist. Direction correct: Fisher information (derived concept) depends on log-likelihood (the object it operates on). Note: Fisher info can also be defined via the score function's variance, but score-function itself depends on log-likelihood.

Which-came-first (Law 7): the log-likelihood log L(θ;x) is the antecedent object of Fisher's maximum-likelihood framework, and the score is defined from it by differentiation: U(θ) = ∂/∂θ log L(θ;x). The score is the log-likelihood's parameter-derivative — derived by a named operation on the earlier object. Filed against the nearest antecedent (the log-likelihood, not the raw likelihood, since the score is the log-derivative; the log-likelihood -> likelihood link already exists on the board, so the longer leap is not filed).