likelihood-function

The likelihood function L(θ|x) = p(x|θ) IS the statistical model evaluated at the observed data. Remove the statistical model and the likelihood has no mathematical definition — there is no p(x|θ) without a model family. The model specifies the parametric family; the likelihood is the model read as a function of θ for fixed x. Operational cessation: without the model, the likelihood ceases to exist as a mathematical object. Direction correct: likelihood is the derived object, model is the foundational one.

Pinned senses against both accepted carves (Law 11d): likelihood-function's carve is 'a human-made function L(theta|x) = p(x|theta) that maps each parameter value theta to the probability (or probability density) of observing the fixed data x under that parameter', with domain the parameter space, codomain the non-negative reals, and the rule fixing the model family at x. That is function's carve stated verbatim: a domain, a codomain, an assignment rule, and single-valuedness (exactly one L(theta|x) per theta). Law 9 test: a competent speaker calls the likelihood a function - its name carries the kind. No nearer accepted kind exists on the board (no parameter-space-mapping entry), so function is the nearest rung; not a ladder leap.

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.

Posterior distribution p(θ|x) is computed via Bayes rule as the product of likelihood p(x|θ) and prior p(θ). Remove the likelihood function and the posterior has no data-feeding term — Bayes rule cannot operate. This is an operational dependency, not merely conceptual association.

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.

Predictive-distribution computes the probability of new data by integrating over parameter uncertainty, marginalizing the likelihood-function against the posterior. Remove the likelihood-function and predictive-distribution has no data-model to predict from — constitutive dependency per Law 8.

The score function U(θ;x) = ∂/∂θ log L(θ;x) operates on the likelihood function. Remove the likelihood and the score function has no mathematical object to differentiate — it ceases to exist. The log-likelihood is a transformation of the likelihood, but the score's fundamental dependency is on the likelihood itself.