likelihood

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Relations to this entry

• maximum-likelihood-estimation← DEPENDS_ON

  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).

• expectation-maximization← DEPENDS_ON

  EM's E-step computes Q(θ|θ^(t)) = E[log p(x,z|θ)] — the expected complete-data log-likelihood. The M-step maximizes this over θ. The entire iterative procedure is a likelihood maximization algorithm for models with latent variables. Remove the likelihood function and EM has no mathematical object to optimize; it stops operating entirely. Constitutive present-tense dependency (Law 8b/8c).

• marginal-likelihood← DERIVED_FROM

  Marginal likelihood p(x) = ∫ p(x|θ)π(θ)dθ is derived from the likelihood function p(x|θ) by integrating over the parameter space weighted by the prior. Likelihood was formalized by Fisher in the 1920s; marginal likelihood (model evidence) came later as the Bayesian extension. The likelihood is the base construct; marginal likelihood is its integrated variant.

• fisher-information← DERIVED_FROM

  Fisher information was derived from studying the likelihood function: it quantifies the curvature of the log-likelihood, measuring how much information the observable random variable carries about an unknown parameter. Fisher (1925) derived this from the likelihood's second derivative — which-came-first: likelihood function predates Fisher information.

• sufficient-statistic← DEPENDS_ON

  A sufficient statistic T(X) is defined by the factorization P(X|theta) = g(T(X),theta)·h(X) — the likelihood function. Remove likelihood and the concept of sufficiency has no mechanism to operate on. Present-tense operational necessity: sufficient statistics extract information from likelihood, not from data alone.

• log-likelihood← DERIVED_FROM

  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).