Likelihood function

A function of model parameters proportional to the probability of observed data under those parameters. It compares parameter values by how well they account for the observed data.

Probability mass function: A function assigning probabilities to the possible outcomes of a discrete random variable. For discrete data, likelihood is the probability mass function evaluated at the observation and treated as a function of parameters.
Statistical model: A set of probability distributions describing possible data-generating processes. A likelihood is defined by selecting a model that assigns probabilities or densities to data.
Confidence interval: A procedure that constructs a random interval intended to cover a fixed parameter at a stated frequency. Likelihood-based procedures can produce confidence intervals by identifying parameter values compatible with the data.
Bayesian probability: An interpretation of probability as a measure of uncertainty about propositions or parameters. A likelihood is not a probability distribution over parameter values; Bayesian probability supplies that interpretation through a prior and posterior.
Probability density function: A function whose integral over a region gives the probability of a continuous random variable falling there. For continuous data, likelihood uses the density at the observed value, not the probability of that exact value.
Parameter space: The set of values allowed for the parameters of a statistical model. Likelihood comparisons are made among parameter values admitted by this space.
Profile likelihood: A likelihood for a parameter of interest formed by maximizing over nuisance parameters. It isolates how well each value of a target parameter fits when other parameters are optimally adjusted.
Frequentist statistics: An approach to statistics that evaluates procedures through their behavior under repeated sampling. Likelihood can be used in frequentist methods, but it is not itself a long-run probability statement about parameters.
Log-likelihood: The natural logarithm of a likelihood function, often used to simplify calculations and optimization. Taking logarithms turns products of independent observations into sums without changing the maximizing parameter.
Random variable: A variable whose possible values are assigned probabilities by a probability model. The data are modeled as outcomes of random variables before their observed values are fixed.

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