The ratio of likelihoods of observed data under two competing statistical models or hypotheses; used as a test statistic and as a measure for model comparison.

Principle

Neyman–Pearson framework: for simple hypotheses the likelihood ratio yields the most powerful test at a given significance level; monotone likelihood ratio properties underpin ordering of evidence.

Demonstration

Given observations from N(μ,σ^2) with known σ, testing H0: μ=μ0 versus H1: μ=μ1 uses LR = L(data|μ1)/L(data|μ0); the log‑likelihood ratio scales with the sample mean difference and forms the rejection criterion.

Misapplication

Interpreting the likelihood ratio as the posterior probability of a hypothesis without incorporating priors (confusing LR with Bayes factor or posterior odds), or applying it when models are not properly specified or likelihoods undefined for parts of sample space.

Consequence

Provides an interpretable test statistic with known asymptotic distributions in many cases (e.g., Wilks' theorem), gives a basis for likelihood‑based confidence regions and model selection procedures.

Reversal

Bayes factors use marginal likelihoods that integrate over parameter priors rather than pointwise likelihoods; inverting roles of hypotheses swaps numerator and denominator and reverses the evidence interpretation.

Boundary

Requires well‑specified likelihood functions on the same sample space and care with nuisance parameters and parameter identifiability; undefined when denominators vanish or when models use incompatible data-generating assumptions.

Semantic Tension

Often confused with p‑values, posterior odds or Bayes factors; unlike p‑values LR compares models directly via likelihoods, and unlike Bayes factors it omits prior weighting unless extended.

Synthesis

The likelihood ratio is the core likelihood‑based measure comparing how well two hypotheses explain observed data; when used with appropriate calibration it yields powerful tests, guides model choice and quantifies relative support.