##  [Likelihood Ratio](/likelihood-ratio-0) 

 Definition

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

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

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

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

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

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

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

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

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.