# generalized likelihood ratio test (GLRT)

I am having trouble in understanding the generalized likelihood ratio test (GLRT). Can anyone explain what it is to me, or point me toward an easy-to-understand reference? Is it a supervised or unsupervised method? How GLRT relates to Bayesian approaches?

Likelihood-ratio tests are a mainstay of classical hypothesis testing. The idea is to form the likelihoods of the two hypotheses under consideration, and choose the one with the highest likelihood if their ratio is sufficiently large. Hypotheses come in two flavors: simple, and composite. Simple tests are those for which the hypothesis uniquely defines the distribution; e.g., the mean is that, or the variance is that. If it is not simple, it is composite; e.g., the mean is not equal to something, or the variance is less than something. Generalized likelihood ratio tests apply to composite hypotheses, and the goal is to find the distributions out of all possible options in the hypothesis space that maximize the likelihoods, and consider their ratio.

In Bayesian statistics one considers the posterior probabilities instead of the likehoods, and the corresponding ratio is termed the Bayes factor.

Supervised/unsupervised learning is something else.

I suggest you consult a textbook for worked examples; I think my explanation does not do the subject justice.

• Thank you so much. By supervised, I mean does GLRT construct a model from historical data and learn distributions/parameters from them and then is applied to a new test record to decide about hypothesis? Because I see that this is used for change point detection. If it works just online and does not need to be integrated with other methods and data, it should raise many false positive in detecting that change point. Does it have any assumption or threshold selection which makes it to be improper in settings that data distributions change periodically? – Arkan Aug 6 '16 at 5:51
• (G)LRTs require knowledge of the likelihood function; not estimates from the data. I suppose you could do that but I don't know what guarantees there are for its power. If you want to find out more the key word is empirical likelihood ratio; cf. e.g. Empirical likelihood ratio test for the change-point problem, or Estimation and hypothesis testing in nonstationary time series. – Emre Aug 6 '16 at 6:07
• Thank you for the link. Can you let me know what are the inputs of that function and how obtained (obtain from a training data)? However, overall I mean that does this GLRT is parametric and need to set some thresholds or any prior assumption about data distribution? – Arkan Aug 6 '16 at 6:23
• The inputs to the GLRT are the likelihood functions, the hypotheses, and the ratio threshold. – Emre Aug 6 '16 at 18:11
• The likelihood is a function of the observed outcomes and distribution parameters; you just take the variates as fixed. – Emre Aug 6 '16 at 20:40

The GLRT is normally applied to a 1) composite hypothesis 2) two or more parameter density functions