#### Likelihood Ratio Technique

I wonder what Fisher, Neyman, and Pearson would say if they see “Technique” after “Likelihood Ratio” instead of “Test.” A presenter’s saying “Likelihood Ratio Technique” for source identification, I couldn’t resist checking it out not to offend founding fathers of the likelihood principle in statistics since “Technique” sounded derogatory to be attached with “Likelihood” to my ears. I thank, above all, the speaker who kindly gave me the reference about this likelihood ratio technique.

On the likelihood ratio for source identification by Sutherland and Saunders (1992) in MNRAS vol. 259, pp. 413-420.

Their computed likelihood ratio (L) correspond to Bayes factor by the form (P(source model)/P(background model)). Considering the fact that it’s binary, source or background, L shares the form of a hazard ratio (L=p(source)/p(not source)=p(source)/(1-p(source)). Since the likelihood can be based on probability density function, the authors defined “Likelihood ratio” literally by taking the ratio of two likelihood functions. Not taking the statistical direction as in the likelihood ratio test and the Neyman-Pearson lemma, naming their method as “likelihood ratio technique” seems proper, and it’s not derogatory any more. The focus of the paper is that estimating the probability density functions of backgrounds and sources more or less empirically without concerns toward general statistical inference. Hitherto, the large Bayes factor, large L (likelihood ratio) of a source, or large posterior probability of a source (p(genuine|m,c,x,y)=L/(1+L)) is just an indicator that the given source is more likely a real source.

In summary, the likelihoods of source and of background (of numerator and of denominator) are empirically obtained based on physics which turned out to have matching parametric distributions well discussed in statistics. What is different from statistics is that the likelihood ratio didn’t lead to testing hypothesis based on Neyman-Pearson Lemma. Computing the likelihood ratio is utilized as an indicator of a source. Well, often times, it’s hard to judge the real content of an astronomical study by its name, title, or abstract due to my statistically oriented stereotypes.

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