Letter to the Editor STATISTICAL INFERENCE-BAYESIAN AND NON-BAYESIAN
To the Editor: In his interesting editorial "Statistical Inference-Bayesian and Non-Bayesian" [1], Dr. Ewens failed to mention the middle road (i.e., likelihood) which is in practice followed by many geneticists. Likelihood is fundamental to both the Bayesian and non-Bayesian positions; it is fundamental to the former because it is by means of the likelihood-function that the data change the prior distribution into the posterior distribution, using Bayes' theorem, and to the latter because the method of maximum likelihood is one of the commonest methods of estimation. Most papers on linkage evaluation, for example, now simply quote the log-likelihood function (sometimes called the support function) for the recombination fraction. Likelihood methods, apart from incorporating some of the least objectionable features of the other approaches to inference, are characteristically simple to use and have a defensible logical base of their own. For the likelihood function is essentially the probability of the data expressed as a function of the parameter (such as a recombination fraction) about which information is sought, and anyone who adopts a parameter value other than that which maximizes the likelihood has to answer the awkward question, "Why have you chosen a value which would lead to the observed data with a lower probability than does the maximum-likelihood value?" I trace the history of likelihood, and defend its use, in my book Likelihood [2], and thus need give no further details here. But I might just repeat two quotations, one from a Bayesian and one from a non-Bayesian, which indicate the respect which likelihood commands. F. P. Ramsey, who is regarded as one of the founders of subjective Bayesianism, wrote in 1928 ([2], p. 27): "In choosing a system [i.e., a hypothesis], we have to compromise between two principles: subject always to the proviso that the system must not contradict any facts we know, we choose (other things being equal) the simplest system, and (other things being equal) we choose the system which gives the highest chance to the facts we have observed. This last is Fisher's 'Principle of Maximum Likelihood' and gives the only method of verifying a system of chances." And R. A. Fisher himself, on whose work so much of the non-Bayesian theory depends, expressed his doubts about that theory in 1938 ([2] p. 100): "A worker with more intuitive insight than I might perhaps have recognized that likelihood must play in inductive reasoning a part analogous to that of probability in deductive problems." Many geneticists who are not fundamentally interested in questions of statistical inference have been led by their own excellent intuition to the use of likelihood methods; 107
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LETTER TO THE EDITOR
a little background reading will reinforce their intuition and teach them they have not been alone in appreciating its merits. A. W. F. EDWARDS Department of Community Medicine University of Cambridge Cambridge CBJ 2ES, England REFERENCES 1. EWENS WJ: Statistical inference -Bayesian and non-Bayesian. Am J Hum Genet 28:420422, 1976
2. EDWARDS AWF: Likelihood. Cambridge, Cambridge Univ. Press, 1972
To the Editor: In his interesting editorial "Statistical Inference-Bayesian and Non-Bayesian" [1], Dr. Ewens failed to mention the middle road (i.e., likelihood) which is in practice followed by many geneticists. Likelihood is fundamental to both the Bayesian and non-Bayesian positions; it is fundamental to the former because it is by means of the likelihood-function that the data change the prior distribution into the posterior distribution, using Bayes' theorem, and to the latter because the method of maximum likelihood is one of the commonest methods of estimation. Most papers on linkage evaluation, for example, now simply quote the log-likelihood function (sometimes called the support function) for the recombination fraction. Likelihood methods, apart from incorporating some of the least objectionable features of the other approaches to inference, are characteristically simple to use and have a defensible logical base of their own. For the likelihood function is essentially the probability of the data expressed as a function of the parameter (such as a recombination fraction) about which information is sought, and anyone who adopts a parameter value other than that which maximizes the likelihood has to answer the awkward question, "Why have you chosen a value which would lead to the observed data with a lower probability than does the maximum-likelihood value?" I trace the history of likelihood, and defend its use, in my book Likelihood [2], and thus need give no further details here. But I might just repeat two quotations, one from a Bayesian and one from a non-Bayesian, which indicate the respect which likelihood commands. F. P. Ramsey, who is regarded as one of the founders of subjective Bayesianism, wrote in 1928 ([2], p. 27): "In choosing a system [i.e., a hypothesis], we have to compromise between two principles: subject always to the proviso that the system must not contradict any facts we know, we choose (other things being equal) the simplest system, and (other things being equal) we choose the system which gives the highest chance to the facts we have observed. This last is Fisher's 'Principle of Maximum Likelihood' and gives the only method of verifying a system of chances." And R. A. Fisher himself, on whose work so much of the non-Bayesian theory depends, expressed his doubts about that theory in 1938 ([2] p. 100): "A worker with more intuitive insight than I might perhaps have recognized that likelihood must play in inductive reasoning a part analogous to that of probability in deductive problems." Many geneticists who are not fundamentally interested in questions of statistical inference have been led by their own excellent intuition to the use of likelihood methods; 107
108
LETTER TO THE EDITOR
a little background reading will reinforce their intuition and teach them they have not been alone in appreciating its merits. A. W. F. EDWARDS Department of Community Medicine University of Cambridge Cambridge CBJ 2ES, England REFERENCES 1. EWENS WJ: Statistical inference -Bayesian and non-Bayesian. Am J Hum Genet 28:420422, 1976
2. EDWARDS AWF: Likelihood. Cambridge, Cambridge Univ. Press, 1972
