why is the likelihood not a pdf?

The return of an old debate on X validated. Can the likelihood be a pdf?! Even though there exist cases where a [version of the] likelihood function shows such a symmetry between the sufficient statistic and the parameter, as e.g. in the Normal mean model, that they are somewhat exchangeable w.r.t. the same measure, the question is somewhat meaningless for a number of reasons that we can all link to Ronald Fisher:

  1. when defining the likelihood function, Fisher (in his 1912 undergraduate memoir!) warns against integrating it w.r.t. the parameter: “the integration with respect to m is illegitimate and has no definite meaning with respect to inverse probability”. The likelihood is “is a relative probability only, suitable to compare point with point, but incapable of being interpreted as a probability distribution over a region, or of giving any estimate of absolute probability.” And again in 1922: “[the likelihood] is not a differential element, and is incapable of being integrated: it is assigned to a particular point of the range of variation, not to a particular element of it”.
  2. He introduced the term “likelihood” especially to avoid the confusion: “I perceive that the word probability is wrongly used in such a connection: probability is a ratio of frequencies, and about the frequencies of such values we can know nothing whatever (…) I suggest that we may speak without confusion of the likelihood of one value of p being thrice the likelihood of another (…) likelihood is not here used loosely as a synonym of probability, but simply to express the relative frequencies with which such values of the hypothetical quantity p would in fact yield the observed sample”.
  3. Another point he makes repeatedly (both in 1912 and 1922) is the lack of invariance of the probability measure obtained by attaching a dθ to the likelihood function L(θ) and normalising it into a density: while the likelihood “is entirely unchanged by any [one-to-one] transformation”, this definition of a probability distribution is not. Fisher actually distanced himself from a Bayesian “uniform prior” throughout the 1920’s.

which sums up as the urge to never neglect the dominating measure!

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