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!

racism, discrimination and statistics – examining the history [at the RSS]

The Royal Statistical Society is holding an on-line round table on “Racism, discrimination and statistics – examining the history” on 30 October, at 4pm UK time. The chair is RSS President Deborah Ashby and the speakers are

• John Aldrich – chair of the RSS History Section
• Angela Saini – science journalist
• Stephen Senn – Fisher Memorial Trust

in the name of eugenics [book review]

In preparation for the JSM round table on eugenics and statistics, organised by the COPSS Award Committee, I read the 1985 book of Daniel Kevles, In the Name of Eugenics: Genetics and the Uses of Human Heredity, as recommended by Stephen Stiegler. While a large part of the book was published in The New Yorker, in which Kevles published on a regular basis, and while he abstains from advanced methodological descriptions, focussing more on the actors of this first attempt at human genetics and of the societal consequences of biased interpretations and mistaken theories, his book is a scholarly accomplishment, with a massive section of notes and numerous references. This is a comparative history of eugenics from the earliest (Francis Galton, 1865) to the current days (1984) since “modern eugenics” survived the exposure of the Nazi crimes (including imposed sterilizations that are still enforced to this day). Comparative between the UK and the US, however, hardly considering other countries, except for a few connections with Germany and the Soviet Union, albeit in the sole perspective of Muller’s sojourn there and the uneasy “open-minded” approach to Lysenkoism by Haldane. (Japan is also mentioned in connection with Neel’s study of the genetic impact of the atomic bombs.) While discussing the broader picture, the book mostly concentrates on the scientific aspects, on how the misguided attempts to reduce intelligence to IQ tests or to a single gene, and to improve humanity (or some of its subgroups) by State imposed policies perceived as crude genetic engineering simultaneously led to modern genetics and a refutation of eugenic perspectives by most if not all. There is very little about statistical methodology per, beside stories on the creation of Biometrika and the Annals of Eugenics, but much more on the accumulation of data by eugenic societies and the exploitation of this data for ideological purposes. Galton and Pearson get the lion’s share of the book, while Fisher does not get more coverage than Haldane or Penrose. Overall, I found the book immensely informative as exposing the diversity of scientific and pseudo-scientific viewpoints within eugenism and its evolution towards human genetics as a scientific endeavour.

a conversation about eugenism at JSM

Following the recent debate on Fisher’s involvement in eugenics (and the renaming of the R.A. Fisher Award and Lectureship into the COPSS Distinguished Achievement Award and Lectureship), the ASA is running a JSM round table on Eugenics and its connections with statistics, to which I had been invited, along with Scarlett Bellamy, David Bellhouse, and David Cutler. The discussion is planned on 06 August at 3pm (ET, i.e., 7GMT) and here is the abstract:

The development of eugenics and modern statistical theory are inextricably entwined in history.  Their evolution was guided by the culture and societal values of scholars (and the ruling class) of their time through and including today.  Motivated by current-day societal reckonings of systemic injustice and inequity, this roundtable panel explores the role of prominent statisticians and of statistics more broadly in the development of eugenics at its inception and over the past century.  Leveraging a diverse panel, the discussions seek to shed light on how eugenics and statistics – despite their entangled past — have now severed, continue to have presence in ways that affect our lives and aspirations.

It is actually rather unclear to me why I was invited at the table, apart from my amateur interest in the history of statistics. On a highly personal level, I remember being introduced to Galton’s racial theories during my first course on probability, in 1982, by Prof Ogier, who always used historical anecdotes to enliven his lectures, like Galton trying to measure women mensurations during his South Africa expedition. Lectures that took place in the INSEE building, boulevard Adolphe Pinard in Paris, with said Adolphe Pinard being a founding member of the French Eugenics Society in 1913.

stained glass to go