Pompilj continued to undertake research at the University of Rome and, in 1942, he was appointed as a lecturer. However, Italy had entered World War II on the side of Germany in June 1940 with the initial aim of expanding its colonies in North Africa. Pompilj was called up for military service and, after being captured by the British, was interred in a POW camp in North Africa. From there he was transferred to the POW camp at Yol in India. Yol, so called after being the site of the Young Officers Leave camp established in 1849, is situated 10 km southeast of Dharamsala and about 15 km northeast of Kangra town. Conditions in the camp were good and there was a small library in which Pompilj found Alec Aitken's book Statistical Mathematics (1939). There was another Italian mathematician held as a prisoner at Yol who had been undertaking research in mathematical statistics for his thesis before being called up for military service. He was able to continue to work on his thesis at Yol and Pompilj was pleased to be able to help him. This was a rather unusual way for an expert in algebraic geometry to become interested in statistics but this was to lead to Pompilj becoming one of the leading statisticians in Italy after the war ended.
In  Giorgio Dall'Aglio writes about the background:-
Thus Pompilj encountered mathematical statistics; a providential contact, which would have been difficult in his country. The Italian statistical school was then dominated by Corrado Gini, who was critical about the development that mathematical statistics was taking in Anglo-Saxon countries, and drove the Italian school towards descriptive statistics. This direction was also favoured by the nationalism of the Fascist period and by the separation induced by the war. So there was in Italy a deep ditch between statistics and probability, although the latter had famous scholars like Francesco Paolo Cantelli and Bruno de Finetti.However, back in Italy after release from the POW camp in India in 1946, Pompilj's publications over the next couple of years were mainly on geometry except for the paper Sulla regressione Ⓣ (1946). In fact he published eight papers in 1946, the other seven being on geometry. In 1947 he published two papers, one on geometry and one on statistics; see THIS LINK.
In 1948 Pompilj entered a competition for a chair of geometry at the University of Rome. He was successful and appointed as professor of geometry. However, in the same year he began teaching statistics in addition to his geometry teaching. Giorgio Dall'Aglio explains in the interview  how Pompilj, as a professor of geometry, began teaching statistics:-
An encounter with Corrado Gini definitively changed the course of his scientific and academic life. Gini was an eminent statistician who, due to the autarchic attitude of Fascism and the war, knew little about and did not appreciate the huge developments abroad. However, he recommended the hiring of Pompilj at the Faculty of Statistics in 1948. It is interesting to note that though Pompilj was officially assigned the chair of geometry, he in fact had to teach probability (which previously was taught only to students of actuarial science) and mathematical statistics. His main accomplishment no doubt is that he was able to foster the academic developments of these new, emerging fields of mathematics.We note that in the 1950s, Pompilj was teaching both statistics and geometry courses at the University of Rome. For the rest of this biography we follow .
Thanks to his mathematical training, Pompilj had the skills to reduce the gap between mathematics and statistics that had been created in Italy. He brought Italian research in statistics closer to those of other countries, especially the Anglo-Saxon ones, reformulating and refining, in the context of the theory of random variables, many of the criticisms that Corrado Gini had to the significance of the theory.
He constantly strove to spread statistical methods for data processing, paying particular attention to the correctness of the interpretation, to the discussion of common and often subtle errors in the applications, to assumptions that were not adequately clarified with the resulting ambiguity in interpretation. He expended great effort in teaching, not only in the Faculty of Statistical Sciences, but also at non-university organisations and institutions, including the Italian Association for Market Studies. At the Central Institute of Statistics, he organised courses in statistical methodology for researchers and employees of public and private corporations, involving eminent teachers and organising the publication of the lectures. He made fundamental contributions to the setting up of the sample surveys carried out Central Institute of Statistics. He created the Istituto di Calcolo delle Probabilità in the Faculty of Statistical Sciences, which became an authoritative reference for all research that required assistance in the proper use of statistical procedures.
Pompilj was also a pioneer in the field of operational research, developing the first significant applications in collaboration with the Italian Navy. He was among the founders of the Italian Association of Operational Research and he created the School of Advanced Operational Research in the Faculty of Statistical Sciences in the University of Rome. As the result of collaboration between the Institute of the Calculus of Probability and the Municipality of Rome a ground-breaking study of the traffic in Rome took shape. Sadly, it was immediately shelved and forgotten, a demonstration of the fact that "mathematical treatment is not made for managers with little skill" [G Pompilj, Funzione e limiti della matematizzazione Ⓣ, in Studi di mercato, II (1962), page14]. For a list of works by Pompilj, see THIS LINK.
He constantly worked to make the successes of the Italian statistical school known abroad, which was unjustly neglected. An unfinished treatise on random variables, composed during a research visit to Pittsburgh, was destined to "gather together in a single exposition that took into account the various contributions of various schools, the attained results from some researchers (and particularly those of the Rome Institute of the Calculus of Probability) in developing, reordering and completing certain contributions of C Gini to the general theory of distributions" [G Pompilj, Le variabili casuali Fasc. I: Assiomatizzazione del calcolo delle probabilità Ⓣ (Rome 1967), page 3].
The first of Pompilj's research contributions was in the field of algebraic geometry and continued themes dear to the Italian school, already studied by Guido Castelnuovo, Federigo Enriques and Francesco Severi, who he had as teachers in Rome. The topics covered by Pompilj were principally: multiple planes; Cremona transformations of the plane that possess special curves of fixed points; and families of hyperelliptical or trigonal surfaces. He also investigated Abelian varieties from an algebraic point of view, functional equivalence, surfaces whose canonical system is degenerate, and irregular surfaces. He was, together with Alfredo Franchetta (1916-2011), the last student of Enriques. We note that Franchetta continued to undertake research in algebraic geometry and taught at the University of Rome, the University of Turin, the University of Palermo and finally at the University of Naples.
Following the change in his research interests :-
... it was said that the desertion of Pompilj had resulted in a grave loss for geometry. But we must take into account the contribution that he has given to the fields in favour of which he neglected his early studies, both with research and with his work on promotion and dissemination. And one can certainly say that what he has done, leaves little room to regret what he could have done in other fields.In his first statistical work, Sulla regressione Ⓣ, in Rendiconti di matematica e delle sue applicazioni (5) 5 (1946), 186-219, he began to develop a geometrical approach to the theory of random variables to which he returned several times in the following years. He wrote in Teoria affine delle variabili casuali Ⓣ in L'Industria X (1956), page 143:-
The geometric approach to which I refer, leads to studying the properties of random variables that are invariant under the action of some subgroup or whole group of affinities.Formulated in this way, the study of the properties of a finite set of random variables is part of the famous Erlangen Program of Felix Klein. This is the most obvious link between the new and the old interests of Pompilj. Thanks to this geometric point of view, Pompilj, together with contributions from some of his students, was able to interpret and to extend, in the context of the theory of random variables, the statistical indicators introduced by Gini, and to deepen his ideas on the theory of distributions.
In the work Teoria statistica della significatività e conformità dei risultati sperimentali agli schemi teorici Ⓣ, in Statistica, Milano 8 (1948), 7-42, and Sulla significatività delle costanti statistiche Ⓣ, in Bollettino dell'Unione matematica italiana (3) 4 (1949), 112-117, Pompilj formulated a 'theory of conformity', an alternative to that of significance. According to Pompilj it does not make sense to consider the problem of plausibility or significance of a statistical model because, as Pompilj writes in Logica della conformità Ⓣ, in Archimede IV (1952), page 27:-
... it is not legitimate to consider whether a model is true or false since you can always say that, strictly speaking, every model is false, because it does not coincide with reality.His criticisms begin from a detailed analysis of the meaning and implications of Bayes' Theorem, the foundation of the statistical induction process. Pompilj's main criticism of the theory of significance, taken from those of Gini, is to confuse the 'a posteriori' probability P(A | B) with the probative P(B | A). His theory of conformity develops the minimum inferential theory necessary for applications to, in particular, experimental design theory . Unlike the theory of significance, the theory of conformity is limited to measuring the conformity of the data collected under a certain hypothesis, refraining from giving an opinion on the probability of the hypothesis itself :-
It was claimed that it is a step backwards when compared to other concepts, forsaking, therewith, arriving at conclusions. But as these conclusions are based on personal evaluations, not widely shared, one has the opportunity not to insist upon them.For Pompilj it is necessary to depart from Bayes' Theorem to also clarify the mechanisms that operate in the interpretation of facts. In Lineamenti di una teorica della persuasione Ⓣ, in Archimede III (1951), 135-143, he discusses numerous examples of how:-
... the same facts are taken, depending on the initial ideas, in support of contrasting hypothesises ... The fact is that in many cases one has good information, objective so to speak, on the probative probabilities, while nothing is known, or almost nothing, on the 'a priori' probabilities, which are then, in turn, by each fixed in a more subjective way.In other words, citing Sei personaggi in cerca d'autore Ⓣ by Luigi Pirandello, "facts are like an empty sack that does not stand without interpretation" and as the interpretation is all in the 'a priori' judgements that we assign to the different interpretative hypothesises, that is why the same facts can be mathematically evaluated in many different ways.
In his concept of probability, Pompilj was close to a subjective concept, but did not completely marry it to his theses. He did not assume the 'degree of confidence in the occurrence of an event' as a definition of probability, but only as a measure of the probability itself :-
Pompilj was a convinced advocator of a "universal solidarity" in the world in which we live "in the same sense that everything influences and is influenced by everything else". ['Le variabili casuali' Ⓣ - fasc. 1, page 12]. This principle, which he repeatedly reiterated in the presentation of the conceptual foundations for the processing and the interpretation of experimental results, it was one aspect of his overall vision of things which was not suffering sharp and isolated divisions, i.e. he liked the connections and overlaps of different fields; this had a positive influence on his work. From this also came his conviction of the vital link between theory and application.Pompilj had numerous pupils and collaborators that appreciated him for his exceptional generosity and rich humanity, that were not affected despite the numerous family tragedies that marred his life. Following the death of his father in World War I and that of his brother Guido came the death of his first wife Ornella - by whom he had three children - in 1958, and later on, the death of his son Frido.
He died in Rome on 9 July 1968, due to complications following an operation for an aneurysm.
Let us end with quoting from Giorgio Dall'Aglio :-
Pompilj asked Alesandro Faedo to recommend a young researcher in analysis. I was chosen, and at the beginning of 1956 I moved to the Faculty of Statistics (the full name was "Facoltà di Scienze Statistiche Demografiche e Attuariali") in Rome. Thus, I had the double fortune of being offered an excellent path for my scientific career as well as finding a remarkable mentor in Giuseppe Pompilj ... Together with his students, Pompilj also did a considerable amount of more applied work. In particular, he very much supported statistical research related to medicine and science in general. One of his most important achievements was his collaboration with the Istituto Centrale di Statistica in the launch of sample surveys. He fought just as hard for the promotion of operations research, creating the Scuola di Perfezionamento in Ricerca Operativa. Pompilj was amongst the founders of the Italian Association in Operations Research. Pompilj's relevance is proven by the fact that a representative of the Istituto di Calcolo delle Probabilità was nominated as a permanent member of the scientific committee. I too, after him, received the honour of this position; but I was not interested, and quit soon after my appointment. ... In only twenty years, he had revolutionized statistics in Italian universities and created a solid group of students. More than ten of them attained a university chair; an unbelievable success considering that just one had succeeded before his death and that most of the appointed chairs were in statistics, where he never really had achieved academic authority. I remember a chat I had once with Jimmie Savage, who remarked upon the scarce scientific works by Pompilj. I explained to him that most of the research conducted by his students (me in the first place) were in fact, started by him. Pompilj was also awarded a Gold Medal of the President of the Republic for his contribution to science and culture. Let me conclude by mentioning an anecdote which demonstrates Pompilj's generous personality. Two years after my arrival, he asked me to write the third chapter of a booklet 'Piano degli Esperimenti' Ⓣ (1959) which would discuss the plans of experiments. Although my contribution was only a small portion of the book, it was published with two authors; moreover, he said that he had received 300, 000 ITL, and one third was for me. This type of generosity was not common in Italian universities.
Article by: J J O'Connor and E F Robertson