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==Carlo Emilio Bonferroni==

Carlo Emilio Bonferroni (28 January 1892 – 18 August 1960) was an Italian mathematician who worked on probability theory. Carlo Emilio Bonferroni was born in Bergamo on 28 January 1892 and died on 18 August 1960 in Florence. He studied in Turin, held a post as assistant professor at the Turin Polytechnic, and in 1923 took up the chair of financial mathematics at the Economics Institute in Bari. In 1933 he transferred to Florence where he held his chair until his death. Bonferroni is best known for the Bonferroni inequalities (a generalization of the union bound), and for the Bonferroni correction in statistics (which he did not invent but which utilizes his inequalities).

Biography
main article:“Bonferroni biography". 

Carlo Emilio Bonferroni was born in Bergamo on 28 January 1892. He was talented at music and became an excellent pianist and composer at an early age. However, under the influence of his father, he gradually changed his interests toward mathematics.

In 1910, Bonferroni attended the University of Turin. As an undergraduate, he could find an error in the book Elements de la theorie des probabilities of Emile Borel about a coin tossing problem. Borel admitted his mistake and corrected it in the second edition.

After gaining the doctoral degree, Bonferroni studied abroad at the Univerisity of Vienna and the Eidgenossische Technische Hochschule for a year.

From 1914 to 1918 during World War I, Bonferroni served as an officer in the engineers in the Italian army.

Teaching Career and Study After World War I, Bonferroni became an assistant professor at the Turin Polytechnic to teach analysis, geometry, and mechanics. During this five years, he worked with Filadelfo Insolera who triggered his intense interest in financial mathematics.

In 1923, he moved to Bari and became a formal professor of financial mathematics at the Financial Mathematics and Economics Institute there.

From 1926 to 1933, he served as rector of the University of Bari. In 1928, Bonferroni attended a study conference, named International Congress of Mathematicians, that at that time was the only one to cover topics about both mathematics and all kinds of its branches—Statistics, Mathematical Economy, Calculus of Probability, and Actuarial Science.

Later in 1933, Bonferroni left Bari and moved to Florence, holding his chair at the University of Florence until his death. During his rest life there, he built his own probability theory, now is known as Boole’s inequality (1936), that explains how to solve the probability of at least one event happens from a finite set of events.

Bonferroni gained recognition by his probability theory after the publish of Maurice Frechet’s book ''Les probabilités associées a un système d'événements compatibles et dépendants. Première partie: d'événements en nombre fini fixe that introduces problems within dependent events in 1940, and the book of William Feller, An Introduction to Probability Theory and its Applications, (1950)'', which adopts lots of examples to explain probability theory, including the use of Boole’s inequality.

Bonferroni continued his study and teaching of mathematics and statistics for the rest of his life. He died on Augest 18, 1960 in Florence.

Probability theory
Probability theory is a branch of mathematics, concerning about problems of probability. It uses probability space, which is assigned the value to between 0 and 1. The outcomes of the probability space is sample space, and any subset of this outcome is called event.

Background
Bonferroni first pointed out the concept of Boole's inequality in his article "Il calcolo delle assicurazioni su gruppi di teste, Studi in Onore del Professore Salvatore Ortu Carboni (1935) to mainly introduce some specific application of his theory.

In 1936,Bonferroni officially published Boole's inequality in his "Teoria statistica delle classi e calcolo delle probabilità". He defined this theory in an abstract way:

"The author establishes above all a symbolic calculus which enables the expression in a rapid and uniform manner of the various probabilities of survival and death amongst a group of assured, expressed as a function of a particular type assumed as primary. This calculus does not require the hypothesis that the assured lives should be independent, as is usual in treatments of this problem. He establishes a noteworthy law of duality between the probability of survival and that of death, introducing as a consequence of the notation some new results."

Definition
For any finite or countable set of events, the probability that at least one of the events happens is no greater than the sum of the probabilities of the individual events.

Formula: For events A1, A2, A3...Ai in a finite set, we have:

P(⋃iAi)≤∑iP(Ai).P(⋃iAi)≤∑iP(Ai).

Background
Bonferroni correction is named by Bonferroni for his use of Boole’s inequalityconcerning about multiple comparison problems.

Statistical hypothesis testing is based on rejecting the null hypothesis. The testing is inaccurate when multiple hypotheses are conducted since the probability of rarely-happened events increase with the increase of the total amount of events being dealing with. Thus, Bonferroni corrections work on the situation when multiple hypotheses exist.

Definition
~type I error: an incorrect rejection of a true null hypothesis.

~V: the number of type I error.

~Familywise error rate (FWER): the probability of making type I error.

~Alpha: the probability of making type I error that we can accept

We have:

FWER= Pr(V<=1) or FWER=1-Pr(V=0)

The result of FWER is less than alpha so that we can control inaccuracy under alpha. In this way, the inaccurate part brought by rare events can be expected and calculated.

Critisicm
With respect to FWER control, the Boole's correction can be conservative if there are a large number of tests and/or the test statistics are positively correlated.

The correction increase the probability of producing false negatives, i.e., reducing statistical power.

Publications
1927: "Teoria e probabilità. In Annuario del R Istituto Superiore di Scienze Economiche e Commerciali di Bari per L'anno Accademico" ( Given as the inaugural lecture for the academic year 1925-1926 on November 22, 1925)

1935: "Il calcolo delle assicurazioni su gruppi di teste, Studi in Onore del Professore Salvatore Ortu Carboni" (he mentioned the concept of Boole's inequality at the first time with some specific application.

1936: "Teoria statistica delle classi e calcolo delle probabilità", Pubbl. d. R. Ist. Super. di Sci. Econom. e Commerciali di Firenze (in Italian) (mainly introduces Boole's inequality in a abstract way)

Carlo Benedetti
Further information: Encounters with the Statistical School: a conversation with Carlo Benedetti 

In 1933, Bonferroni began to teach at the University of Florence. According to Encounters with the Statistical School: a conversation with Carlo Benedetti written by G M Giorgi, one of Bonferroni students, Carlo Benedetti, provided him high praise as a helpful teacher, a great scientist with noticeable characteristics, and a romantic and kind person.

Benedetti said that it was Bonferroni who led him to the world of mathematics, and gave him helpful suggestions on his future career about mathematics. When Benedetti wanted to devote all his life to mathematics, Bonferroni told him that the study of mathematics was so exclusive that “people who had not attended a mathematics faculty did not have many opportunities ” to join in. Therefore, Benedetti reconsidered his decision and finally changed his focus to the study of statistics.

From Benedetti’s description, Bonferroni studied widely from pure mathematics to mathematical statistics and to actuarial mathematics. He always simplified complex problems with his own style, was usually fascinated by printing mistakes and errors.

Benedetti stated that except for Bonferroni’s achievements as an excellent pianist and composer, his passion for music was also reflected in his study life-- “[Bonferroni’s] mathematical papers in fact you can recognize a lightness and refinement which is almost musical.”To people, Bonferroni was sensitive, kind, and refined. Benedetti described that Bonferroni was affectionate towards him like a father.

Michael Dewey and Eugene Seneta
Michael Dewey and Eugene Seneta thought that Bonferroni received less recognition compared with his promoted contributions. From their point of view, Bonferroni’s books lack dissemination— “most of his works do not exist in typeset versions” ; Bonferroni handwrote texts to his students to reduce their cost on the teaching material, so most of his books are printed from his handwriting, and few become typeset versions which are easy to be popularized.