Celebratio Mathematica

I had only oc­ca­sion­al con­tact with Dav­id Black­well through the years. But I al­ways found him to be a warm, gra­cious per­son with a friendly greet­ing. He entered the Uni­versity of Illinois at Urb­ana–Cham­paign in 1935 at the age of six­teen and re­ceived a bach­el­or’s de­gree in math­em­at­ics in 1938 and a mas­ter’s de­gree in 1939. Black­well wrote a doc­tor­al thes­is on Markov chains with Joseph L. Doob as ad­visor in 1941. Two earli­er al­most-con­tem­por­ary doc­tor­al stu­dents of J. L. Doob were Paul Hal­mos, with a doc­tor­al de­gree in 1938, and War­ren Am­brose, with the de­gree in 1939.

Black­well was a postdoc­tor­al fel­low at the In­sti­tute for Ad­vanced Study for a year from 1941 (hav­ing been awar­ded a Ros­en­wald fel­low­ship). There was an at­temp­ted ra­cist in­ter­ven­tion by the then-pres­id­ent of Prin­ceton, who ob­jec­ted to the hon­or­if­ic des­ig­na­tion of Black­well as a vis­it­ing fel­low at Prin­ceton (all mem­bers of the In­sti­tute had this des­ig­na­tion). He was on the fac­ulty of Howard Uni­versity in the math­em­at­ics de­part­ment from 1944 to 1954. Ney­man sup­por­ted the ap­point­ment of Dav­id Black­well at the Uni­versity of Cali­for­nia, Berke­ley, in 1942, but this fell through due to the pre­ju­dices at that time (see [e4]). However, in 1955 Dav­id Black­well was ap­poin­ted pro­fess­or of stat­ist­ics at UC Berke­ley and be­came chair of the de­part­ment the fol­low­ing year.

Black­well wrote over ninety pa­pers and made ma­jor con­tri­bu­tions in many areas — dy­nam­ic pro­gram­ming, game the­ory, meas­ure the­ory, prob­ab­il­ity the­ory, in­form­a­tion the­ory, and math­em­at­ic­al stat­ist­ics. He was an en­ga­ging per­son with broad-ran­ging in­terests and deep in­sights. He was quite in­de­pend­ent but of­ten car­ried out re­search with oth­ers. In­ter­ac­tion with Gir­shick prob­ably led him to re­search on stat­ist­ic­al prob­lems of note. Re­searches with K. Ar­row, R. Bell­man, and E. Barankin fo­cused on game the­ory. Joint work with A. Thomasi­an (a stu­dent of his) and L. Breiman was on cod­ing prob­lems in in­form­a­tion the­ory. He also car­ried out re­searches with col­leagues at UC Berke­ley, such as Dav­id Freed­man, Lester Du­bins, J. L. Hodges, and Peter Bick­el. The Rao–Black­well the­or­em deal­ing with the ques­tion of op­tim­al un­biased es­tim­a­tion is due to him.

He was elec­ted the first Afric­an Amer­ic­an mem­ber of the Na­tion­al Academy of Sci­ences, USA, and re­ceived many oth­er awards. He was a dis­tin­guished lec­turer. We’re thank­ful that he sur­vived the dif­fi­culties that Afric­an Amer­ic­ans had to en­dure in a time of great bi­as (in his youth). He was a per­son of sin­gu­lar tal­ent in the areas of stat­ist­ics and math­em­at­ics.

I shall de­scribe lim­ited as­pects of the re­search of Black­well and Du­bins [e2] on reg­u­lar con­di­tion­al dis­tri­bu­tions (see Doob [e1] for a dis­cus­sion of con­di­tion­al prob­ab­il­ity). This was an area that Black­well of­ten found of in­terest. Giv­en a meas­ur­able space \( (\Omega, \mathcal{B}) \) with \( \mathcal{A} \) a sub \( \sigma \)-field of the \( \sigma \)-field \( \mathcal{B} \), call \( P \) defined on \( \Omega \times \mathcal{B} \) a reg­u­lar con­di­tion­al dis­tri­bu­tion (r.c.d.) \tex­tit{for \( \mathcal{A} \) on \( \mathcal{B} \)} if for all \( \omega \in \Omega \), \( B \in \mathcal{B} \),

1. \( P(\omega, \cdot) \) is a prob­ab­il­ity meas­ure on \( \mathcal{B} \).

2. For each \( B \in \mathcal{B} \), \( P(\cdot, B) \) is \( \mathcal{A} \)-meas­ur­able and re­lated to the prob­ab­il­ity dis­tri­bu­tion via \[ \int_A P(\omega, B)\,dP(\omega) = P(A \cap B) \] for \( A \in \mathcal{A} \), \( B \in \mathcal{B} \).

Such reg­u­lar con­di­tion­al dis­tri­bu­tions do not al­ways ex­ist. But as­sum­ing ex­ist­ence, call it prop­er if \[ P(\omega, A) = 1 \] whenev­er \( \omega \in A \in \mathcal{A} \).

The prob­ab­il­ity meas­ure \( P \) on \( \mathcal{A} \) is called ex­treme if \( P(A) = 0 \) or 1 for all \( A \in \mathcal{A} \). An \( \mathcal{A} \)-atom is the in­ter­sec­tion of all ele­ments of \( \mathcal{A} \) that con­tain a giv­en point of \( \Omega \). If for \( A \in \mathcal{A},\, P(A) = 1 \), \( P \) is said to be sup­por­ted by \( A \).

Then we have:

This res­ult shows that, for \( \Omega \) the in­fin­ite product of a sep­ar­able met­ric space con­tain­ing more than one point, neither the tail field, the field of sym­met­ric events, nor the in­vari­ant field ad­mit a prop­er r.c.d (reg­u­lar con­di­tion­al dis­tri­bu­tion). They weak­en the count­able ad­dit­iv­ity con­di­tion of an r.c.d. to fi­nite ad­dit­iv­ity and add (1) to ob­tain the no­tion of a nor­mal con­di­tion­al dis­tri­bu­tion and ar­rive at suf­fi­cient con­di­tions for ex­ist­ence. Later re­lated re­search by Berti and Rigo [e3] con­siders the r.c.d.s with ap­pro­pri­ate weak­en­ings of the concept of prop­er.