#### by Morris H. DeGroot

#### “I expected to be an elementary-school teacher”

**DeGroot:**
How did you originally get interested in statistics and probability?

**Blackwell:**
I think I have been interested in the concept of probability ever since I was an
undergraduate at Illinois, although there wasn’t very much probability or statistics around.
Doob
was there but he didn’t teach probability. All
the probability and statistics were taught by a very nice old gentleman named Crathorne; you probably never heard of him. But he was a very good friend of
Henry Rim
and, in fact, they collaborated on a college
algebra book. I think I took all the courses that Crathorne taught:
two undergraduate courses and one first-year graduate course. Anyway, I have
been interested in the subject for a long time, but after I got my Ph.D. I didn’t expect to get professionally interested in statistics.

**DeGroot:**
But did you always intend to go on to graduate school?

**Blackwell:**
No. When I started out in college I expected to be an elementary-school teacher. But
somehow I kept postponing taking those education courses.
[*Laughs.*]
So I ended up getting a master’s degree, and then I got a fellowship to continue my work there at Illinois.

**DeGroot:**
So your graduate work wasn’t particularly in the area of statistics or probability?

**Blackwell:**
No, except of course that I wrote my thesis under Doob in probability.

**DeGroot:**
What was the subject of your thesis?

**Blackwell:**
Markov chains. There wasn’t very much original in it. There was
one beautiful idea, which was Doob’s idea and which he gave to me. The thesis was
never published as such.

**DeGroot:**
But your first couple of papers pertained to Markov chains.

**Blackwell:**
The first couple of papers came out of my thesis, that’s right.

**DeGroot:**
So after you got your degree…

**Blackwell:**
After I got my degree, I sort of expected to work in probability,
real variables, measure theory, and such things.

**DeGroot:**
And you *have* done a good deal of that.

**Blackwell:**
Yes, a fair amount. But it was
Abe Girshick
who got me interested in statistics.

**DeGroot:**
In Washington?

**Blackwell:**
Yes. I was teaching at Howard, and the mathematics environment
was not really very stimulating, so I had to look around beyond the university just
for whatever was going on in Washington that was interesting mathematically.

**DeGroot:**
Not just statistically, but mathematically?

**Blackwell:**
I was just looking for anything interesting in mathematics that was going on in Washington.

**DeGroot:**
About what year would this be?

**Blackwell:**
I went to Howard in 1944. So this would have been during the year 1944–1945.

**DeGroot:**
Girshick was at the Department of Agriculture?

**Blackwell:**
That’s right. And I heard him give a lecture sponsored by the Washington Chapter of the American Statistical Association. That’s a pretty lively chapter. I first met
George Dantzig
when he gave a lecture there around that same time. His
lecture had nothing to do with linear programming, by the way. In fact, I first became
acquainted with the idea of a randomized test by hearing Dantzig talk about it. I
think that he was the guy who invented a test function, instead of having just a
rejection region that is a subset of the sample space. At one of those meetings Abe
Girshick spoke on sequential analysis. Among other things, he mentioned
Wald’s
equation.

**DeGroot:**
That’s the equation that the expectation of a sum of random variables
is __\( E(N) \)__ times the expectation of an individual variable?

**Blackwell:**
Yes. That was just such a remarkable equation that I
didn’t believe it. So I went home and thought I had constructed a counterexample.
I mailed it to Abe, and I’m sure that he discovered the error. But he didn’t write
back and tell me it was an error; he just called me up and said, let’s talk about it.
So we met for lunch, and that was the start of a long and beautiful association that
I had with him.

**DeGroot:**
Would you regard the Blackwell and Girshick book
[e3]
as the culmination
of that association?

**Blackwell:**
Oh, that was a natural outgrowth of the association. I learned a
great deal from him.

**DeGroot:**
Were you together at any time at Stanford?

**Blackwell:**
Yes, I spent a year at Stanford. I think it was 1950–1951. But
he and I were also together at other times. We spent several months together at Rand.
So we worked together in Washington, and then at Rand, and then at Stanford.

#### “I wrote 105 letters of application”

**DeGroot:**
Tell me a little about the years between your Ph.D. from Illinois
in 1941 and your arrival at Howard in 1944. You were at a few other schools in between.

**Blackwell:**
Yes. I spent my first postdoctoral year at the Institute for Advanced Study. Again, I continued to show my interest
in statistics. I sat in on
Sam Wilks’
course in Princeton during that year.
Henry Scheffé
was also sitting in on that class. He had just completed his Ph.D. at Wisconsin.
Jimmie Savage
was at the Institute for that year. He was at some of Wilks’ lectures, too. There were a lot of statisticians about our age around Princeton
at that time.
Alex Mood
was there.
George Brown
was there.
Ted Anderson
was there. He was in Wilks’ class that year.

**DeGroot:**
He was a graduate student?

**Blackwell:**
He was a graduate student, just completing his Ph.D. So that was
my first postdoctoral year. Also, I had a chance to meet
von Neumann
that year. He was
a most impressive man. Of course, everybody knows that. Let me tell you a little
story about him.

When I first went to the Institute, he greeted me, and we were talking, and
he invited me to come around and tell him about my thesis. Well, of course, I thought
that was just his way of making a new young visitor feel at home, and I had no intention
of telling him about my thesis. He was a big, busy, important man. But then a couple
of months later, I saw him at tea and he said, “When are you coming around to tell
me about your thesis? Go in and make an appointment with my secretary.” So I did,
and later I went in and started telling him about my thesis. He listened for about
ten minutes and asked me a couple of questions, and then he started telling
*me* about *my* thesis. What you have really done is this, and probably
this is true, and you could have done it in a somewhat simpler way, and so on. He was
a really remarkable
man. He listened to me talk about this rather obscure subject and in ten minutes
he knew more about it than I did. He was extremely quick. I think he may have wasted
a certain amount of time, by the way, because he was so willing to listen to second-
or third-rate people and think about their problems. I saw him do that on many occasions.

**DeGroot:**
So, from the Institute you went where?

**Blackwell:**
I went to Southern University in Baton Rouge, Louisiana.
That’s a state school and at that time it was *the* state university in Louisiana for
blacks. I stayed there just one year. Then the next year, I went to Clark College
in Atlanta, also a black school. I stayed there for one year. Then I went to Howard
University in Washington and stayed there for ten years.

**DeGroot:**
Was Howard at a different level intellectually from these other schools?

**Blackwell:**
Oh yes. It was the ambition of every black scholar in those days
to get a job at Howard University. That was the best job you could hope for.

**DeGroot:**
How large was the math department there in terms of faculty?

**Blackwell:**
Let’s see. There were just four regular people in the math department. Two professors.
I went there as an assistant professor. And there was one instructor. That was it.

**DeGroot:**
Have you maintained any contact with Howard through the years?

**Blackwell:**
Oh, yes. I guess the last time I gave a lecture there was about
three years ago, but I visited many times during the years.

**DeGroot:**
Do you see much change in the place through the years?

**Blackwell:**
Yes, the math department now is a livelier place than it was when I was there. It’s much bigger and the current chairman,
Jim Donaldson,
is very good and very active. There are some interesting things going on there.

**DeGroot:**
Did you feel or find that discrimination against blacks affected
your education or your career after your Ph.D.?

**Blackwell:**
It never bothered me; I’ll put it that way. It surely shaped my
expectations from the very beginning. It never occurred to me to think about teaching
in a major university since it wasn’t in my horizon at all.

**DeGroot:**
Even in your graduate-student days at Illinois?

**Blackwell:**
That’s right. I just assumed that I would get a job teaching in
one of the black colleges. There were 105 black colleges at that time, and I wrote
105 letters of application.

**DeGroot:**
And got 105 offers, I suppose.

**Blackwell:**
No, I eventually got three offers, but I accepted the first one
that I got. From Southern University.

**DeGroot:**
Let’s move a little further back in time. You grew up
in Illinois?

**Blackwell:**
In Centralia, Illinois. Did you ever get down to Centralia or that
part of Illinois when you were in Chicago?

**DeGroot:**
No, I didn’t.

**Blackwell:**
Well, it’s a rather different part of the world from northern Illinois.
It’s quite southern. Centralia in fact was right on the border line of segregation.
If you went south of Centralia to the southern tip of Illinois, the schools were
completely segregated in those days. Centralia had one completely black school, one
completely white school, and five “mixed” schools.

**DeGroot:**
Well that sounds like the boundary all right. Which
one did you go to?

**Blackwell:**
I went to one of the mixed schools, because of the part of town
I lived in. It’s a small town. The population was about 12,000 then, and it’s still
about 12,000. The high school had about 1,000 students. I had very good high-school
teachers in mathematics. One of my high-school teachers organized a mathematics club,
and used to give us problems to work. Whenever we would come up with something that
had the idea for a solution, he would write up the solution for us, and send it in our name to a journal
called *School Science and Mathematics*. It was a great thrill to see your name
in the magazine. I think my name got in there three times. And once my
*solution* got printed. As I say, it was really Mr. Huck’s write-up based on my idea.
[*Laughs.*]

**DeGroot:**
Was your family encouraging about your education?

**Blackwell:**
It was just sort of assumed that I would go to college. There
was no “Now be sure to study hard” or anything like that. It was just taken for granted
that I was going to go to college. They were very, very supportive.

#### Some favorite papers

**DeGroot:**
You were quite young when you received your Ph.D. You were 21 or so?

**Blackwell:**
22. There wasn’t any big jump. I just sort of did everything a little faster than normal.

**DeGroot:**
And you’ve been doing it that way ever since. You’ve published about
80 papers since that time. Do you have any favorites in that list that you particularly
like, or that you feel were particularly important or influential?

**Blackwell:**
Oh, I’m sure that I do, but I’d have to look at the list and think
about that. May I look?

**DeGroot:**
Sure. This is an open-book exam.

**Blackwell:**
Good. Let’s see… Well, my first statistical paper, called “On an equation of Wald”
[e1],
grew out of that original conversation with Abe Girshick. That’s a paper that I
am still really very proud of. It just gives me pleasant feelings every time I think
about it.

**DeGroot:**
Remind me what the main idea was.

**Blackwell:**
For one thing it was a proof of Wald’s theorem under, I think,
weaker conditions than it had been proved before, under sort of *natural* conditions.
And the proof is *neat*. Let me show it to you.
[*Goes to blackboard.*]

Suppose that __\( X_1,X_2,\dots \)__ are i.i.d. and you have a stopping rule __\( N \)__,
which is a random variable. You want to prove that __\( E(X_1 + \dots + X_n) = E(X_1)E(N) \)__. Well, here’s my idea. Do it over and over again. So you have
stopping times __\( N_1,N_2,\dots \)__, and you get
__\begin{align*}
S_1 &=X_1+\dots+ X_{N_1},
\\
S_2&=X_{N_1+1}+\dots+ X_{N_1+N_2},
\\
&\dots
\end{align*}__
Consider __\( S_1+ \dots+ S_k=X_1+\dots + X_{N_1+\dots+N_k} \)__.
We can write this equation as
__\[
\frac{S_1+\dots + S_k}{k}=
\biggl(\frac{X_1+\dots+X_{N_1+\dots + N_k}}{N_1+\dots + N_k}\biggr)
\biggl(\frac{N_1+\dots + N_k}{k}\biggr).
\]__
Now let __\( k \to\infty \)__. The first term on the right is a subsequence of
the __\( X \)__ averages. By the strong law of large numbers, this converges to __\( E(X_1) \)__.
The second term on the right is the average of __\( N_1,\dots,N_k \)__. We are assuming that
they have a finite expectation, so this converges to that expectation
__\( E(N) \)__. Therefore, the sequence
__\[
\frac{S_1+\dots + S_k}{k}
\]__
converges a.e. Then the converse of the strong law of large numbers
says that the expected value of each __\( S_i \)__ must be finite, and that
__\[
\frac{S_1+\dots + S_k}{k}
\]__
must converge to that expectation __\( E(S_1) \)__. Isn’t that neat?

**DeGroot:**
Beautiful, beautiful.

**Blackwell:**
So that’s the proof of Wald’s equation, just by invoking the strong
law of large numbers and its converse. I think I like that because that was the first
time that *I* decided that I could do something original. The papers based on my thesis
were nice, but those were really Doob’s ideas that I was just carrying
out. But here I had a really original idea, so I was very pleased with that paper.
Then I guess I like my paper with
Ken Arrow
and Abe Girshick, “Bayes and minimax solutions of sequential decision problems”
[e2].

**DeGroot:**
That was certainly a very influential paper.

**Blackwell:**
That was a serious paper, yes.

**DeGroot:**
Then was some controversy about that paper, wasn’t there?
Wald
and
Wolfowitz
were doing similar things at more or less the same time.

**Blackwell:**
Yes, they had priority. There was no question about that, and I think we did give
inadequate acknowledgment to them in our work. So they were very much disturbed about
it, especially Wolfowitz. In fact, Wolfowitz was cool to me for more than 20 years.

**DeGroot:**
But certainly your paper was different from theirs.

**Blackwell:**
We had things that they didn’t have, there was no doubt about
that. For instance, induction backward—calculation backward—that
was in our paper and I don’t think there is any hint of it in their work. We did
go beyond what they had done. Our paper didn’t seem to bother Wald too much, but
Wolfowitz was annoyed.

**DeGroot:**
Did you know Wald very well, or have much contact with him?

**Blackwell:**
Not very well. I had just three or four conversations with him.

#### Important influences

**DeGroot:**
I gather from what you said that
Girshick
was a primary influence on you in the field of statistics.

**Blackwell:**
Oh yes.

**DeGroot:**
Were there other people that you felt had a strong influence on you?
Neyman,
for example?

**Blackwell:**
Not in my statistical thinking. Girshick was certainly *the* most
important influence on me. The other person who had just one influence, but it was
a very big one, was
Jimmie Savage.

**DeGroot:**
What was that one influence?

**Blackwell:**
Well, he explained to me that the Bayes approach was the right
way to do statistical inference. Let me tell you how that happened. I was at Rand,
and an economist came in one day to talk to me. He said that he had a problem. They
were preparing a recommendation to the Air Force on how to divide their research
budget over the next five years and, in particular, they had to decide what fraction
of it should be devoted to long-range research, and what fraction of it should be
devoted to more immediate developmental research.

“Now,” he said, “one of the things that this depends on is the probability
of a major war in the next five years. If it’s large, then of course that would
shift the emphasis toward developing what we already know how to do, and if it’s small then there
would be more emphasis on long-range research. I’m not going to ask you to tell me a number,
but if you could give me any guide as to how I could go about finding such a number I would
be grateful.” Oh, I said to him, that question just doesn’t make sense. Probability
applies to a long sequence of repeatable events, and this is clearly a unique situation.
The probability is either 0 or 1, but we won’t know for five years, I pontificated.
[*Laughs.*]
So the economist looked at me and nodded and said, “I was afraid you were
going to say that. I have spoken to several other statisticians and they have all
told me the same thing. Thank you very much.” And he left.

Well, that conversation bothered me. The fellow had asked me a reasonable, serious question and I had given him a frivolous, sort of flip, answer, and I wasn’t happy. A couple of weeks later Jimmie Savage came to visit Rand, and I went in and said hello to him. I happened to mention this conversation that I had had, and then he started telling me about deFinetti and personal probability. Anyway, I walked out of his office half an hour later with a completely different view on things. I now understood what was the right way to do statistical inference.

**DeGroot:**
What year was that?

**Blackwell:**
About 1950, maybe 1951, somewhere around there. Looking back on
it, I can see that I was emotionally and intellectually prepared for Jimmie’s message,
because I had been thinking in a Bayesian way about sequential analysis, hypothesis testing, and other statistical
problems for some years.

**DeGroot:**
What do you mean by thinking in a Bayesian way? In terms of prior distributions?

**Blackwell:**
Yes.

**DeGroot:**
Wald used them as a mathematical device.

**Blackwell:**
That’s right. It just turned out to be clearly a very natural
way to think about problems, and it was mathematically beautiful. I simply regretted
that it didn’t correspond with reality.
[*Laughs.*]
But then what Jimmie was telling
me was that the way that I had been thinking all the time was really the right way
to think, and not to worry so much about empirical frequencies. Anyway, as I say,
that was just one very big influence on me.

**DeGroot:**
Would you say that your statistical work has mainly used the Bayesian
approach since that time?

**Blackwell:**
Yes; I simply have not worked on problems where that approach
could not be used. For instance, all my work in dynamic programming just has that
Bayes approach in it. That is *the* standard way of doing dynamic programming.

**DeGroot:**
You wrote a beautiful book called *Basic Statistics*
[◊]
that was really based on the Bayesian approach, but as I
recall you never once mentioned the word “Bayes” in that book. Was that intentional?

**Blackwell:**
No, it was not intentional.

**DeGroot:**
Was it that the terminology was irrelevant to the concepts that you were trying to get across?

**Blackwell:**
I doubt if the word “theorem” was ever mentioned in that book.
That was not originally intended as a book, by the way. It was simply intended as
a set of notes to give my students in connection with lectures in this elementary statistics course.
But the students suggested that it should be published and a McGraw-Hill man said
that he would be interested. It’s just a set of notes. It’s short; I think it’s less than 150 pages.

**DeGroot:**
It’s beautiful. There are a lot of wonderful gems in those 150 pages.

**Blackwell:**
Well, I enjoyed teaching the course.

**DeGroot:**
Do you enjoy teaching from your own books?

**Blackwell:**
No, not after a while. I think about five years after the book was published
I stopped using it. Just because I got bored with it. When you reach the point
where *you’re* not learning anything, then it’s probably time to change something.

**DeGroot:**
Are you working on other books at the present time?

**Blackwell:**
No, except that I am *thinking about* writing a more elementary
version of parts of your book on optimal statistical decisions, because I have been using it
in a course and the undergraduate students say that it’s too hard.

**DeGroot:**
Uh, oh. I’ve been thinking of doing the same thing.
[*Laughs.*]
Well, I am just thinking
generally, in terms of an introduction to Bayesian statistics for undergraduates.

**Blackwell:**
Very good. I really hope you do it, Morrie. It’s needed.

**DeGroot:**
Well, I really
hope you do it, too. It would be interesting. Are there courses that you particularly
enjoy teaching?

**Blackwell:**
I like the course in Bayesian statistics using your book. I like
to teach game theory. I haven’t taught it in some years, but I like to teach that
course. I also like to teach, and I’m teaching right now, a course in information
theory.

**DeGroot:**
Are you using a text?

**Blackwell:**
I’m not using any one book.
Pat Billingsley’s book *Ergodic Theory and Information*
comes closest to what I’m doing. I like to teach measure theory.
I regard measure theory as a kind of hobby, because to do probability and statistics
you don’t really need very much measure theory. But there are these fine, nit-picking
points that most people ignore, and rightly so, but that I sort of like to worry
about.
[*Laughs.*]
I know that it is not important, but it is interesting to me to worry
about regular conditional probabilities and such things. I think I’m one of only three people in our department who rally takes measure theory seriously.
Lester [Dubins]
takes it fairly seriously, and so does
Jim Pitman.
But the rest of the people just sort of ignore it.
[*Laughs.*]

#### “I would like to see more emphasis on Bayesian statistics”

**DeGroot:**
Lets talk a little bit about the current state of statistics. What
areas do you think are particularly important these days? Where do you see the field
going?

**Blackwell:**
I can tell you what I’d like to see happen. First, of course,
I would like to see more emphasis on Bayesian statistics. Within that area it seems to me that one promising
direction which hasn’t been explored at all is Bayesian experimental design.
In a way, Bayesian statistics is much simpler than classical statistics in that, once you’re
given a sample, all you have to do are calculations based on that sample. Now, of course, I say
“all you have to do”—sometimes those calculations can be horrible. But if you are trying to
design an experiment, that’s not all you have to do. In that case,
you have to look at all the different samples you might get, and evaluate every one
of them in order to calculate an overall risk, to decide whether the experiment is worth
doing and to choose among the experiments. Except in very special situations, such
as when to stop sampling, I don’t think a lot of work has been done in that area.

**DeGroot:**
I think the reason there hasn’t been very much done is because
the problems are so hard. It’s really hard to do explicitly the calculations that
are required to find *the* optimal experiment. Do you think that perhaps the computing
power that is now available would be helpful in this kind of problem?

**Blackwell:**
That’s certainly going to make a difference. Let me give you a simple
example that I have never seen worked out, but I am sure could be worked out. Suppose
that you have two independent Bernoulli variables, say, a proportion among males
and a proportion among females. They are independent, and you are interested in estimating
the sum of those proportions or some linear combination of those proportions.
You are going to take a sample in two stages. First of all you can ask, how large
should the first sample be? And then, based on the first sample, how should you allocate
proportions in the second sample?

**DeGroot:**
Are you going to draw the first sample from the total population?

**Blackwell:**
No. You have males and you have females, and you have a total
sample effort of size __\( N \)__. Now you can pick some number __\( n \leq N \)__ to be your sample
size. And you can allocate those __\( n \)__ observations among males and females. Then, based
on how that sample comes out, you can allocate your second sample. What is the best
initial allocation, and how much better is it than just doing it all in one stage?
Well, I haven’t done that calculation but I’m sure that it can be done. It would
be an interesting kind of thing and it could be extended to more than two categories.
That’s an example of the sort of thing on which I would like to see a lot of work
done—Bayesian experimental design.

One of the things that I worry about a little is that I don’t see theoretical statisticians having as much contact with people in other areas as I would like to see. I notice here at Berkeley, for example, that the people in Operations Research seem to have much closer contact with industry than the people in our department do. I think we might find more interesting problems if we did have closer contact.

**DeGroot:**
Do you think that the distinctions between applied and theoretical statistics are still as rigid as
they were years ago, or do you think that the
field is blending more into a unified field of statistics in which such distinctions are not
particularly meaningful? I see the emphasis on data analysis which is coming about,
and the development of theory for data analysis and so on, blurring these distinctions
between theoretical and applied statistics in a healthy way.

**Blackwell:**
I guess I’m not familiar enough with data analysis and what
computers have done to have any interesting comments on that. I see what
some of our people and people at Stanford are doing in looking
at large-dimensional data sets and rotating them so that you can see lots of three-dimensional
projections and such things, but I don’t know whether that suggests interesting theoretical
questions or not. Maybe that’s not important, whether it suggests interesting theoretical
questions. Maybe the important thing is that it helps contribute to the solution
of practical problems.

#### Infinite games

**DeGroot:**
What kind of things are you working on these days?

**Blackwell:**
Right now I am working on some thongs in information theory,
and still trying to understand some things about infinite games of
perfect information.

**DeGroot:**
What do you mean by an infinite game?

**Blackwell:**
A game with an infinite number of moves. Here’s an example. I write
down a 0 or a 1, and you write down a 0 or a 1, and we keep going indefinitely. If
the sequence we produce has a limiting frequency, I win. If not, you win. That’s a trivial game because I can force it to have a limiting frequency just by doing
the opposite of whatever you do. But that’s a simple example of an infinite game.

**DeGroot:**
Fortunately, it’s one in which I’ll never have to pay off to you.

**Blackwell:**
Well, we can play it in such a way that you would have to pay off.

**DeGroot:**
How do we do that?

**Blackwell:**
You must specify a strategy. Let me give you an example. You know
how to play chess in just one move: You prepare a complete set of instructions so that
for every situation on the chess board you specify a possible response. Your one
move is to prepare that complete set of instructions. If you have a complete set
and I have a complete set, then we can just play the game out according to those instructions. It’s just one move.
So in the same way, you can specify a strategy in this infinite game. For every finite
sequence that you might see up to a given time as past history, you specify your next
move. So you can define this function once and for all, and I can define a function, and
then we can mathematically asses those functions. I can prove that
there is a specific function of mine such that, no matter what function you specify, the set will
have a limiting frequency.

**DeGroot:**
So you could extract money from me in a finite amount of time.
[*Laughs.*]

**Blackwell:**
Right. Anyway it’s been proved that all such infinite games with Borel payoffs are determined, and
I’ve been trying to understand the
proof for several years now. I’m still working on it, hoping to understand it and simplify it.

**DeGroot:**
Have you published papers on that topic?

**Blackwell:**
Just one paper, many years ago. Let me remind myself of the title
[*checking his files*],
“Infinite games and analytic sets”
[e5].
This is the only paper I’ve published on infinite games;
and that’s one of my papers that I like very much, by the way. It’s an application
of games to prove a theorem in topology. I sort of like the idea of connecting those
two apparently not closely related fields.

**DeGroot:**
Have you been involved in applied projects or applied problems through
the years, at Rand or elsewhere, that you have found interesting and that have stimulated
research of your own?

**Blackwell:**
I guess so. My impression though is this: When I have looked at
real problems, interesting theorems have sometimes come out of it. But never anything
that was helpful to the person who had the problem.
[*Laughs.*]

**DeGroot:**
But possibly to somebody else at another time.

**Blackwell:**
Well, my work on comparison of experiments was stimulated by some work by
Bohnenblust,
Sherman,
and
Shapley.
We were all at Rand. They called their original paper
“Comparison of reconnaissances,”
and it was *classified* because it arose out of some question
that somebody had asked them. I recognized a relation between what they were doing
and sufficient statistics, and proved that they were the same in a special case.
Anyway, that led to this development which I think is interesting theoretically,
and to which you have contributed.

**DeGroot:**
Well, I have certainly used your work in that area. And it has spread
into diverse other areas. It is used in economics in comparing distributions of income, and
I used it in some work on comparing probability forecasters.

**Blackwell:**
And apparently people in accounting have made some use of
these ideas. But anyway, as I say, nothing that I have done has ever helped the person who raised the question. But there is no doubt in my mind that you
do get interesting problems by looking at the real world.

#### “I don’t have any difficulties with randomization”

**DeGroot:**
One of the interesting topics that comes out of a Bayesian view
of statistics is the notion of randomization, and the role that it should play in statistics.
Just this little example you were talking about before with two proportions made me think about that.
We just assume that we are drawing the observations at random from within each
subpopulation in that example,
but perhaps basically because we don’t have much choice. Do you
have any thoughts about whether one should be drawing observations at random?

**Blackwell:**
I don’t have any difficulties with randomization. I think it’s probably a good idea. The strict theoretical idealized Bayesian would of course never
need to randomize. But randomization probably protects us against our own biases.
There are just lots of ways in which people differ from the ideal Bayesian. I guess
the ideal Bayesian, for example, could not think about a theorem as being probably
true. For him, presumably, all true theorems have probability 1 and all
false ones have probability 0. But you and I know that’s not the way we think. I think
of randomization as being a protection against your own imperfect thinking.

**DeGroot:**
It is also to some extent a protection against others. Protection
for you as a statistician in presenting your work to the scientific community, in
the sense that they can have more belief in your conclusions if you use some randomization
procedure rather than your own selection of a sample. So I see it as involved with
the sociology of science in some way.

**Blackwell:**
Yes, that’s an important virtue of randomization. That reminds
me of something else, though. We tend to think of evidence as being valid only when
it comes from random samples or samples selected in a probabilistically specified
way. That’s wrong, in my view. Most of what we have learned, we have learned just
by observing what happens to come along, rather than from carefully controlled experiments.
Sometimes statisticians have made a mistake in throwing away experiments because they
were not properly controlled. That is not to say that randomization isn’t a good
idea, but it is to say that you should not reject data just because they have been
obtained under uncontrolled conditions.

**DeGroot:**
You were the Rouse Ball Lecturer at Cambridge in 1974. How did that
come about and what did it involve?

**Blackwell:**
Well, I was in England for two years, 1973–1975, as the director
of the education-abroad program in Great Britain and Ireland for the University of
California. I think that award was just either
Peter Whittle’s
or
David Kendall’s
idea of how to get me to come up to Cambridge to give a lecture. One of the things which
delighted me was that it was named the Rouse Ball Lecture because it gave me an opportunity
to say something at Cambridge that I liked—namely, that I had heard of Rouse Ball
long before I had heard of Cambridge.
[*Laughs.*]

**DeGroot:**
Well, tell me about Rouse Ball.

**Blackwell:**
He wrote a book called *Mathematical Recreations and Essays*.
You may have seen the book. I first came across it when I was a high-school
student. It was one of the few mathematics books in our library. I
was fascinated by that book. I can still picture it. Rouse Ball was a 19th century
mathematician, I think.
[Walter William Rouse Ball,
1850–1925.] Anyway, this is a lectureship that they have named after him.

**DeGroot:**
I guess there aren’t too many Bayesians on the statistics faculty
here at Berkeley.

**Blackwell:**
No. I’d say,
Lester
and I are the only ones in our department. Of course, over in operations Research,
Dick Barlow
and
Bill Jewell
are certainly sympathetic to the Bayesian approach.

**DeGroot:**
Is it a topic that gets discussed much?

**Blackwell:**
Not really; it used to be discussed here, but you very soon discover
that it’s sort of like religion; that it has an appeal for some people and not for
other people, and you’re not going to change anybody’s mind by discussing it. So
people just go their own ways. What has happened to Bayesian statistics surprised
me. I expected it either to catch on and just sweep the field, or to die. And I was
rather confident that it would die. Even though to me it was the right way to think,
I just didn’t think that it would have a chance to survive. But I thought that, if
it did, then it would sweep things. Of course, neither one of those things has happened.
Sort of a steady 5–10\% of all the work in statistical inference is done from a Bayesian
point of view. Is that what you would have expected 20\,years ago?

**DeGroot:**
No, it certainly doesn’t seem as though that would be a
stable equilibrium. And maybe the system is still not in equilibrium. I see the Bayesian
approach growing, but it certainly is not sweeping the field by any means.

**Blackwell:**
I’m glad to hear that you see it growing.

**DeGroot:**
Well, there seem to be more and more meetings of the Bayesians, anyway.
The actuarial group that met here at Berkeley over the last couple of days to discuss credibility
theory seems to be a group that just naturally accepts the Bayesian approach in
their work in the real world. So there seem
to be some pockets of users out there in the world, and I think maybe that’s what has kept the Bayesian approach alive.

**Blackwell:**
There’s no question in my mind that, if the Bayesian approach does
grow in the statistical world, it will not be because of the influence of other statisticians but because of the influence
of actuaries, engineers, business
people, and others who actually like the Bayesian approach and use it.

**DeGroot:**
Do you get a chance to talk much to researchers outside of statistics on campus, researchers in substantive areas?

**Blackwell:**
No, I talk mainly to people in Operations Research and Mathematics, and
occasionally Electrical Engineering. But the things in Electrical Engineering
are theoretical and abstract.

#### “The word ‘science’ in the title bothers me a little”

**DeGroot:**
What do you think about the idea of this new journal, *Statistical Science*, in
which this conversation will appear? I have the impression that you think the I.M.S.
is a good organization doing useful things, and there is really no need to mess with it.

**Blackwell:**
That is the way I feel. On the other hand, I must say that I felt exactly the
same way about splitting the *Annals of Mathematical Statistics* into
two journals, and that split seems to be working. So I’m hoping that the new journal
will add something. I guess the word “science” in the title bothers me a little.
It’s not clear what the word is intended to convey there, and you sort of have the
feeling that it’s there more to contribute a tone than anything else.

**DeGroot:**
My impression is that it *is* intended to contribute a tone. To give
a flavor of something broader than just what we would think of as theoretical statistics.
That is, to reach out and talk about the impact of statistics on the sciences and
the interrelationship of statistics with the sciences, all kinds of
sciences.

**Blackwell:**
Now, I’m all in favor of that. For example, the relation of statistics
to the law is to me a quite appropriate topic for articles in this journal. But somehow
calling it “science” doesn’t emphasize that direction. In fact, it rather suggests
that that’s *not* the direction. It sounds as though it’s tied in with things that are
supported by the National Science Foundation, and to me that restricts it.

**DeGroot:**
The intention of that title was to convey a broad impression rather
than a restricted one. To give a broader impression than just statistics and probability,
to convey an applied flavor and to suggest links to all areas.

**Blackwell:**
Yes. It’s analogous to computer science, I guess. I think *that*
term was rather deliberately chosen. My feeling is that the I.M.S. is just a beautiful
organization. It’s about the right size. It’s been successful for a good many years.
I don’t like to see us become ambitious. I like the idea of just sort of staying the
way we are, an organization run essentially by amateurs.

**DeGroot:**
Do you have the feeling that the field of statistics is moving away
from the I.M.S. in any way? That was one of the motivations for starting this journal.

**Blackwell:**
Well, of course, statistics has always been substantially bigger than the I.M.S. But you’re suggesting that the I.M.S. represents a smaller and smaller fraction of statistical activity.

**DeGroot:**
Yes, I think that might be right.

**Blackwell:**
You know, Morrie, I see what you’re talking about happening in
mathematics. It’s less and less true that all mathematics is done in mathematics
departments. On the Berkeley campus, I see lots of interesting mathematics being
done in our department, in Operations Research, in Electrical Engineering, in Mechanical Engineering, some in Business Administration, a lot in the Economics Department by
Gerard Debreu
and his colleagues; a lot of really interesting, high-class mathematics
is being done outside mathematics departments. What you’re suggesting is that statistics
departments and the journals in which they publish are not necessarily the centers
of statistics the way they used to be, that a lot of work is being done outside.
I’m sure that’s right.

**DeGroot:**
And perhaps *should* be done outside statistics departments.
That used to be an unhealthy sign in the field, and we worked hard in statistics
departments to collect up the statistics that was being done around the campus. But
I think, now that the field has grown and matured, that it is probably a healthy thing
to have some interesting statistics being done outside.

**Blackwell:**
Yes. Consider the old problem of pattern recognition. That’s a
statistical problem. But to the extent that it gets solved, it’s not going to be
solved by people in statistics departments. It’s going to be solved by people working
for banks and people working for other organizations who really need to have a device
that can look at a person and recognize him in lots of different configurations.
That’s just one example of the cases where we’re somehow too narrow to work on a lot of serious statistical
problems.

**DeGroot:**
I think that’s right, and yet we have something important to contribute
to those problems.

**Blackwell:**
I would say that we *are* contributing, but indirectly. That is, people
who are working on the problems have studied statistics. It seems to me that a lot of the engineers I talk to are very familiar with the basic concepts
of decision theory. They know about loss functions and minimizing expected risks and such
things. So, we have contributed, but just indirectly.

**DeGroot:**
You are in the National Academy of Sciences…

**Blackwell:**
Yes, but I’m very inactive.

**DeGroot:**
You haven t been involved in any of their committees or panels?

**Blackwell:**
No, and I’m not sure that I would want to be. I guess I don’t like the
idea of an official committee making scientific pronouncements. I like people to form opinions about scientific matters just on the basis of
listening to individual scientists. To have one group with such overwhelming prestige bothers
me a little.

**DeGroot:**
And it s precisely the prestige of the Academy that they rely on, when reports get issued
by these committees.

**Blackwell:**
Yes. So I think it’s just great as a purely honorific organization,
so to speak. To meet just once a year, and elect people more or less at random. I
think everybody that’s in it has done something reasonable and even pretty good,
in fact. But on the other hand, there are at least as many people *not* in it who have
done good things as there are in it. It’s kind of a random selection process.

**DeGroot:**
So you think it’s a good organization as long as it doesn’t do anything.

**Blackwell:**
Right I’m proud to be in it, but I haven’t been active. It’s sort
of like getting elected to Phi Beta Kappa—it’s nice if it happens to you…

#### “I play with this computer”

**DeGroot:**
Do you feel any relationship between your professional
work and the rest of your life, your interests outside of statistics? Is there any
influence of the outside on what you do professionally, or are they just
sort of separate parts of your life?

**Blackwell:**
Separate, except my friends are also my colleagues. It’s only
through the people with whom I associate outside that there’s any
connection. It’s hard to think of any other real connection.

**DeGroot:**
It’s not obvious what these connections might be for anyone. One’s political views or social views seem to be pretty much independent of the technical
problems we work on.

**Blackwell:**
Yes. Although it’s hard to see how it could *not* have an influence,
isn’t it? I guess my life seems all of a pace to me but yet it’s hard to see where the connections are.
[*Laughs.*]

**DeGroot:**
What do you see for your future?

**Blackwell:**
Well, just gradually to wind down, gracefully I hope. I expect
to get more interested in computing. I have a little computer at home, and it’s a
lot of fun just to play with it. In fact, I’d say that I play with this computer
here in my office at least as much as I do serious work with it.

**DeGroot:**
What do you mean by play?

**Blackwell:**
Let me give you an example. You know the algorithm for calculating
square roots. You start with a guess and then you divide the number by your guess
and take the average of the two. That’s your next guess. That’s actually Newton’s method
for finding square roots, and it works very well. Sometimes doing statistical work,
you want to take the square root of a positive-definite matrix. It occurred to me
to ask whether that algorithm works for finding the square root of a positive-definite
matrix. Before I got interested in computing, I would have tried to solve
it theoretically. But what did I do? I just wrote up a program and put it on the
computer to see if it worked.
[*Goes to blackboard.*]

Suppose that you are given the matrix __\( M \)__ and want to find __\( M^{1/2} \)__. Let __\( G \)__ be
your guess of __\( M^{1/2} \)__. Then you new guess is __\( (G+ MG^{-1})/2 \)__. You just iterate this and
see if it converges to __\( M^{1/2} \)__. Now, Morrie, I want to show you what happens.
[*Goes to terminal.*]

Let’s do it for a __\( 3{\times}3 \)__ matrix. We’re going to find the square root of a
positive-definite __\( 3{\times}3 \)__ matrix. Now, if you happen to have in mind a particular __\( 3{\times}3 \)__ positive-definite matrix whose square root you want, you could enter it directly. I don’t happen to have one in mind, but I do know a theorem: If you take any nonsingular
__\( 3{\times}3 \)__ matrix __\( A \)__, then __\( AA^{\prime} \)__ is going to be positive definite. So I’m just going to enter any __\( 3{\times}3 \)__ nonsingular matrix
[*putting some numbers into the terminal*]
and let __\( M = AA^{\prime} \)__. Now, to see how far off your guess __\( G \)__ is at any stage, you
calculate the Euclidean norm of the __\( 3{\times}3 \)__ matrix __\( M - G^2 \)__. That’s what I call the error.
Let’s start out with the identity matrix __\( I \)__ as our initial guess. We get a big error,
29 million. Now let’s iterate. Now the error has dropped down to 7 million. It’s going to keep being divided by 4 for a long time.
[*Continuing the iterations for a while.*]
Now notice, we’re not bad. There’s our guess, there’s its square, there’s what we’re
trying to get. It’s pretty close. In fact the error is less than one.
[*Continuing.*]
Now the error is really small. Look at that, isn’t that beautiful? So there’s just
no question about it. If you enter a matrix at random and it works, then that sort
of settles it.

But now wait a minute, the story isn’t quite finished yet. Let me just continue
these iterations… Look at that! The error got bigger, and it keeps getting bigger.
[*Continuing.*]
Isn’t that lovely stuff?

**DeGroot:**
What happened?

**Blackwell:**
Isn’t that an interesting question, what happened? Well, let me
tell you what happened. Now you can study it theoretically and ask, should it converge?
And it turns out that it will converge if, and essentially only if, your first guess
commutes with the matrix __\( M \)__. That’s what the theory gives you. Well, my first guess
was __\( I \)__. It commutes with everything. So the procedure theoretically converges. However, when you calculate,
you get round-off errors. By the way, if your first guess commutes, then all subsequent
guesses will commute. However, because of round-off errors, the matrices that you actually
get don’t quite commute. There are two ways to do this. We could take __\( MG^{-1} \)__ or we
could have taken __\( G^{-1}M \)__. Of course, if __\( M \)__ commutes with __\( G \)__, then it commutes with __\( G^{-1} \)__
and it doesn’t matter which way you do it.
But if you don’t calculate __\( G \)__ exactly at some stage, then it will not quite commute.
And in fact, what I have here on the computer is a calculation at each stage of
the noncommutativity norm. That shows you how different __\( MG^{-1} \)__ is from __\( G^{-1}M \)__. I
didn’t point those values out to you, but they started
out as essentially 0, and then there was a 1 in the 15th place,
and then a 1 in the 14th place, and so on. By this stage, the noncommutativity norm
has built up to the point where it’s having a sizable influence on the thing.

**DeGroot:**
Is it going to diverge, or will it come back down after
some time?

**Blackwell:**
It won’t come back down. It will reach a certain size, and sometimes
it will stay there and sometimes it will oscillate. That is, one __\( G \)__ will go into a
quite different __\( G \)__, but then that __\( G \)__ will come back to the first one. You get periods,
neither one of them near the truth. So that’s what I mean by just playing, instead
of sitting down like a serious mathematician and trying to prove a theorem. Just
try it out on the computer and see if it works.
[*Laughs.*]

**DeGroot:**
You can save a lot of time and trouble that way.

**Blackwell:**
Yes. I expect to do more and more of that kind of playing. Maybe I get lazier as
I get older. It’s fun, and it’s an interesting toy.

**DeGroot:**
Do you find yourself growing less rigorous in your mathematical work?

**Blackwell:**
Oh, yes. I’m much more interested in the ideas, and in truth under not-completely-specified
hypotheses. I think that has happened to me over the last 20\,years. I can certainly
notice it now.
Jim MacQueen
was telling me about something that he had discovered.
If you take a vector and calculate the squared correlation between that vector and
some permutation of itself, then the average of that squared correlation over all
possible permutations is some simple number. Also, there was some extension of this
result to __\( k \)__ vectors. He has an interesting algebraic identity. He told me about it,
but instead of my trying to prove it, I just selected some numbers at random and
checked it on the computer. Also, I had a conjecture that some stronger
result was true. I checked it for some numbers selected at random, and it turned out to be true for him and *not* true for what I
had said. Well, that
just settles it. Because suppose you have an algebraic function
__\( f(x_1,\dots,x_n) \)__,
and you want to find out if it is identically 0. Well, I think it’s true that any
algebraic function of __\( n \)__ variables is either identically 0 or the set of __\( x \)__’s for which
it is 0 is a set that has measure 0. So you can just select __\( x \)__’s at random and evaluate
__\( f \)__. If you get 0, it’s identically 0.
[*Laughs.*]

**DeGroot:**
You wouldn’t try even a second set of __\( x \)__’s?

**Blackwell:**
I did.
[*Laughs.*]

**DeGroot:**
Getting more conservative in your old age.

**Blackwell:**
Yes.
[*Laughs.*]
I’ve been wondering whether in teaching statistics the typical
set-up will be a lot of terminals connected to be a big central computer or a lot of small personal computers.
Let me turn the interview around. Do you have any thoughts about which way that is going or which way
it ought to go?

**DeGroot:**
No, I don’t know. At Carnegie-Mellon we are trying to have both
worlds by having personal computers but having them networked with each other. There’s a plan at Carnegie-Mellon that each student will have to have a personal computer.

**Blackwell:**
Now when you say each student will have to have a personal computer, where will it
be physically located?

**DeGroot:**
Wherever he lives.

**Blackwell:**
So that they would not actually use computers in class on the
campus?

**DeGroot:**
Well, this will certainly lessen the burden on the computers
that are on campus, but in a class you would have to have either terminals
or personal computers for them.

**Blackwell:**
Yes. I’m pretty sure that in our department in five years we’ll
have several classrooms in which each seat will be a work station for a student,
and in front of him will be either a personal computer or a terminal. I’m not sure which,
but that’s the way we’re going to be in five years.

#### “I wouldn’t dream of talking about a theorem like that now”

**DeGroot:**
A lot of people have seen you lecture on film. I know of at least one film
you made for the American Mathematical Society that I’ve seen a few times. That’s a beautiful film, “Guessing at Random.”

**Blackwell:**
Yes. I now, of course, don’t think much of those ideas.
[*Laughs.*]

**DeGroot:**
There were some *minimax* ideas in there…

**Blackwell:**
Yes, that’s right. That was some work that I did before I became
such a committed Bayesian. I wouldn’t dream of talking about a theorem like that
now. But it’s a nice result…

**DeGroot:**
It’s a nice result and it’s a beautiful film. Delivered
so well.

**Blackwell:**
Let’s see… How does it go? If I were doing it now I would
do a weaker and easier Bayesian form of the theorem. You were given an arbitrary sequence
of 0s and 1s, and you were going to observe successive values, and you had to predict
the next one. I proved certain theorems about how well you could do against every possible sequence.
Well, *now* I would say that you have a probability distribution on the set of all
sequences. It’s a general fact that if you’re a Bayesian,
you don’t have to be clever. You just *calculate*. Suppose that somebody generates an arbitrary sequence of
0s and 1s and it’s your job after seeing each finite segment to predict the next
coordinate, 0 or 1, and we keep track of how well you do. Then I have to be clever and invoke the
minimax theorem to devise a procedure that asymptotically does
very well in a certain sense. But now if you just put a prior distribution on
the set of sequences, any Bayesian knows what to do. You just calculate
the probability of the next term being a 1 given the past history. If it’s more than __\( 1/2 \)__ you
predict a 1, if it’s less than __\( 1/2 \)__ you predict a 0. And that simple procedure has the corresponding
Bayesian version of all the things that I talked about in that film. You just know
what is the right thing to do.

**DeGroot:**
But how do you know that you’ll be doing well in relation to the
reality of the sequence?

**Blackwell:**
Well, the theorem of course says that you’ll do well for all sequences except
a set of measure zero according to your own prior distribution, and that’s all a
Bayesian can hope for. That is, you have to give up something, but it just makes
life so much *neater*. You just know that this is the right thing to do.

I encountered the same phenomenon in information theory. There is a very good theory about how to transmit over a channel, or how to transmit over a sequence of channels. The channel may change from day to day, but if you know what it is every day, then you can transmit over it. Now suppose that the channel varies in an arbitrary way. That is, you have one of a finite set of channels, and every day you’re going to be faced with one of these channels. You have to put in the input, and a guy at the other end gets an output. The question is, how well can you do against all possible channel sequences?

You don’t really know what the weather is out there, so you don’t know what
the interference is going to be. But you want to have a code that transmits well
for all possible weather sequences. If you just analyze the problem crudely, it turns out
that you can’t do *anything* against all possible sequences. However, if you select
the code in a certain random way, your overall error probability will be small for each weather sequence.
So, you see, it’s a nice theoretical result but it’s unappealing. However, you can get exactly
the same result if you just put a probability distribution on the sequences. Well,
the weather could be any sequence, but you expect it to be sort of this way or
that. Once you put a probability distribution on the set of sequences,
you no longer need random codes. And there is a deterministic code that gives you
that same result that you got before. So either you must behave in a random way, or you must
put a probability distribution on nature.

[*Looking over a copy of his paper, Blackwell, Breiman and Thomasian: “The capacities of certain channel classes under random coding”*
[e4].]
I don’t think we did the nice
easy part. We behaved the way
Wald
behaved. You see, the minimax theorem says
that if for every prior distribution you can achieve a certain gain, then there is
a random way of behaving that achieves that gain for every parameter value. You don’t need the prior distribution; you can throw it away. Well,
I’m afraid that in this paper, we invoked the minimax theorem. We
said, take any prior distribution on the set of channel sequences. Then you can achieve
a certain rate of transmission for that prior distribution. Now you invoke the minimax
theorem and say, therefore, there is a randomized way of behaving which enables you
to achieve that rate against every possible sequence. I now wish that we had *stopped*
at the earlier point.
[*Laughs.*]
For us, the Bayesian analysis was just a preliminary
which, with the aid of the minimax theorem, enabled us to reach the conclusions we
were seeking. That was Wald’s view and that’s the view that we took in that paper.
I’m sure I was already convinced that the Bayes approach was the right approach, but
perhaps I deferred to my colleagues.

**DeGroot:**
That’s a very mild compromise. Going *beyond* what was necessary
for a Bayesian resolution of the problem.

**Blackwell:**
That’s right. Also, I suspect that I had
Wolfowitz
in mind. He
was a real expert in information theory, but he wouldn’t have been interested in
anything Bayesian.

**DeGroot:**
What about the problem of putting prior distributions on spaces
of infinite sequences, or function spaces? Is that a practical problem and is there
a practical solution to the problem?

**Blackwell:**
I wouldn’t say for infinite sequence, but I think it’s a very important
practical problem for large finite sequences, and I have no idea how to solve it. For
example, you could think that the pattern-recognition problem that I was talking
about before is like that. You see an image on a TV screen. That’s just a long finite
sequence of 0s and 1s. And now you can ask how likely it is that that sequence of
0s and 1s is intended to be the figure 7, say. Well, with some you’re certain that
it is, and some you’re certain that it isn’t, and with others there’s a certain probability that it is and
a probability that it isn’t. The problem of describing that probability distribution
is a very important problem. And we’re just not close to knowing how to describe
probability distributions over long finite sequences that correspond to our opinions.

**DeGroot:**
Is there hope for getting such descriptions?

**Blackwell:**
I don’t know. But again it’s a statistical problem that is not
going to be solved by professors of statistics in universities. It might be solved
by people in artificial intelligence, or by researchers outside universities.

#### “Just tell me one or two interesting things”

**DeGroot:**
There’s an argument that says that, under the Bayesian approach, you have to seek
the optimal decision and that’s often just too hard to find. Why not settle for some other approach
that requires much less structure, and get a reasonably good answer out of it, rather than an optimal answer?
Especially in these kinds of problems where we don’t know how to
find the optimal answer.

**Blackwell:**
Oh, I think everybody would be satisfied with a reasonable
answer. I don’t see that there’s more of an emphasis in the Bayesian approach on
optimal decisions than in other approaches. I separate Bayesian inference from Bayesian
decision. The inference problem is just calculating a posterior distribution, and
that has nothing to do with the particular decision that you’re going to make. The same
posterior distribution could be used by many different people making different decisions.
Even in calculating the posterior distribution, there is a lot of approximation.
It just can’t be done precisely in interesting and important cases. And I don’t think
anybody who is interested in applying Bayes’ method would insist on something that’s precise to the fifth decimal place. That’s just the conceptual framework in which you
want to work, and which you want to approximate.

**DeGroot:**
That same spirit can be carried over into the decision problem, too.
If you can’t find the optimum decision, you settle for an approximation to it.

**Blackwell:**
Right.

**DeGroot:**
In your opinion, what have been the major breakthroughs in the field
of statistics or probability through the years?

**Blackwell:**
It’s hard to say… I think that theoretical statistical thinking was just completely dominated by
Wald’s
ideas for a long time.
Charles Stein’s
discovery that __\( \bar{X} \)__ is inadmissible was certainly important.
Herb Robbins’
work on empirical Bayes was also a big step, but possibly in the wrong direction.

You know, I don’t view myself as a statesman or a guy with a broad view of the field or anything like that. I just picked directions that interested me and worked in them. And I have had fun.

**DeGroot:**
Well, despite the fact that you didn’t choose the problems for their impact or because of
their importance, a lot of people have gained a lot from your work.

**Blackwell:**
I guess that’s the way scholars *should* work. Don’t worry about
the overall importance of the problem; work on it if it looks interesting. I think there’s probably a sufficient correlation between interest and
importance.

**DeGroot:**
One component of the interest is probably that others are interested
in it, anyway.

**Blackwell:**
That’s a big component. You want to tell somebody about it after you’ve done it.

**DeGroot:**
It has not always been clear that the published papers in our
more abstract journals did succeed in telling anybody about it.

**Blackwell:**
That’s true. But if you get the fellow to give a lecture on it,
he’ll probably be able to tell you something about it. Especially if you try to restrict
him: Look, don’t tell me everything. Just tell me *one or two* interesting things.

**DeGroot:**
You have a reputation as one of the finest lecturers in the field. Is that your style
of lecturing?

**Blackwell:**
I guess it is. I try to emphasize that with students. I notice
that when students are talking about their theses or about their work, they want
to tell you everything they know. So I say to them: You know much more about this
topic than anybody else. We’ll never understand it if you tell it all to us. Pick
just one interesting thing. Maybe two.

**DeGroot:**
Thank you, David.