"Mises Redux" -- Redux: Fifteen Arguments against Finite Frequentism
Author(s): Alan Hájek
Source: Erkenntnis (1975-), Vol. 45, No. 2/3, Probability, Dynamics and Causality (Nov., 1996),
pp. 209-227
Published by: Springer
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ALAN HAJEK
MISES REDUX" - REDUX: FIFTEEN ARGUMENTS AGAINST
FINITE FREQUENTISM
ABSTRACT. According to finite frequentism, the probability of an attributeA in a finite
class B
reference
fifteen
arguments
is the relative
against
frequency
this position.
1.
The most
of actual
occurrences
of A within
B.
I present
INTRODUCTION1
of probability
is frequentism.
interpretation
a
the
that
coin
lands
heads when
says:
Roughly,
probability
frequentism
tossed is the number of times that the coin lands heads, divided by the total
widely
accepted
of times that the coin is tossed; the probability
that a radium atom
so decay, divided
is
of
atoms
in
the
number
radium
1500
that
years
decays
so
on.
of
the
number
and
total
radium
This
should
sound familiar
atoms;
by
?
?
all too familiar
for somehow
this notion still pervades much scientific
it
But
should be rejected, as Iwill argue here ?
about
probability.
thinking
fifteen times over.
number
or philosophically
To philosophers
inclined scientists,
the demise of
is familiar, I admit, even though it hasn't quite been universally
frequentism
are many of the arguments
that Iwill present here
accepted.2 Familiar too
?
were
some
of
them
Richard
indeed,
inspired by
Jeffrey's "Mises Redux"
I
to
it
will
be
useful
have
them
hope
(1977)?though
gathered in one place.
in this paper are new, as far as I am aware. So even if
arguments
the fact that there is bad news for frequentism
is old news, I hope it is
news
how
much
bad
there
newsworthy
just
really is.
Other
The
stance
important
here.
that one
Iwill
takes on
begin
issues
by saying
of philosophical
some
friendly
is
methodology
about
the
role
things
of intuition in the philosophical analysis of objective probability, by way
of preparation for the unfriendly things that Iwill say about frequentism
as such an analysis.
two frequentist analyses ?finite fre?
Iwill distinguish
space limitations require
quentism and hypothetical frequentism.
Although
me to confine my discussion
to the former, many of the arguments
that I
will adduce will count equally against both.
69
Erkenntnis 45: 209-227, 1997.
1997
Kluwer
Academic Publishers. Printed in theNetherlands.
?
210
ALAN HAJEK
OBJECTIVE PROBABILITY: INTUITIONS AND ANALYSIS
2.
and science. The
is, I claim, a concept of both commonsense
Probability
on
street
and
it
the
understands
person
recognizes
(at least to some extent)
in locutions such as "the probability
that this coin lands heads is 1/2"; the
and understands
it (at least to some extent)
in locu?
recognizes
to be spin 'up' with probability
such as "this electron
is measured
1/2". Commonsense
and science are joined by a two-way street: scientific
scientist
tions
theories
are, after all, invented by people who share the folk's conceptual
and who
seek to refine it; and commonsense,
in turn, partly
apparatus,
some
as
of
these
scientific
ideas become popu?
refinements,
incorporates
concept of gravity, for example, was once a part of neither
nor science, and now it is part of both. I believe
the same is
true of the concept of probability.
larized.
The
commonsense
seem
economists...
statisticians, physicists,
Many computer scientists,
or
is
relative
to speak as if probability
ifs
end
of
buts,
simply
frequency?no
a
as
We
to
mistaken.
would
do
better
think
of
it
is
This
surely
putative
story.3
one which both informs
notion of probability,
analysis of our pretheoretical
scientific notion of probability.
by a more
sophisticated
a
term that one is free to define
not
after
is
technical
all,
just
'Probability',
to our
as one pleases. Rather,
is answerable
it is a concept whose
analysis
intuitions, a concept that has various associated platitudes
(for example: "if
and
is informed
has probability
greater than 0, then X can happen"). Thus, it is unlike
or 'material
like 'complete metric
space' or 'Granger causation'
are
with
definitions
which
there
which
there
for
conditional',
stipulative
X
terms
platitudes. What
'probability'
arguing, and no associated
or
or 'if ... then',
like 'space'
'causation'
is like, instead, are concepts
It is fair game to
of analyses.
that can be the subject matter
concepts
to
it
is
fair
and
such
game
certainly
dispute frequentism.
analyses;
dispute
about my project
I say this early on, to forestall any possible puzzlement
from various computer
that I have already encountered
here (a puzzlement
...
statisticians
scientists,
).
is at best an analysis of objective probability,
Furthermore,
frequentism
is no
sensible
for example, from
objective chance (to be distinguished,
or degree of belief.) As we will
it
see, however,
probability,
subjective
cannot be even that.
have some sort
and relative frequency
This is not to deny that probability
sometimes
called
connection.
theorists, logical probabilists,
propensity
Subjectivists,
that an event with probability
all agree, for example,
and so on presumably
to occur roughly half of the time, in some senses of
half should be expected
and 'roughly' (I would
words
the
say this is another
'should',
'expected',
can
a
out
relative frequency
I concede
that finding
platitude). Moreover,
70
of close
FIFTEENARGUMENTSAGAINST FINITEFREQUENTISM
211
the only
way of finding out
a
I do not deny the existence of some interesting
the value of
probability.
their identification.
the two; I am only disputing
between
relationship
notion
So far I have taken as a starting point our commonsensical
as an analysis
of that
and I have regarded frequentism
of probability,
from another direction.
(a bad one). But we could come to frequentism
often
the best
be
?
and
even
sometimes
of probability. More
is, as I said at the outset, an interpretation
a
the
of
of
axioms
it
is
theory?
putative interpretation
probability
precisely,
So starting with a primitive,
those provided by Kolmogorov.
traditionally,
Frequentism
function P, defined over a certain set-theoretic
substructure,
uninterpreted
we
come to a
and
is non-negative,
which
normalized,
additive,
might
a
to play.
P.
Here
have
role
of
intuitions
again,
frequentist understanding
to do with our intuitive notion of
that have nothing
toKolmogorov's
axioms, and so in some sense provide
probability
an interpretation
think of mass, or length, or volume, which are
of them
can be suitably normalized.
and additive, and which
clearly non-negative
commonsense
tells us
They are not even in the running, however, because
For many
quantities
conform
thatprobability is simply something else. (Just try substituting any of them
into the platitudes above!) Incidentally, this also shows that it is too glib to
as long as
of probability
is provided
say that a satisfactory
understanding
we find a concept of importance to science that conforms to the axioms-for
that does not narrow down the field enough. In any case, the more strictly
our commonsensical
concept would
project of analysing
as the project of analysing our commonsensical
concept of
even
we
if
the
had
done
remain
would
causation,
say,
already
job for the
as
in
it
science.
appears
concept
philosophical
remain, much
3.
VERSIONS OF FREQUENTISM
to distinguish
two variants of actual frequentism,
It is necessary
and these
to
from hypothetical
actual
the prob?
frequentism. According
frequentism,
an
or
to
event
is
of
attribute
be
its
identified
actual
relative
with
ability
for all the requisite
there is no need to 'leave the actual world',
frequency:
if there happen to be infinitely many events or
facts are right here. Now,
attributes of the requisite
sort, then we cannot simply count the number
and divide this by the total number of trials, since this will
of 'successes'
take the indeterminate
relative
frequency
In that case, we take the limit of the
trial, as n tends to infinity. Hypothetical
form oo/oo.
up to the nth
frequentism keeps the intuition that probability is such limiting relative
frequency, but applies when the actual world does not furnish the infinitely
It thus identifies probability with a counterfactual
trials required.
many
71
212
ALAN HAJEK
relative
limiting
frequency:
trials.
the limiting
relative
frequency
if there were
infinitely many
I cannot discuss
since
the problems with hypothetical
here; and
frequentism
the infinite variant of actual frequentism
suffers from many of the
same problems, my discussion
of it is best left for another occasion
also.4
So let me focus solely, then, on the version of actual frequentism
in which
trials of the relevant sort - for short, finite
there are only finitely many
frequentism:
FINITE FREQUENTISM: The probability of an attributeA in a finite
class B
reference
within
is the relative
frequency
of actual
occurrences
of A
B.
Venn (1876), in his discussion of the proportion of births of males and
is nothing but that proportion"
(p. 84, his
"probability
at
its purest. Reichenbach
This I take to be finite frequentism
in the end looked more
inclinations
also, although his account
concludes:
females,
emphasis).
had such
accounts were pursued
in
finite frequentist
like hypothetical
frequentism;
and Putnam among others.
detail by Cramer, Hempel
thus be thought
Of course, all of this was a long time ago. It might
interest as far as the philosophical
is at best of historical
that frequentism
I have already emphasized
literature is concerned?its
currency elsewhere
?
more
in
these
times. Up to a
of
much
scrutiny
enlightened
unworthy
more
I guess; but only up to a point. As I indicated
this is true enough,
inclined
still has its proponents
among philosophically
earlier, frequentism
some
even
of
the
criticisms
and
Furthermore,
statisticians,
philosophers.
point,
accounts
that
here have some force against more sophisticated
presented
have grown out of frequentist
soil, for example Lewis'
(1994) 'best system'
as
at
I
the
will
to
chance,
argue
point.
appropriate
objective
approach
Indeed,
account of objective
chance could be
if an empiricist
and I suspect that many of
look a lot like frequentism,
could be adapted accordingly
against any such account.
scruples should not give
frequentist with serious empiricist
one wonders
that didn't
given
these arguments
Any aspiring
up on finite frequentism
comes at a considerable
lightly. The move
to hypothetical
say,
frequentism,
price, one that an empiricist should be
is really the only version that upholds
to pay. Finite frequentism
unwilling
that might make frequentists
inclinations
scientific
the anti-metaphysical,
at
first
In
us
first
in the
of
blush, it is an attractive theory. It
any case,
place.
are
it apparently
well understood;
is a reductive analysis, whose primitives
makes
the epistemology
metaphysical
of probability
72
straightfoward;
unlike
the classical
FIFTEENARGUMENTSAGAINST FINITEFREQUENTISM
213
it appears to be about the world; and it
and logical theories of probability,
seems to be inspired by actual scientific practice.
at second blush, it does not look nearly so good: it runs afoul
However,
of many
important
intuitions
that we
have
Or
so I will
of them?
It might
about probability.
argue.
4.
THE ARGUMENTS
are almost
We
make
ready for the arguments.
that I am uneasy
suspicious
Why
about
so many
them, substituting
quantity
for
buckets'
'leaky
metaphor
philosophical
one is better than fifteen leaky
to paraphrase him, one watertight
arguments:
ones. Suffice to say that I think the arguments here are pretty watertight,
and some of them are pretty decisive on their own.
you
for quality.
It recalls
Flew's
reason for giving so many arguments
is this. You might agree with
cannot be an analysis of our concept of (objective)
think that it is, so to speak, a success?
however,
you might
probability;
One
me
that frequentism
one that captures an important and
of the concept,
ful partial
analysis
central strand in our thinking about probability
(even if there are other
even
In
I feel some temptation
such strands).
moments,
my ecumenical
to concede
this: perhaps we should let a thousand flowers bloom, with
one of them. Or you might
think that frequentism
is a
of
of
the
surrogate for
concept
cleaned-up
probability?a
good explication
even
a messy,
one
confused
suitable for
vague, and
concept,
ambiguous,
use in science and clear-headed
discourse.
But I do think that the many
frequentism
being
successively
chip away at frequentism's
possible domain,
its
interest.
steadily reducing
reason for giving so many arguments
is that it shows just how
Another
are
for retrenching frequentism
in favor of some close
dim
the prospects
arguments
here
to itmight prompt one to
relative of it. A single class of counterexamples
in order to save it.What I hope tomake clear, by piling
add a single epicycle
are not
on ever more arguments
is that the problems
against frequentism,
an artifact
with
of some particular presentation
cosmetic
surgery. No
a little clever
of it, ones that would go away
the problems with frequentism
run deep.
the title and fanfare, I don't want to be too fussy about how the
are counted. Not all of the arguments are completely
indepen?
arguments
dent of each other; in fact, several of them might be regarded as stemming
Despite
from a single intuition (that probability
On
of 'counterfactual
independence').
73
can obey a certain sort
I might
the other hand, elsewhere
statements
214
ALAN HAJEK
the one
two or more
that could be
heading
arguments
the arguments
in a way that I hope is natural.
separated. Iwill distinguish
Iwill begin with some general arguments
that I think are telling against
form
of
then, arguments
any
frequentism;
specifically
against finite fre?
combine
under
quentism.
GENERAL PROBLEMS CONCERNING ANY VERSION OF FREQUENTISM
1. The Reference
We
Class Problem
events
think that various
and indeed we
bilities,
have unconditional
straightforwardly
proba?
even have theories that tell us what some of these
are. But it seems that frequentism
delivers only conditional
- or
in any case, relativized probabilities.5
Von Mises
probabilities
(1957)
"It is only the notion of probability
in a given collective
writes:
which
probabilities
is unambiguous" (p. 20). Suppose I am interested inmy probability of
dying by age 60. What I want is an unconditional probability. I can be
reference classes: the set of all living things; the set of all
set
the set of all non-smoking
all males;
the
of
males who exercise
humans;
the set of all philosophers;
the set of all Woody
Allen
fans
occasionally;
... Each of these reference
classes will have its own associated
relative
for death by age 60. But I'm not interested inmy probability
of
frequency
in various
placed
death qua philosopher,
say. To repeat, Iwant
Here we confront the notorious
'reference
an unconditional
probability.
a given event
class problem':
or attribute has more than one relative frequency;
to the fre?
and according
or
means
event
more
one
that
the
this
attribute
has
than
quentist,
probability.
is some
have been quick to mock Car
irony here. Frequentists
on
the grounds that itmust always be relativized
napian logical probability
to a choice of language, and no single language seems to be the canoni?
cal one. But a parallel problem
is practically
alluded to in the very name
must
'relative frequency'
to a choice
frequencies
always be relativized
no
of reference class, and
single reference class seems to be the canonical
There
one.
I see only one possible way out for the frequentist. He should insist that
is really conditional;
all probability
and that a putative unconditional
prob?
is really elliptical for a conditional
statement
ability statement
probability
is tacit. He could maintain
in which
the condition
that probability
theory
could
between
still do a lot of work
conditional
for us. For example, knowledge
of inequalities
we
be
all
that
need
in order to
might
probabilities
control
our environment
ficially.
(When you
in desirable ways,
see that the conditional
74
modifying
our behavior
probability
of death by age 60,
bene?
FIFTEENARGUMENTSAGAINST FINITEFREQUENTISM
215
is substantially
you
greater than it is given non-smoking,
given smoking,
- at
see a good reason to quit smoking
least when you have ruled out other
such as the existence of a common cause,
for this correlation,
explanations
also
on the sole basis
of conditional
rather than seeing the reference
it.
embrace
In short,
information.)
probability
as
a
class problem
the frequentist
problem,
could
a way out of the reference class prob?
Perhaps this gives the frequentist
that
this
'eliminativism'
he
admit
should
lem. But
regarding unconditional
is somewhat
science seems to abound
radical, if only because
probability
with
statements
longer pretend
And of course he can no
probability.
an interpretation of Kolmogorov's
axioms.
of unconditional
to be giving
2. Typing Events
may Change
the Probability
princi?
Every event, in all itsmyriad detail, is unique. This is just Leibniz's
are
to
events.
of
So
if
of
the
indiscernibles,
you
identity
applied
going
ple
to group an event with others, you will have to allow differences
between
them. But
difference
to be a guarantee
that these differences
to the probability. The thought must be that whatever
there needs
make
no
the differ?
over
are, they are not relevant. The bulge in the carpet then moves
to the notion of 'relevance'.
It had better not be probabilistic
relevance, on
pain of circularity. But what is it, then?
ences
drains from finite frequentism
Plausibility
especially when the putative
or too small. Unfortunately
reference
for the
class is too heterogeneous,
in tandem, so that solving one tends to
these problems work
frequentist,
can be enhanced by raising the admis?
the other. Homogeneity
exacerbate
sion standards into a reference class, demanding greater similarity between
or events; but that reduces
that can be admitted.6
the individuals
events
3. Probabilities
Distant Events
of Local
Events
the number
can be Counterfactually
of
individuals
Independent
or
of
Let me first continue my plea on behalf of commonsense,
with a little homi?
Sometimes
arguments against a philo?
ly on philosophical
argumentation.
to
show
that
the
sophical position attempt
position has internal difficulties
some
are of this form). On the
of
my arguments against frequentism
(and
other hand, sometimes
intuitions
arguments begin with commonsensical
that are supposed to be dear to us, and then deploy these intuitions against
a philosophical
position with which
they clash. For example, we might
begin with the following
hand is a matter solely
the fire, my
hand,
commonsensical
intuition: this fire's burning my
involving a small region of space-time
containing
and little else; and then deploy
this intuition against
75
216
ALAN HAJEK
Hume's
often
with
tion
and
are
'Intuition-based'
regularity account of causation.
arguments
as 'internal-difficulty'
not as damaging
and
when
faced
arguments;
an intuition-based
a proponent
of the philosophical
argument,
posi?
in question might simply retort that the argument is
question-begging,
that we should revise our intuitions. Nevertheless,
on the
depending
and the weight
that we attach to common
strength of those intuitions,
can
some
such
still
have
sense,
arguments
pull on us. Much
philosophy?
the greater part of it, Iwould
in just this way. (Think espe?
say-proceeds
famous
the Chinese Room,
cially of philosophy's
thought experiments:
and so on.) The
Twin Earth,
argument
that Iwant
to turn to now
is of this
form.
is a radium
Here
1/2. If radium
atoms
atom;
distant
its probability
of decaying
in 1500 years is
from it in space and time had behaved differ?
ently, this probability would still have been 1/2.1 submit that probability
statements
about
things can be (and perhaps
typically
are) counterfactually
independent of what other things ofthat kind happen to do; and probability
statements
about
can be (and perhaps
typically are) counter
of distant events of the same kind. But according
to
this is not so.
local events
factually
independent
relative frequentism,
statements about a thing at a time can be
And I submit that probability
of the behavior ofthat
counterfactually
independent
thing at other times.
The chance that this coin lands heads now does not depend on how the
? as
itwere, the coin doesn't have to 'wait and
coin will land in the future
it happens to do in the future in order to have a certain chance of
now. To put the point crudely, though vividly:
heads
it's almost as if
landing
in something
like backward causation from future
the frequentist believes
see' what
Put more carefully:
the frequentist believes
that
on the chance that the
of the coin places constraints
is counterfactually
coin lands heads now9 and in that sense, that chance
on
the future behavior.
Indeed, if the coin has yet to be tossed,
dependent
to the
determines
that current chance, according
the future behavior
fully
results
to current
the future
chances.
behavior
I think the frequentist has things backwards:
surely it is
frequentist.7
of landing heads that gives rise to its statistics, rather
the coin's probability
than the other way round.
finite
And
Hume's
so
the same fate as
suffers much
Frequentism
The fact that this flame burned my hand does
other flames happen to be contiguous with other
it is in general.
theory of causation.
not depend on whether
hands getting burned. The
this argument is that probability,
matter
than that.
private
some
of my arguments have force
I
earlier
that
said
briefly:
Digressing
more
that
of objective
analyses
probability
against certain
sophisticated
like causation,
intuition behind
is a far more
76
FIFTEENARGUMENTSAGAINST FINITEFREQUENTISM
217
of frequentism,
could be thought of as refinements
(1994).
notably Lewis'
that
It runs roughly as follows. The laws of nature are those regularities
are theorems of the best theory: the true theory of the universe
that best
of
(that is, the probability
strength, and likelihood
simplicity,
course of history, given the theory).
If any of the laws are
these laws say they are. It seems
then the chances are whatever
probabilistic,
balances
the actual
to Lewis,
the probability
that this radium atom decays
that according
in
1500 years does depend on what other, perhaps distant atoms (and perhaps
?
at least this will be so under the
not just radium atoms) happen to do
are
that
there
sufficiently many such atoms. For if many such
assumption
atoms
say, much
had decayed,
earlier
than they actually
did
(something
the best theory will admit has positive chance), then plausibly the best
?
it would
have
decay laws
to have reasonable
In particular, plausibly
likelihood.
law for radium would have been different, and hence so too the
had different
have
theory would
to in order
needed
the decay
radioactive
for this particular radium atom.
decay probability
I think that the next two arguments also carry some weight
against the
and perhaps even the two after that (with some smallish
Lewis analysis,
of (finite) frequentism.
but let us return to our discussion
modifications);
Concern
4. An Argument
from
Let me
a variation
pursue
on the
'counterfactual
I am
theme.
independence'
'argument from
(1980) famous
inspired here by Kripke's
to which
Lewis'
counterpart
theory, according
against
across
identified
worlds:
genuinely
possible
entities
concern'
cannot
be
to Lewis]
ifwe say 'Humphrey might have won the
[According
we are not talking
election (if only he had done such-and-such)',
about something
that might have happened
toHumphrey
but to
someone
a
else,
"counterpart".
Probably, however, Humphrey
someone else, no matter how much
could not care less whether
resembling
him, would
have been victorious
in another possible
world, (p. 45)
from concern
Arguments
of B9 then our concerns
A
cannot
defeasible,
contingent
the form:
"If A were
the correct
analysis
such-and-such;
they are not; hence
be the correct analysis of BV To be sure, such arguments
are
our
concerns
and what
to be is, I suppose,
a highly
happen
matter.
philosophical
contingent
have
-1
would
be
as thought experiments
can be a source of
our
are surely highly
to
them
responses
though
Still, much
insight?even
believe
such arguments
77
can be also. And
they may
serve a
218
ALAN HAJEK
distinctive
trading as they do not only on our beliefs,
function,
our desires
(fears, regrets,...
but also on
).
It is natural to think thatmy probability of dying by a certain age is a
of me
property
though von Mises
(or perhaps me plus my immediate environment).
Natural,
goes out of his way to deny it: "We can say nothing about
the probability of death of an individual even ifwe know his condition of
life and health in detail. The phrase
it refers
'probability of death', when
to a single person, has no meaning
at all for us" (p. 11). Also: "It is utter
nonsense
to say, for instance, thatMr. X, now aged forty, has the
probability
0.011 of dying in the course of the next year" (pp. 17?18). But surely it is
that Mr. X really cares about, when
just such 'nonsense'
about his probability
of death. Of course, he may well
about his fellow citizens
too, and he may be concerned
he is concerned
care
unselfishly
to find out how
high the death rate is among people of his type. But to the extent that
the other people can drop out of the
his concerns are directed to himself,
can drop
say that Humphrey's
counterparts
picture (much as Kripke would
concern about losing the
out of the picture when
it comes to Humphrey's
election). The statistics about others like him may give him good evidence
as to his own chance of dying, but the fact
cares about
that he ultimately
? one
a
of
is a fact about himself
by
expressed
meaningful
'probability
death' statement that refers to a single person.
GENERAL PROBLEMS CONCERNING ACTUAL FREQUENTISM (BOTH FINITE
AND INFINITE)
5. Actual
Here
is another
variant
though sufficiently
once
of Autrecourt
object,
Commits
Frequentism
one cannot
of
the
said words
the existence
to a Surprisingly
Rich Ontology
'counterfactual
argument,
independence'
separate treatment, I think. Nicolas
to the effect that, from the existence
of one
to merit
different
deduce
one
the existence
of others. But
of a non-trivial
to finite
according
for an event does
probability
of other, similar events in the actual world. The fact that
for dying by age 60 proves that I am not alone
I have a non-trivial probability
some probability
to
the
Or consider
in the world,
according
frequentist.
frequentism,
imply the existence
own mind:
that itwill deteriorate by age
for example,
to
this means
the finite frequentist,
0.1. According
60 with probability
like mine experience
such
that a tenth of the people out there with minds
?
are
course
that
there
other
such
of
60
which
deterioration
implies
by age
for you!
there's a quick argument against solipsism
minds. Now
statement
about my
the falsehood
here. It's not that implying
I'm being a little facetious
? on
the contrary. And of course the would-be
is a bad thing
of solipsism
78
FIFTEENARGUMENTSAGAINST FINITEFREQUENTISM
219
will simply deny that there are any non-trivial
solipsist-frequentist
prob?
abilities
about my mind. What's
of
though, is that statements
troubling,
about a mind, an object, or an event, seem to be simply irrel?
probability
evant to the existence
of other minds,
other objects, other events of the
same sort, right here in the actual world. Moreover,
we can even put lower
on how many such entities there are ? for example, at least 9 other
in the case just considered,
and of course some multiple
minds
of 10minds
in total. It is often true that the required things do indeed exist, and in the
bounds
numbers required (at least 9 other minds, for example). But sometimes
? see
the problem of the single case below; and in any case,
do not
they
these
things simply don't seem to be implied by the corresponding probability
statements.
6. Actual
=
Frequentism
Operationalism
about Probability
of probability
The finite frequentist
sounds a lot like an opera?
definition
tional definition.
Like the operational
of temperature
in terms
definitions
or mental
states in terms of actu?
of actual thermometer measurements,
in terms of the results of
al behavior, we have probability
being defined
some actual 'measurement'
(put in scare quotes, since the results might not
this
be
in
case, the results of trials of the relevant sort.
observed):
always
is mistaken
for the probability
of the probability
itself.
?
course.
so
hard
of
has hit
And rightly
the argu?
times,
Operationalism
are well known. To rehearse just one of these arguments: we want
ments
can be misleading
to be able to say that measurements
("the thermometers
were poorly calibrated"),
but an operational
definition
doesn't
let us say
'Measurement'
that. Likewise,
if the frequentist
has his way, we can't say that the chance
of the coin landing heads really was
1/2, but that there was an unusually
high proportion of tails in the actual
be a very natural thing to say.
7. Chance
Why
do we
is Supposed
believe
to Explain
in chances?
sequence
of tosses. And
Stable Relative
Because
we
yet that could
Frequencies
observe
that various
relative
are stable; and that is exactly what we would expect
frequencies
are
if there
chances with similar values. We posit chances
in
underlying
order to explain the stability of these relative frequencies.
But there is no
of events
explaining
to be done
not
be those
just is actual relative frequency: you can't
explain something by reference to itself. Here I am echoing a well-known
argument due to Armstrong
(1983) against the 'naive regularity theory' of
are
lawhood
laws
(that
simply true universal
generalizations).
Compare:
we posit laws of nature in order to explain regularities,
so they had better
simply
if chance
regularities,
as a naive
79
regularity
theory
of lawhood
220
would
ALAN HAJEK
it. (Indeed,
have
the demise
of frequentism
to that of the
is parallel
in many
naive
respects.)
regularity theory
I have presented firstly some general arguments
that work equally well
?
of frequentism
that I have mentioned
and
against any of the versions
refinements
indeed, in some cases, even against more sophisticated
thereof;
and then some further arguments
against actual frequentism,
irrespective
of the size of the reference classes. But finite frequentism
also has its own
all really stemming from simple mathematical
characteristic
problems,
ratios of (finite) natural numbers.
facts about
PROBLEMS SPECIFIC TO FINITE FREQUENTISM
8. Attributes
Chance
with No Occurrences
have Undefined
Relative
Frequencies.
Gaps
are undefined
frequencies
no
value. But
determinate
0/0 has
Relative
probabilities
for attributes
I contend
that have no occurrences:
can have
that such attributes
nonetheless.
each with a single die. In the first world,
Imagine two different worlds,
the die is tossed a number of times, but in the second it is never tossed.
There's a sense inwhich both dice can be said to have well-defined
proba?
to finite frequentism,
bilities for landing 6, say, but according
only the first
in the first world,
the die is weighed,
does. By analogy,
both
dice have masses.
it is never weighed;
nonetheless,
von
considerations
similar
Mises
adduces
Ironically,
but in the second
tomine
about dice
that probability
is relative
to argue for his opposite
conclusion
a
a
a
of
of
6
is
"The
given die and
physical property
probability
frequency:
to itsmass, specific heat, or electrical resistance"
is a property analogous
(p.
ought to
14). Exactly! But taking the analogy at face value, the conclusion
is an intrinsic property of chance devices
be that probability
(such as dice)
to
a
those other
theorist
The
say.
analogy
might
something
propensity
von
to
of
because
his
Mises
thoroughgoing
appealed
properties presumably
in order
on operationalism.
He regarded the mass of a die,
bordering
positivism,
as the limit of a sequence of ever improving measurements
for example,
of the mass. But who among us now would want to say that?
All
this may
devolve
into a clash of intuitions,
as so many
philosophical
debates do (which is not to say that the debates areworthless); but I think
that the intuitions
examples
on my side are perhaps
section.
in the next
80
even more
compelling
in the
FIFTEENARGUMENTSAGAINST FINITEFREQUENTISM
9. IfB Occurs
Once,
221
0 or 1: Local Determinism
A has Probability
toss a certain
suppose that we
the relative frequency
Now
of heads
coin exactly once. It lands heads. Then
is 1. But we don't want to be committed
to saying that the probability
of heads is 1, since we want to allow that it
device. (Change the example to Stern-Gerlach
could be an indeterministic
measurements
of electron spin, if you think that coin tosses are determin?
inwhich
there is only the one coin, if you think
istic; and imagine a world
there are any, are relevant.)
that the results of tossing other coins, when
an
once
event that only happens
to any sensible
And in general,
(according
for typing it) does not automatically
do so with
an a priori argument for local pockets of determinism
standard
Such
probability
is surely
1.
too
good to be true!8
now a radioactive
atom that obeys an
as
so
but
it
there
is exactly one such
decay law,
happens,
atom in the entire history of the universe (cf. Lewis'
(1994) "unobtainium").
of decay is 0 or 1, over any time interval,
Are we to say that its probability
Of
course
it isn't true. Consider
indeterministic
simply
because
for each
such
interval
the relative
frequency
of decays
is
either 0 or 1? So with probability 1 it decays exactly when it does? This
contradicts
our supposition
that it obeys
an indeterministic
decay
law. An
innocuous
surely.
supposition,
are most naturally regarded as being unrepeatable ?
experiments
game, a horse race, a presidential
election, a war, a death, certain
in the very early history of the universe. Nonetheless,
it
chancy events
seems natural to think of non-extreme
some
to
of
probabilities
attaching
Many
a football
them. This, then, is another notorious
called problem
of the single case.
10. Universal
Generalizations
statements
Certain
are
for example, universal
think that non-trivial
'single
problem
for frequentism:
and Existential
Statements
case'
of their very
claims.
in virtue
and existential
generalizations
the so
logical form:
Some people
to such statements ?
attach
probabilities
that all ravens are black is 0.9', or 'the
be, 'the probability
that there exist tachyons is 0.1 '. If there is sense to be made of
probability
such probabilities,
then it is not the frequentist who can make
it, for such
(objective)
as it might
to be true or false. How
only get one opportunity
a universal
or an existential
in which
generalization,
statements
cases
true? What
is the reference
do you count
is
statement,
instances are 'all ravens are
class, 0.9 of whose
one
I suppose
black'
instances?
could imagine counting possible worlds:
the
of
all
10%
(in
limit?)
possible worlds are 'there exist tachyons'-worlds.
But this is hardly an attractive proposal,
and in any case, it is certainly not
finite frequentism.
81
222
ALAN HAJEK
some further, non-fre
frequentist might acknowledge
sense
covers
cases
that
such
of
probability
("let a thousand flowers
quentist
...
bloom
that frequentism
still holds sway elsewhere.
The
"), insisting
to
sorts
certain
of
statements
this
is
of
argument
identify
probability
point
An
ecumenical
that people have found quite intelligible, even though they are (virtually)
on a frequentist
unintelligible
are not so ecumenical.9
analysis.
And
various
real-life
frequentists
So far, the problems have involved very low numbers of instances of the
attributes in question, namely 0 or 1. So the reaction might be: "frequentism
was never meant
to handle cases inwhich
there are no statistics, or only a
single data point; but in decent-sized
in the catchy
intuition encapsulated
samples itworks just fine." This is the
but all-too-vague
slogan "Probability
The reaction
is wrong-headed:
problems
is long run relative frequency."
remain even if we let our finite number
11. Intermediate
The
problem
guaranteed
terministic
Probabilities
'
of trials be as large as we
in an Deterministic
like.
World
are
of the single case was that certain relative frequencies
are
of
inde?
the
results
to be extreme (0 or 1), even when
they
because
for frequentism,
This is an embarrassment
processes.
such indeterminism is thought to be incompatiblewith extreme (objective)
?
cannot be probabilities.
hence those relative frequencies
probabilities
on
it would
its head: determinism,
let's turn this thought
Now
seem, is
a
in
determin?
intermediate
with
probabilities:
(objective)
incompatible
are
so
0 or 1.
chances
all
is
and
istic world,
objective
chancy,
nothing
But
determinism
is no obstacle
Remember
to there being relative frequencies
the probability
Venn's example:
that
of a
these values.
of male births among all births. This
is simply the proportion
a
that determines
but the process
is
1/2;
roughly
presumably
proportion
nonetheless.
sex
well
be
deterministic
could
baby's
lie between
male
birth
12. Finite
Frequentism
Consider
a coin
with
Generates
Spurious
Biases
that is perfectly
fair, meaning
by this that it lands heads
Yet itmight not come
for
tails.
equal to 1/2, and likewise
1/2 of the time in actual tossing. In fact, itwould be highly
probability
heads
up
exactly
to do so in a huge number of tosses, say 1,000,000.
unlikely
to
itwould be more than unlikely
If the number of tosses is 1,000,001,
?
thinks
So the finite frequentist
it would be downright
do so
impossible.
fair. Put
in saying that the coin is perfectly
then be wrong
that we would
an
coin
that
is
it
to
finite
any
analytictnxth
frequentism,
simply: according
a
there's
Now
is
biased.
an
of
times
number
odd
that is tossed
startling bit
of a priori
reasoning for you.
82
FIFTEENARGUMENTSAGAINST FINITEFREQUENTISM
223
we do not need to leave our finite frequentist
to
arm-chairs
Likewise,
of all n-sided
dice that are tossed a number of
'discover'
the biasedness
times that is not divisible by n (a coin can be regarded, after all, as just the
=
If only
2). And so on for other chance processes.
special case inwhich n
so
matters
could be settled
all empirical
easily!
to the possible biases of the coin, or
there is a 'graininess'
Furthermore,
the dice. Toss them n times; the relative frequencies must all be multiples
assure us that various coins
of 1/n. So not only can the finite frequentist
and dice are biased
he can even put severe constraints on the possible
extents
He
of the biases!
should
that the coin
to reply to all of this: "When we say
the temptation
mean
we
is fair,
that it lands heads with probability
really
to
and
for tails."10 Firstly, there is no
likewise
1/2,
equal
resist
approximately
that the fair coin will
guarantee
even approximately
half the
a
one
fair
that
coin,
Secondly,
imagine
genuinely
an
moreover
odd number of times; but the finite frequentist
is tossed
thinks
an
that this is on a par with
red
uncolored
imagining
object
namely,
we
time.
land heads
can at least
imagining gibberish. Finally, we should not let too much hinge on the
choice of the example. Consider
if you prefer certain Stern-Gerlach
spin
are perhaps
which
'fairer' than the coin is; or consider the
measurements,
half-life of radium; or whatever
your favorite example might be.
the longer the finite run of coin tosses (or whatever),
the
it is that the relative frequency
the
value
that
exactly equals
To be sure, the probability
that the relative frequency
is near
Ironically,
more unlikely
it 'should'.
the value that it 'should' equal increases. But if frequentism
is supposed
near enough is not good enough.
be an analysis of probability,
13. Finite
Frequentism
Generates
Spurious
to
Correlations
Let us say thatA is spuriously correlated with B
if P(A\B)
^ P(A)9
and yet A and B are not causally
related. The finite frequentist will see
over
correlations
all
the
spurious
place. We can be pretty sure that, say, the
of people who die by the age of 60 is not exactly the
relative frequency
same in general as it is among people who wear green shirts. In fact, we
can be absolutely
sure that this is so if the smaller sample size happens not
to divide the larger one. To see the point, pretend that there are 10 people
in our sample reference class, and that 7 of them wear green shirts. (Note
that 7 does not divide 10.) Then all relative frequencies within thewhole
the green shirt sample
of 1/10, while within
sample must be a multiple
a
no
must
be
of
1/7.
there
is
of 1/10
Now,
way for a multiple
they
multiple
to equal a multiple
cases
of 1/7 (apart from the trivial
of 0 and 1, which
are uninteresting).
The finite frequentist
translates this as: there is no way
83
224
ALAN HAJEK
a correlation - presumably,
to agree. In other words,
for the probabilities
?
a spurious one
between death by the age of 60, and the wearing
of green
it is startling that such results can
in this case. Again,
shirts, is guaranteed
be derived
aprioril
14. All Irrational
Probabilities,
'GoMissing'
a good
There's
Rational
and Infinitely Many
Probabilities,
sense
in which
most
of the numbers
between
0 and
1 are
irrational
aren't). Yet a finite
relative
Thus,
(uncountably many are, only countably many
can never take an irrational value.
frequency
to probabilities
is necessarily
which
gives such values
its
of
to finite frequentism,
subject matter.
irrespective
any
theory
false, according
That's certainly a
to finite
according
For example,
law for radium is false for all time periods
frequentism,
for decay?which
is to say that it is false
that have irrational probabilities
almost everywhere.
and very similar to one that we
1 (? la Reichenbach,
Reply number
an irrational value as closely as we like,
saw above): we can approximate
a
we
have
sufficiently
large (finite) number of trials.
provided
quick
of quantum
the radioactive
refutation
Counter-reply:
that probability
again,
mechanics!
this misses
the point. The
relative frequency,
thesis before
us is not
but that it is relative
is approximately
an identification
of
with
relative frequen?
have
probability
frequency.
we
can
values as
that
it
Of
course,
probability
approximate
cy.
implies
?
values
as
we
like with relative frequency
anything approximates
closely
but it is a much
itself as closely as we like!
stronger claim. The point
We
relative frequen?
in justifying
might be appropriate
approximation
but our topic is
for discovering
tism as good methodology
probabilities;
not
its
the analysis of probability,
methodology.
are such things as
Reply number 2: Bite the bullet, and deny that there
about
irrational probabilities.
Counter-replies:
laws
probabilistic
radioactive
No
experiment
Firstly, this would
is more complicated
could
mean
ever reveal
their existence.
that the truth about various
than we
think. For
the
instance,
than
smooth
rather
involve step functions,
decay laws would
curves. Secondly,
the reply smells of positivism.
Thirdly, we
exponential
even
irrational probabilities,
that instantiate
can imagine possible worlds
not
want
one
do
We
of
them.
not
to
be
if the actual world turns out
surely
is not only false, but logically false.11
to say that quantum mechanics
are ruled out by finite
a
infinitesimal
the
probabilities
way,
fortiori
(By
?
of
version
and
yet such prob?
indeed, by any
frequentism
frequentism
an
For
to
role
have
abilities may nonetheless
sympathetic
play.
important
84
FIFTEENARGUMENTSAGAINST FINITEFREQUENTISM
discussion
pp.
of infinitesimal
see for example
probabilities,
Skyrms
225
(1980),
177-187.)
to finite frequentism,
Moreover,
according
infinitely many rational prob?
also. This is related to the point Imade earlier about
abilities
'go missing'
the 'graininess'
rational values
of finite
for a given sample size. All
frequencies,
the endpoints of the grains will be ineligi?
to the finite frequentist.
according
relative
that fall between
ble as probability
values,
Note that the last few arguments did not require any assumptions
about
what we take to be the relevant reference classes. As I indicated before, I
about including in the reference class of a certain coin, the
have misgivings
results of tossing other, very different or distant coins. But even waiving
the last arguments
still go through. Include if you like
those misgivings,
the results of various other coins when determining
the probability
for this
coin; indeed, include if you like the results of all coins that ever were tossed,
are tossed, and ever will be tossed. Since there will still be only finitely
many trials in the reference class, still the frequentist will have to say: all
to be rational, and in fact, all multiples
will be guaranteed
probabilities
of a certain finite fraction;
spurious correlations with
and it is discoverable
chosen factors can be guaranteed;
that if the total number
15. Non-Frequentist
Symmetry, Derivation
of tosses
is odd,
Considerations
from
the coins
Enter
other appropriately
from the arm-chair
are biased.
our Probabilistic
Judgments:
Theory...
cases above as fatal for finite frequentism,
cannot easily be bitten. We know that
bullets
that
they provide
so
cannot
to be biased, because we
coins and dice
easily be 'shown'
have independent
sometimes
grip on what their various chances are. We
We
should
regard
the various
because
of radioactive decay cannot so easily be 'shown' to
know that probabilities
be rational, because quantum mechanics
There are other
says otherwise.
?
sources of our probability
besides
relative
for
judgments
frequencies
and derivation from scientific theories
symmetry considerations,
example,
that we
to. When
subscribe
there's a conflict between
already
one
of
other
and
these
the latter often wins.
sources,
frequency
5.
In this space,
relative
CONCLUSION
I could not give voice
to various
responses
to these arguments
on behalf of finite frequentists (although I did give voice to quite a few).
reject the starting points of some of the arguments,
that were
other arguments
'intuition-based';
they would
They would
doubtless
particularly
those
85
ALAN H?JEK
226
That should
grant me, remaining untroubled
by their conclusions.
seem
to
most
debates
go the same way.
philosophical
hardly be surprising:
I do, however,
think that finite frequentism
is about as close to being
perhaps
as a serious
refuted
once we
ever gets. This
position
philosophical
of
how
the
probabilities
separated
question
have
becomes
clear
are discovered
from the question of what probabilities are. (A good way to find out if a
'bachelor'
is to ask him; but we wouldn't want to analyse
to
To
in
the
the form
put my position
question.)
'yes'
reasonable
bad
'Finite frequentism:
methodology,
analysis'.12
is a bachelor
man
as one who
of a slogan:
answers
NOTES
I
This paper is an edited version of the first half of my talk "Thirty Arguments Against
Frequentism", presented at the Luino Conference. Iwanted this paper to reflect that talk,
while
meeting
the reasonable
that this volume
constraints
length
So
required.
I have
omit?
tedmy lengthy discussion of hypothetical frequentism, hoping to present that on another
occasion.
I thank
21 know
this from
the editors
various
for their forbearance.
I have
conversations
had,
though
such
catching
frequentists
out
of the closet and in print is not so easy. Shafer (1976) comes close in his definition of
one of the possible
outcomes
of the time that a particular
the proportion
[of a
the chance
ofthat
outcome"
is called
tends to occur
(p. 9), and closer
experiment]
later.
"tends
to" four pages
still when
he drops the qualification
3
a
for the
mathematical
abstraction
is
but
"The
word
Frieden
Witness
'probability'
(1991):
"
term 'frequency
of occurrence'
(p. 10).
intuitively more meaningful
chance:
"...
random
4
The discussion takes place inmy manuscript "Fifteen Arguments Against Hypothetical
the second
Frequentism",
half
talk at Luino.
of my
5
and a
The distinction, I take it, is between a conditional probability of the form P(B\A)
relativized probability of the form Pa{B). The former presupposes thatP{A) is defined,
the latter does not.
6
somewhat
This resembles
for
the
'best'
of
theory
between
the tension
and strength
simplicity
account
to Lewis'
(1994)
central
the universe,
in the competition
of chance:
raising
the standards for admission is like an increase in strength, with a corresponding loss in
on
More
simplicity.
7
better
It is hardly
corresponding
coin
be-tossed
that shortly.
to propose
has
that
instead
frequencies
an undefined
of
chance
evolve
the chances
over
evolve
relative
time:
landing
for
heads
mean
as the
exactly
that the yet-to
'chance
gap'
over
that would
?
see the
time
objection
below.
81 cannot
tive chance.
pause for further discussion
I admit that the connections
on
the connection
I assume
here
between
determinism
are not uncontroversial.
and objec?
See Lewis
(1986) pp. 117-121 for a fuller treatment.
9
Frieden,
for example
-
see footnote
3.
10
I suppose we already knew that no actual coin lands heads with probability 1/2 exactly,
and tailswith probability 1/2 exactly, if only because some tiny amount of probability goes
to the coin landing on its edge; and perhaps even some (tinier still) amounts of probability
go
to the coin
landing
on each
of the two edges
of
its edge.
II
Here I construe logic broadly to include analytic truths.
86
FIFTEENARGUMENTSAGAINST FINITEFREQUENTISM
227
121am grateful tomany people for discussions of thismaterial, especially Jim Bogen, Alex
Byrne, Fiona Cowie, Ned Hall, David Hubert, Marc Lange, Brian Skyrms, Nigel Thomas,
Jim Woodward,
and Lyle
Zynda.
REFERENCES
Armstrong, D. M.: 1983, What is a Law ofNature? Cambridge University Press.
B. R.:
Frieden,
1991, Probability,
Statistical
Optics,
and Data
Testing,
Springer-Verlag.
Jeffrey,Richard: 1977, 'Mises Redux', inR. E. Butts and J.Hintikka (eds.), Basic Problems
inMethodology and Linguistics.
Kripke, Saul: 1980,Naming and Necessity, Oxford University Press.
Lewis, David: 1986, Philosophical Papers, vol. II,Oxford University Press.
Lewis,
David:
1994,
'Humean
Supervenience
Debugged',
Mind
103,473-490.
Reichenbach, Hans: 1949, The Theory of Probability, University of California Press.
Shafer, Glenn: 1976, AMathematical Theory of Evidence, Princeton University Press.
Brian:
Skyrms,
Venn,
John:
von Mises,
1980, Causal
1876,
Richard:
The Logic
Necessity,
of Chance,
1957, Probability,
Yale
1980.
2nd
ed., Macmillan
and Co.
Truth
and Statistics,
Macmillan.
California Inst, of Technology
Division of Humanities and Social Sciences
228-77
Pasadena CA 91125
U.S.A.
87