# Reviving Frequentism††thanks: Accepted for publication in _Synthese_.

Mario Hubert  California Institute of Technology
Division of the Humanities and Social Sciences

(December 31, 2020)

###### Abstract

Philosophers now seem to agree that frequentism is an untenable strategy to
explain the meaning of probabilities. Nevertheless, I want to revive
frequentism, and I will do so by grounding probabilities on typicality in the
same way as the thermodynamic arrow of time can be grounded on typicality
within statistical mechanics. This account, which I will call _typicality
frequentism_ , will evade the major criticisms raised against previous forms
of frequentism. In this theory, probabilities arise within a physical theory
from statistical behavior of almost all initial conditions. The main advantage
of typicality frequentism is that it shows which kinds of probabilities (that
also have empirical relevance) can be derived from physics. Although one
cannot recover all probability talk in this account, this is rather a virtue
than a vice, because it shows which types of probabilities can in fact arise
from physics and which types need to be explained in different ways, thereby
opening the path for a pluralistic account of probabilities.

###### Contents

  1. 1 Introduction
  2. 2 Typicality Frequentism
    1. 2.1 Typicality and the Arrow of Time
    2. 2.2 Probabilities as Typical Frequencies
      1. 2.2.1 Random Variables and their Relation to Physics
      2. 2.2.2 Probability from Typicality
  3. 3 Defending Typicality Frequentism
    1. 3.1 The Range-Account of Probabilities
    2. 3.2 The Humean Mentaculus
    3. 3.3 Countering Standard Critique of Frequentism
      1. 3.3.1 It’s a Poor _Analysis_ of Probabilities
      2. 3.3.2 The Problem of Ascertainability
      3. 3.3.3 The Reference Class Problem
      4. 3.3.4 Frequentism Is Not Completely Objective
    4. 3.4 Objections and Replies
  4. 4 Conclusion

##  1 Introduction

Frequentism is dead. This seems to be the consensus among contemporary
philosophers. A recent textbook on the philosophy of probabilities phrases it
this way:

> Although the frequency view remains popular outside philosophy – e.g. among
> statisticians – it is not the subject of much, if any, active research.
> (Rowbottom, 2015, p. 112)

Frequentism may be useful for all practical purposes for statisticians,
although it does not convey the true meaning of probabilities, since
philosophers have successfully exposed the underlying unsurmountable problems.
Therefore, active research for developing frequentism has been discontinued.
Hájek (1996) prominently debunked finite frequentism; a decade later followed
his criticism of hypothetical frequentism (Hájek, 2009). Recently, La Caze
(2016) agreed that any version of frequentism is doomed to fail, at least in
providing a comprehensive understanding of probabilities.

I think we can breathe life back into frequentism and develop it into a
serious account of probabilities. I intend to defend frequentism against these
criticisms and modify it in such a way that it incorporates elements of finite
frequentism, hypothetical frequentism, and the classical interpretation of
probabilities. I will call this account _typicality frequentism_ , which
defines, in brief, probabilities as typical long-term frequencies based on the
law of large numbers.

Typicality has been developed within Boltzmann’s reduction of thermodynamics
to statistical mechanics, but the scope of this notion is not particularly
tied to statistical mechanics (Wagner, 2020). Wilhelm (2019) recently showed
how typicality explanations work in general by connecting them with Hempel’s
deductive–nomological model. Ideas along these lines to derive probabilities
from typicality as special kinds of frequencies have been presented by Maudlin
(2007a, 2018) and sketched in the literature on Boltzmann’s statistical
mechanics and the de Broglie–Bohm quantum theory (for a brief overview, see
Goldstein, 2012), but there has been no work contrasting this kind of
frequentism with the traditional theories of frequentism in order to establish
typicality frequentism as a serious alternative theory in its own right.

In my opinion, the biggest methodological error made by the forefathers of
frequentism, like Reichenbach (1949/1971), Venn (1888), and von Mises
(1928/1957), was to interpret probabilities as frequencies from empirical
behavior: they started with how we talk about probabilities and tried to
underly an interpretation in terms of frequencies that supports their
empiricism (Gillies, 2000, Ch. 5). Instead, I propose a strategy from a
physical theory to probabilities: starting with a deterministic fundamental
physical _theory_ and analyze how this theory introduces probabilities from
statistical behavior. Then we may recover how our general use of probabilities
is backed up by physics. But, as it turns out, some ordinary ways of talking
about probability will not be recovered within this approach. To account for
these, we are free to introduce another, complementary interpretation of
probability—becoming pluralists about probability. The method I will be using
to define probabilities is the same statistical method that has been used to
justify the thermodynamic arrow of time in statistical mechanics or the arrow
of time in electrodynamics (North, 2003).

##  2 Typicality Frequentism

The idea behind typicality frequentism is to apply the tools from statistical
mechanics to explain how probabilities arise from deterministic physical
dynamics. Maudlin (2018) is confident about this strategy, “The challenge of
deriving probabilities—or quasi-probabilities, probabilities with
tolerances—from an underlying deterministic dynamics can be met. Typicality is
the conceptual tool by which the trick is done.” An important predecessor of
typicality frequentism, apart from the different versions of frequentism, is
the theory of probability by Johannes von Kries, laid out in his _Principien
der Wahrscheinlichkeitsrechnung_ (1886, _engl._ Principles of Probability
Theory). As von Kries’s view seems to be best characterized as objective
Bayesianism (see p. 16 of the introduction by Eberhardt and Glymour in
Reichenbach, 2008) or as a predecessor of the logical interpretation
(Fioretti, 2001), subjective and objective aspects are intertwined. For my
purpose, I want to lay out in more detail the objective parts of von Kries’s
account, because they contain some essential features of typicality
frequentism, although von Kries criticized the frequentist theories at his
time (Zabell, 2016b, section 3).

Influenced by the physics of the 19th century, it was important to von Kries
to distinguish between laws of nature and initial conditions (Pulte, 2016).
Unlike Laplace, who reduced probability to incomplete knowledge of the initial
conditions, von Kries built up probabilities from objective _variations_ of
initial conditions, and he called the sets of admissible initial conditions
“Spielräume,” which are best translated as “sets of possibilities”.111Eberhard
and Glymour call them “sets of ur-events” because they are the irreducible
basis for von Kries’s probabilities (see Reichenbach, 2008, Introduction,
section 4.2). More quantitatively, von Kries defined the probability for an
event $E$ in the following way (see Pulte, 2016, section 5, for this
reconstruction). Let us say that the event $E$ is brought about by the set of
initial conditions $C$ (given certain laws of nature) and the set of initial
conditions that do _not_ bring about $E$ is $C^{*}$ ($C$ would be then the
“Spielraum” or set of possibilities for $E$). The probability $p$ for $E$ is
then defined, in the Laplacian sense, as the quotient of the favorable initial
conditions leading to $E$ over all possible initial conditions by measuring
the Spielraum and its complement by an appropriate measure $m$:

$p:=\frac{m(C)}{m(C)+m(C^{*})}.$

Although the objective aspect of probabilities mentioned here comes from the
initial conditions of the physical process, it is not a frequentist account.
Moreover, this reconstruction of von Kries’s theory may incline us to think
that the Spielräume are unique and always tied to a physical theory, but von
Kries was in this respect more a pragmatist and sometimes even a skeptic (see
the discussion in Pulte, 2016, section 5). Depending on the knowledge of the
agent building a probabilistic model, the space of possibilities may change
and may not be a space of initial conditions of a physical theory but rather a
more generalized sample space; thus, the measure $m$ may not be unique either.

I share von Kries’s intuition to reduce probabilities to certain basic events,
but I endeavor a more objective account of probabilities always embedded into
physics and using only the tools of physics in defining probabilities. I,
therefore, propose that physics offers a unique space from which to derive
probabilities as typical long-term frequencies. This is the fundamental space
of physics, like _phase space_ or _configuration space_ (depending on the
physical theory). Here, I agree with the method of arbitrary functions (or
more adequately named the _range account of probabilities_ by Rosenthal,
2016), which can be regarded as a modern elaboration of von Kries’s theory.
But I deviate from the range account by incorporating typicality as a central
notion to define probabilities; in a similar fashion, it is possible to
explain the thermodynamic arrow of time as arising from generic initial
conditions of the universe.

###  2.1 Typicality and the Arrow of Time

Scrambled eggs never unscramble, a shattered vase never reassembles itself,
and ice cubes never un-melt. Although our basic physical laws are time-
reversal invariant, that is, the time-reversed process of a physically
possible process is also physically possible, such time-reversed processes are
not observed. Boltzmann proposed a solution to this problem by distinguishing
microscopic from macroscopic behavior and systems with few degrees of freedom
from systems with many degrees of freedom. It is the microscopic behavior that
is time-reversal invariant, and one only finds directed processes on the
macroscopic level, when systems have many degrees of freedom. If we have a
sequence of photos showing a system of few degrees of freedom, like two
rotating molecules, we would not be able to distinguish forward from backward
behavior, but if we have a sequence of photos of a glass bottle thrown on the
ground, we would distinguish one direction as the true one.

Boltzmann gave an explanation in terms of statistics why such behavior is not
observed. The short answer is due to the many degrees of freedom of a
macroscopic process: one had to finely orchestrate all the many microscopic
states of the particles constituting a macroscopic object in order to yield a
time-reversed process. If we don’t interfere this meticulously (and in most
cases we cannot do so), then a familiar directed process comes about. In other
words, almost all initial conditions of a macroscopic system yield the
familiar directed processes; only very special initial conditions yield the
reversed process. So given broken glass on the floor, there are many more
states of particles constituting the pieces of glass such that these pieces
remain on the floor than those states that would converge the pieces into a
brand-new bottle.

This behavior can be phrased by means of typicality: a physical behavior is
called _typical_ , if almost all initial conditions yield this behavior (see,
for instance, Goldstein, 2001; Lebowitz, 2008; Myrvold, 2016; Volchan, 2007).
And a physical behavior is called _atypical_ , if almost none of the initial
conditions yield this behavior. So, it is typical that broken glass remains
broken, and it is atypical that scrambled eggs unscramble.

Boltzmann’s ideas on the irreversibility of physical processes have recently
experienced a renaissance among philosophers in which the notion of typicality
has become central (see, for instance, Barrett, 2017; Frigg, 2009; Lazarovici
and Reichert, 2015, 2019). The notion of typicality, as we introduced it, is
still too imprecise for quantitative use in physics. As Wilhelm (2019) rightly
emphasizes, there are many ways to formalize “almost all.” The right way to do
so in statistical mechanics is by means of a measure over phase space. Phase
space is constructed from a set of particles in three-dimensional space.222For
simplicity’s sake and to be as close to Boltzmann’s reasoning as possible, I
assume Newtonian mechanics as the microscopic theory. Consider $N$ particles
in three-dimensional space (if we have a realistic macroscopic body, $N$ is of
the order of Avogadro’s constant, that is, approximately $10^{23}$). Since a
particle is completely described by its position $\boldsymbol{x}$ and momentum
$\boldsymbol{p}$, we can summarize the complete physical state of $N$
particles as a vector
$\left(\boldsymbol{x}_{1},\boldsymbol{p}_{1},\boldsymbol{x}_{2},\boldsymbol{p}_{2},\dots,\boldsymbol{x}_{N},\boldsymbol{p}_{N}\right)$,
and this vector is one point in phase space, which has roughly $6\times
10^{23}$ dimensions. So every point in phase space, each microstate,
represents a set of $N$ particles with their precise positions and momenta. In
order to get macrostates, one needs to divide phase space $\mathcal{P}$ into
disjoint subsets, where each set represents a macrostate (see Figure 1). So a
macrostate arises from a map $M$ that assigns to every microstate $X$ a
macrostate $M(X)$ corresponding to one of the subsets
$\mathcal{P}_{M}\subseteq\mathcal{P}$ according to the partition—$M(X)$ is the
macrostate of $X$ if $X\in\mathcal{P}_{M}$.

Figure 1: Clusters in phase space according to thermodynamic macrostates and
the measure of typicality, as depicted by Roger Penrose (1989, p. 402).
Thermal equilibrium is by far the biggest macrostate in phase space.

The tool that ultimately explains irreversible behavior is Boltzmann’s
definition of entropy assigned to every point in phase space:

$S_{B}(X):=k_{B}\ln\lvert\mathcal{P}_{M(X)}\rvert,$ (1)

where $k_{B}$ is Boltzmann’s constant and $\ln$ is the natural logarithm. The
main part of Boltzmann’s entropy is $\lvert\mathcal{P}_{M(X)}\rvert$, which
deserves some elaboration. In order to measure the sizes of the subsets
$\mathcal{P}_{M(X)}$, one needs to introduce a measure $\lambda$, which
assigns a number to every such subset. Conventionally, if the system is
finite, one normalizes the measure to $1$ such that the size of the entire
phase space would be $1$. In the entropy formula,
$\lvert\mathcal{P}_{M(X)}\rvert$ denotes the size of $\mathcal{P}_{M(X)}$
according to the appropriate measure $\lambda$, that is,
$\lvert\mathcal{P}_{M(X)}\rvert=\lambda\left(\mathcal{P}_{M(X)}\right)$. The
only purpose of the measure $\lambda$ is to tell us which sets are big and
which are small, in order to identify typical and atypical behavior; in this
sense, it is _a measure of typicality_. Since for real physical systems, like
gases in a box or melting ice cubes, the phase space volume of thermal
equilibrium has by far the largest volume according to the measure of
typicality and so it has the highest entropy, we observe systems that are not
in equilibrium (low entropy $S_{B}$) to reach equilibrium (high entropy
$S_{B}$), whereas we do not see a system going from a high entropy state to a
low entropy state, because the low entropy states are much smaller in phase
space.

Moreover, if we zoom into the phase space region of a low entropy macrostate,
a melting ice cube, for example, almost all microstates will move to a
macrostate with higher entropy and ultimately to equilibrium. It is physically
possible that a low entropy macrostate goes into another low entropy
macrostate (by itself), but there are very few microstates within this macro
region that do that. This is Boltzmann’s explanation why we observe only one
direction of a physical process and not the time-reversed process, although
this behavior is physically possible according to the time-reversal
fundamental laws. The symmetry is broken by a statistical argument, that is,
by distinguishing those initial conditions that yield typical behavior from
those that yield atypical behavior.

In classical mechanics, one normally uses the _Liouville measure_ , a natural
generalization of the standard _Lebesgue measure_ on three-dimensional space
to phase space, as the measure of typicality. But in order to distinguish
small sets from big sets, other measures would do the job as well. Indeed,
every measure that is _absolutely continuous_ with respect to the Liouville
measure will agree on the same physical behavior to be typical or
atypical.333A measure $\mu$ is absolutely continuous with respect to a measure
$\lambda$ (symbolically $\mu\ll\lambda$), if all the null sets of $\lambda$
are null sets of $\mu$, that is $\forall
X\left(\lambda(X)=0\Rightarrow\mu(X)=0\right)$. Moreover, there is a certain
vagueness intended in the notion of typicality that is to be reflected in the
mathematical formalization. The sets $A$ yielding typical behavior are those
that have measure $1$ or close to one, that is, $\lambda(A)=1-\epsilon$, where
$\epsilon$ is a very small number also depending on the application.
Similarly, for atypical behavior where the relevant sets may have a measure
$\lambda(B)=0+\delta$ for some small $\delta$, which depends on the specific
application.444There has been a long debate to make Boltzmann’s argument more
mathematically and conceptually precise. For our purpose, we do not need to
dive into these details (see, e.g., Volchan, 2007; Frigg, 2009; Werndl, 2013;
Lazarovici and Reichert, 2015; Myrvold, 2019). This will become important when
we apply typicality and its mathematical formalizations to develop a new
theory of frequentism.

###  2.2 Probabilities as Typical Frequencies

There are two steps to present the theory of typicality frequentism. First, I
need to elucidate the role of random variables, in a way that differs from
standard accounts of probability theory (section 2.2.1). In typicality
frequentism, random variables are primarily used to bridge the gap between a
_physical_ theory and the mathematics of probability theory. Second, this
account of random variables is needed to interpret the law of large numbers in
such a way to define probabilities as typical long-term frequencies (section
2.2.2).

####  2.2.1 Random Variables and their Relation to Physics

Consider a box with $1000$ balls; the balls are either blue, green, or red.
Let’s say 500 balls are blue, 300 are green, and 200 are red; in other words,
$50\%$ are blue, $30\%$ are green, and $20\%$ are red. With this information
we can build a simple stochastic model. The set of balls forms the _sample
space_ $\Omega:=\\{1,\dots,1000\\}$. From this sample space, we can define a
coarse-graining function $X:\Omega\rightarrow\\{B,G,R\\}$, which assigns to
every ball a color B=blue, G=green, or R=red. Functions of this kind are
usually (and unfortunately misleadingly) called _random variables_. There is
indeed nothing random about them; their only use is to abstract from the
sample space, when one is interested in specific features of the members of
the sample space. Next, one determines the _distribution_ of the random
variable. This is a function $\rho_{X}:F(X)=\\{B,G,R\\}\rightarrow[0,1]$, such
that $\rho_{X}(B)=0.5$, $\rho_{X}(G)=0.3$, and $\rho_{X}(R)=0.2$. This
illustrates the standard way of building a _probability space_ (see Fig.
2).555There are some subtleties when one generalizes this scheme to infinite
sample spaces, like, forming a $\sigma$-algebra. These are treated in standard
textbooks on probability theory and are not the focus of this paper.

The distribution $\rho$ is normally called a _probability distribution_ , for
it assigns “probabilities” to certain sets of the sample space. But this would
be putting the cart before the horse; at this stage, we do not have a theory
of probabilities, just a certain recipe for building a mathematical model.
This particular model of colored balls is conceptually very simple, because
the numbers $50\%$, $30\%$, and $20\%$ are mere proportions of balls having
the same color. Nevertheless, some work is to be done to interpret these
numbers correctly as probabilities, as we will be doing in the next
subsection, when I fully lay out typicality frequentism.

Figure 2: The ingredients of a stochastic model and how they relate to each
other. Random variables $X$ abstract from the sample space $\Omega$ by
assigning every member of $\Omega$ a real number. Abstracting means that $X$
maps many elements in its domain to the same number. The image of $X$ gets
assigned a number in the interval $[0,1]$, which measures the size of the sets
that are mapped by $X$ to the same real number. It’s important for typicality
frequentism that all random variables are ultimately defined on phase space,
which is the fundamental sample space.

The sample space can be in principle any kind of (mathematical) space, and in
general no particular attention is paid to the sample space in textbooks,
because in order to make correct predictions the images of the random
variables and the probability distribution are sufficient. I want to go beyond
a pragmatic attitude toward probability theory, although it is justified by
its success in application, and derive probabilities instead from physical
behavior. This is where von Kries’s idea of sets of possibilities or ur-events
comes in. He wanted to prove that probabilities can be derived from certain
compositions of ur-events—random variables, in his account, are defined on
these spaces. Although he intended an objective theory of probability, von
Kries had to rely on a subjective element in order to justify that ur-events
are equiprobable. This element is _the Principle of Indifference_ , which he
advocated in the form of a principle of _in_ sufficient reason: Two events are
equipossible if at the current state of knowledge there is no reason to
consider one of the events more likely than the other (Reichenbach, 2008, p.
15).

It is, however, possible to retain ur-events without this subjective
ingredient. There is a distinguished sample space among all possible sample
spaces, namely, phase space, on which a typicality measure can be defined,
thereby erasing the principle of insufficient reason.666If one were to embed
this discussion in quantum theory, one would need to replace phase space with
configuration space. I now make the following postulate: _all random variables
are ultimately defined on phase space, because all statistical patterns are
determined by what happens in the fundamental physical space, which are
governed by the laws of physics_. Hence, probability theory ultimately works
because it is embedded into physics, and it is so successful because it
abstracts from many physical details, so that when we apply probability theory
we, in most cases, are not aware of the relations to fundamental physics.

In the above example, the sample space $\Omega$, which distinguishes the
different balls, is a coarse-grained space of phase space $\mathcal{P}_{B}$,
which describes _the balls’_ actual positions and velocities. These two spaces
are also connected by a random variable
$X_{B}:\mathcal{P}_{B}\rightarrow\Omega$. We can even go one floor deeper to
the fundamental phase space. Every ball is a macroscopic object consisting of
zillions of tiny particles. The positions and momenta of these particles are
summarized in the fundamental phase space $\mathcal{P}_{f}$ . Again a random
variable $X_{f}$ connects this fundamental space to $\mathcal{P}_{B}$, that
is, $X_{f}:\mathcal{P}_{f}\rightarrow\mathcal{P}_{B}$.

Of course, this interpretation of probability theory will not be shared by
subjective Bayesians and other schools. My goal is not to provide a framework
that suits all interpretations of probability but rather to interpret
probability theory in such a way that is best suited for a modern version of
frequentism.

####  2.2.2 Probability from Typicality

Let us now apply all this to demonstrate how probability arises from
typicality. Recall Boltzmann’s explanation that we observe certain physical
processes only in one direction: it is typical that ice cubes melt and not
unmelt because the universe started in a low entropy macrostate, where most of
the initial microstates yield a universe in which ice cubes melt. A very
similar kind of explanation can be given for the emergence of probabilities
from a deterministic dynamics. For simplicity’s sake, I’ll restrict myself to
coin tosses, which is a _deterministic_ physical process following the laws of
Newtonian mechanics, but it is, in principle, straightforward to generalize
the main idea to other physical processes and to other deterministic physical
theories.

First, there is an observational fact about coin tosses, as there is an
observational fact about the thermodynamic behavior of ice cubes: When we toss
a coin thousands of times, we see that heads and tails appear approximately
half the time, and the more we toss the closer the fraction of heads and tails
approaches $\nicefrac{{1}}{{2}}$. For instance, Kerrich (1946) noted to have
tossed a coin 10,000 times of which heads appeared 5,067 times, and it’s also
said that Karl Pearson tossed a coin 24,000 times of which heads appeared
12,012 times (see Küchenhoff, 2008, although no source for Pearson’s
experiment is given).

Second, recall from the thermodynamic arrow of time that almost all points in
phase space are in thermal equilibrium, where every point represents the
physical state of a gas in a box or the entire universe. When the system
starts from a low-entropy macrostate, statistical mechanics says that almost
all phase space points within this macrostate will follow a trajectory
according to the Newtonian laws of motion that leads to thermal equilibrium
(for not too long time scales). If, say, $\Gamma$ is this low-entropy
macrostate and $P$ is the property “following a trajectory that leads to
thermal equilibrium,” then the property $P$ is said to be typical in $\Gamma$
(see Wilhelm, 2019, p. 4, for this general framework). (We can imagine the
property $P$ to give a certain color to phase-space points.) Next, let’s say
we are interested in the behavior of a subsystem with respect to the initial
conditions of a larger system, for example, gases in a box with respect to the
initial conditions of the entire universe. Then given the special low-entropy
initial macrostate of the universe, it is typical (within this macrostate)
that subsystems in this universe will reflect thermodynamically time-oriented
behavior. In other words, $\Gamma$ would be the low-entropy initial macrostate
of the entire universe, and the property $P$ would be “subsystems reflect
thermodynamically time-oriented behavior”. (Then again, almost all points in
this macrostate would have the same color).

This relation between the behavior of subsystems and the initial conditions of
the universe is central to typicality frequentism. When we apply this picture
to the coin toss, we need to start with the phase space regions of the entire
universe in which coins exist. The relevant property $P$ is “long-term
frequencies of fair coin-tosses are approximately $\nicefrac{{1}}{{2}}$”. It
turns out that almost all universes share this property.

All this can be mathematically captured by the (weak) law of large
numbers:777Whenever I refer to the law of large numbers, I always mean the
weak law of large numbers.

$\lambda\left(\left\lvert\frac{1}{N}\sum_{k=1}^{N}X_{k}(x)-\frac{1}{2}\right\rvert<\epsilon\right)\approx
1,$

where $\epsilon$ is an arbitrary small real number, $N$ is taken to be very
large, the random variables $X_{k}$ represents the $k$th toss, $X_{k}(x)$ is
the result of the $k$th toss determined _by the initial condition of the
universe $x$_, and $\lambda$ is the measure of typicality. For typical coin
tosses, that is, for most universes in which coins are tossed, which
translates mathematically into $\lambda(\cdot)\approx 1$, the arithmetical
mean of an actual run of tosses does not deviate from $\nicefrac{{1}}{{2}}$
more than $\epsilon$. So in any sufficiently long finite series of flips in
these generic universes the frequency of heads and tails will be in the range
of $50\%\pm\epsilon$ for some specific $\epsilon$.

There is something different and something similar between finite and infinite
sequences. In both cases the fraction of heads and tails lies within
$\pm\epsilon$ from 50%, but $\epsilon$ in the finite case cannot be
arbitrarily small and the actual frequency to be (typically) within the error
bounds, whereas that is the case for infinite (or sufficiently long) sequences
according to the law of large numbers. However small $\epsilon$ is chosen, it
is typical that an infinite series of coin flips will have a limiting
frequency within $50\%\pm\epsilon$. Note also that the finite case has to be
sufficiently large in order to show some robust behavior.888 More precisely,
there are three parameters in the law of large numbers that are fixed
successively. First, one chooses an $\epsilon$, then a $\delta$, and then
sufficiently large $N>N_{0}$ such that:
$\lambda\left(\left\lvert\frac{1}{N}\sum_{k=1}^{N}X_{k}(x)-\frac{1}{2}\right\rvert<\epsilon\right)>1-\delta.$
Such an $N_{0}$ exists, because according to the Chebychev inequality
$\lambda\left(\left\lvert\frac{1}{N}\sum_{k=1}^{N}X_{k}(x)-\frac{1}{2}\right\rvert<\epsilon\right)>1-\frac{1}{\epsilon^{2}N}.$

Moreover, the law of large numbers mathematically says that a series of coin
flips that shows 100% heads and 0% tails after, say, 1,000,000 flips, although
physically possible, would be atypical:

$\lambda\left(\left\lvert\frac{1}{N}\sum_{k=1}^{N}X_{k}(x)-\frac{1}{2}\right\rvert>\epsilon\right)\approx
0.$

For any large $\epsilon$ you choose, there is a very small set of initial
conditions that would give a long series that deviate from $50\%\pm\epsilon$.

One may argue that the law of large numbers is a limit theorem and so doesn’t
say anything about finite cases, nor does it say anything about what is
typical or atypical (I thank an anonymous referee for raising these concerns).
One needs to distinguish between _what_ the limit of a sequence is and _how_
the sequence approaches the limit. Finding out about the right convergence
behavior is, for example, a major task in functional analysis and mathematical
quantum mechanics (Lieb and Seiringer, 2010). Here is a simple example to
illustrate this point. The three sequences $\frac{1}{n}$, $\frac{1}{\ln(n)}$,
and $\frac{1}{\ln(\ln(n))}$ go to $0$ for $n\rightarrow\infty$. These
sequences, however, approach the limit differently: $\frac{1}{\ln(n)}$ goes to
0 more slowly than $\frac{1}{n}$, and $\frac{1}{\ln(\ln(n))}$ even more slowly
than $\frac{1}{\ln(n)}$.999E.g., $\frac{1}{1,000,000}=0.000001$,
$\frac{1}{\ln(1,000,000)}\approx 0.072$, and
$\frac{1}{\ln(\ln(1,000,000))}\approx 0.38$. If one traces a certain
(standard) proof of the weak law of large numbers, one finds the formula
footnote 8, which can be itself proven and which tells us something about the
limit behavior of _finite_ sequences. Then, given how typicality is defined
via a measure, one can indeed rephrase the law of large numbers, as well as
the limit behavior of finite sequences, in terms of typicality. I admit that
this is not how the law of large numbers is standardly understood, but it is a
possible, and I think consistent, way of re-interpreting what the law of large
numbers says (see also Dürr et al., 2017).

After these elaborations, we can finally define what probabilities are in
typicality frequentism:

_Definition of probability:_ Some state of affairs has probability $p$, if,
according to a fundamental physical theory, the physical process $X_{k}$
yielding this state of affairs is in principle infinitely repeatable and the
instances of $X_{k}$ are uncorrelated such that the frequency typically (that
is, in almost all universes) approaches a unique limit $p$, that is, for all
$\epsilon>0$
$\lim\limits_{N\rightarrow\infty}\lambda\left(\left\lvert\frac{1}{N}\sum_{k=1}^{N}X_{k}(x)-p\right\rvert<\epsilon\right)=1.$

This definition has several important parts that I want to comment on:101010I
thank an anonymous referee for raising many of the following points.

  1. 1.

_Definition of probability_ : One may argue that what follows is not a
definition but rather a sufficient condition for probabilities, because what
the definition says requires certain strong idealizations that may not be met.
My reply is twofold. For one, this is a definition of what probabilities are
in typicality frequentism. For another, if one takes a broader view of what
probabilities are in general, then this “definition” is indeed a sufficient
(but not necessary) condition of probabilities, since I am aware that other
ways of talking about probabilities differs from a frequentist account. I,
therefore, advocate a pluralist theory of probabilities that complements
typicality frequentism in areas where typicality frequentism does not give an
account of probabilities.

  2. 2.

_State of affairs:_ I use “state of affairs” instead of “events” that get
assigned probabilities, in order not to confuse events with the standard
technical term in probability theory as a subset of the sample space or, more
precisely, a member of the $\sigma$-algebra. The tossing of a coin or a ball
in roulette would be examples of “states of affairs”.

  3. 3.

_The fundamental physical theory:_ In the ideal case, the fundamental physical
theory I refer to is the Theory of Everything, the unique physical theory that
correctly represents the world. Since we haven’t found this theory yet, other
approximately true deterministic physical theories can do the job, like
Newtonian physics or the de Broglie–Bohm quantum theory. The theory needs to
be approximately true in order to give rise to (at least) the right
statistical pattern. Newtonian physics has been proven successful in the
domain of statistical mechanics, and it is good enough for most macroscopic
applications. It is also very unlikely that Newtonian physics and statistical
mechanics will be completely overthrown by future physical theories. It is
plausible to assume that both theories will be recovered in a kind of
classical limit. A candidate for a deterministic theory on the quantum level
is the de Broglie–Bohm pilot-wave theory, which also allows for extension to
quantum field theory. Another deterministic quantum theory is the many-worlds
theory according to Everett. My introduction of probabilities is closer to the
de Broglie–Bohm theory, but also Hugh Everett III wanted to base probabilities
on typicality (Barrett, 2017). I also think that one can generalize typicality
frequentism to indeterministic theories, which would be a future project and
would also require to distinguish this idea from propensities.

  4. 4.

_Uncorrelated events:_ The events $X_{k}$ (or rather the random variables)
that build up the physical process need to be uncorrelated in order to
converge. Standardly, the law of large numbers requires the events of the
stochastic process to be stochastically independent, which is a stronger
condition than being uncorrelated. If the $X_{k}$’s were correlated (for
example, the individual tosses of a coin), then one would be able to undermine
the law of large numbers, and a unique limit may not exist. Or a limit may
exist but it would not be $p$, where $p$ is technically the expectation value
of $X_{k}$.

  5. 5.

_The frequency:_ The frequency that is supposed to approach the limit $p$ is
the relative (finite) frequency of the the physical process $X_{k}$:
$\frac{1}{N}\sum_{k=1}^{N}X_{k}(x)$. For a coin toss, for example,
$X_{k}\in\\{0,1\\}$ representing when a coin lands heads $X_{k}=0$ or tails
$X_{k}=1$. So, $\frac{1}{N}\sum_{k=1}^{N}X_{k}(x)$ counts the number of tails
and divides it by how often one tossed the coin.
$1-\left(\frac{1}{N}\sum_{k=1}^{N}X_{k}(x)\right)$ would then be the relative
frequency for heads.

  6. 6.

_Almost all universes:_ One may think that one needs to quantify over all
universes in order to determine the probabilities in our universe, and,
therefore, the probabilities in our universe are also determined by what
happens in other universes. The first part is correct—that one needs to
quantify over all possible universes—but this doesn’t mean that the
probabilities here are determined by the goings-on in the other universes.
Rather, one needs to compare what happens here to what happens there, and the
appropriate tool for this comparison is the measure of typicality. If we are
in such a world in which the assumptions of the law of large numbers hold,
then the probabilities are particularly robust and regular, because most of
the other universes show the same statistical pattern.111111There is one
caveat: even if all the assumptions of the law of large numbers were fulfilled
it is still possible for a sequence to have a different limit or no limit at
all; the initial conditions leading to these sequences have measure zero
though. The “atypical” worlds widely diverge from the typical ones and also
widely diverge among themselves. There is no unifying or regular behavior to
be expected in these “atypical” universes.

We need to distinguish between the definition of probability and the empirical
significance of this number. While the number $p$, is defined in terms of
infinite sequences, which cannot be instantiated in the real world, the
empirical content of this number arises from its relation to finite sequences:

_Empirical significance of $p$:_ The relative frequencies of finite sequences
obey the restrictions given by the law of large numbers; that is, the observed
frequencies of sufficiently long finite series typically lie in an interval
$p\pm\epsilon$, where $\epsilon$ is a positive number approximately equal to
$0$.

Let me add the following comments:121212I also thank here an anonymous referee
for raising these issues.

  1. 1.

_Status of the empirical significance of $p$:_ I take it to be a true
statement about the observed relative frequencies, and that this statement
follows from the definition of probabilities. It is, therefore, rather a
corollary than a criterion, since a criterion would be something closer to an
axiom.

  2. 2.

_Sufficiently long series:_ The above definition of probabilities presupposes
that the physical process is “in principle infinitely repeatable”, but, of
course, it doesn’t and cannot say how often the real process is actually
repeated. The probability $p$ is empirically significant because it gives
bounds for the real observed (and expected) relative frequencies. It may be
unsatisfactory that the real process needs to be “sufficiently long” without a
precise numerical length. The appropriate length of the series depends on many
factors of the real physical set-up and the overall physical process.

  3. 3.

_Interval $p\pm\epsilon$:_ For real world cases, one has tolerances for the
relative frequencies. The question is now how robust these tolerances are. A
small “uncertainty” of $p$ would also be consistent with the observed
frequency being within the interval. First, I assume the real $p$, the one
given by the true Theory of Everything, to be unique. Second, I assume that
the approximately true candidate theories that are not the Theory of
Everything, like Newtonian physics or the de Broglie–Bohm theory, etc., would
give $p$’s that are very similar. So in this case, there may be a tiny
interval or at least a point-like spread of $p$’s. And we would have to say
that there are several ”candidate probabilities”. Perhaps one of them hits the
true probability; I assume, however, that these “candidate probabilities” are
very close to the true one and for all practical purposes indistinguishable.

In typicality frequentism, there are actually three ideas mingled together
from other interpretations of probability. The first ingredient is similar to
the classical interpretation of probability, which adds up different equally
probable events according to the principle of indifference. Everything in
typicality frequentism hinges on a proper way of counting that leads us to
distinguish typical from atypical behavior based on big and small sets, whose
elements are intrinsically “equally likely” to occur. Second, the definition
of probabilities in terms of a limit that cannot be carried out in the real
world is reminiscent of hypothetical frequentism. Third, in order to make
these typical frequencies empirically meaningful one needs to introduce
tolerances for finite sequences in order to have realistic frequency bands for
actual processes, but in contrast to finite frequentism probabilities in
typicality frequentism are _not defined_ by finite sequences.

There are two ways to undermine the long-term frequency of
$\nicefrac{{1}}{{2}}$. Either one is in one of those special universes that
yield a different statistical pattern for fair coin tosses, or one were able
to replicate, say, with a sophisticated tossing machine, the exact conditions
in every toss. The special universes that yield atypical coin behavior may
reflect all kinds of long-term coin pattern: there are initial conditions of
the universe that lead to 95% heads and also to no regular behavior at all.
Because of these diverse behaviors, there is no way to put these special
universes under one umbrella. It is, however, appropriate to talk of a
probability of 100% showing heads in the tossing machine example. In order to
get probabilities diverging from 100% or 0%, physics requires significant
variations in the ways a coin is tossed (as is realistic), and these
variations in fact yield robust statistical patterns.

As Gillies (2000, Ch. 5) describes, the problem of connecting limiting
frequencies with actual finite frequencies had been raised by de Finetti
against von Mises (1928/1957):

> It is often thought that these objections may be escaped by observing that
> the impossibility of making the relations between probabilities and
> frequencies precise is analogous to the practical impossibility that is
> encountered in all the experimental sciences of relating exactly the
> abstract notions of the theory and the empirical realities. The analogy is,
> in my view, illusory: in the other sciences one has a theory which asserts
> and predicts with certainty and exactitude what would happen if the theory
> were completely exact; in the calculus of probability it is the theory
> itself which obliges us to admit the possibility of all frequencies. In the
> other sciences the uncertainty flows indeed from the imperfect connection
> between the theory and the facts; in our case, on the contrary, it does not
> have its origin in this link, but in the body of the theory itself […]. (de
> Finetti, 1937, p. 77)

The criticism against frequentism is (i) that limiting frequencies as
predicted in infinite series are not observed, (ii) that there is no precise
way to give an interval for the empirical frequencies, and (iii) that if an
interval is given it is still possible that the actual observed frequency may
lie outside this interval. Von Mises argued (see Gillies, 2000, p. 103), as
described in the first sentence of de Finetti’s quote, that the problem of
connecting limiting frequencies with actual frequencies is no different from
connecting the idealized predictions of a scientific theory with actual
observations, something ubiquitous and practically unproblematic in all of the
natural sciences. To which, de Finetti replied that this analogy is invalid
because a scientific theory would in principle be able to make exact
predictions if it were to capture sufficiently all the relevant details of the
world, whereas probability theory, even in the best case, would allow
significant deviations from its predictions, both from the limiting
frequencies, as well as from finite frequencies.

I think, de Finetti’s criticism of von Mises is correct, and von Mises indeed
overlooked the disanalogy between probability theory and the standard
application of scientific theories. The imprecision of probability theory has
a different origin than the imprecision of applying scientific theories or
scientific models to real world cases. The main problem for von Mises was to
justify were the imprecision of his frequentism comes from. Since his theory
was solely based on empirical facts, the truthmakers for the predictions of
probability theory need to be empirical facts too. But how can these
exceptions be empirically made true if they are rarely or never observed in
the first place?

Hájek (2009, pp. 217–8) makes the same argument as de Finetti when he says,
“There is no Fact of what the Hypothetical Sequences Look Like”. He imagines a
coin that is just tossed once and happens to have landed Heads. Hájek then
asks about the coin:

> How would it have landed if tossed infinitely many times? Never mind
> that—let’s answer a seemingly easier question: how would it have landed on
> the second toss? Suppose you say "Heads". Why Heads! The coin equally could
> have landed Tails, so I say that it would have. We can each pound the table
> if we like, but we can’t both be right. More than that: neither of us can be
> right. For to give a definite answer as to how a chancy device would behave
> is to misunderstand chance. (Hájek, 2009, p. 217)

Again, this argument is valid for the traditional version of frequentism, but
in typicality frequentism a physical theory tells us “how the coin would have
landed on the second toss”. The truthmakers for the predicted frequencies come
from a physical theory, in particular, from the distributions of initial
conditions of the micro-constituents of the involved physical bodies and
ultimately of the entire universe itself—of course, this move would be
contested by an empiricist like von Mises. Typicality frequentism, thus,
explains why probability theory is intrinsically imprecise and that this
imprecision cannot be improved, but at least to certain degree quantified and
grounded.

##  3 Defending Typicality Frequentism

Typicality Frequentism combines ideas from finite frequentism, hypothetical
frequentism, and the classical interpretation of probabilities. Finite
frequencies (with error bounds) describe actual outcomes of a series of a
chancy process; hypothetical frequencies in terms of infinite series are used
to define what probabilities are; and the principle of indifference, which is
the central piece of the classical interpretation, is replaced by a measure of
typicality to count events on the sample space. It seems, therefore, that the
critique raised against either of these interpretations of probability is
again effective to undermine typicality frequentism. The principle of
indifference has been rightly dismissed when an agent is truly
ignorant—although it may be successfully used for symmetry arguments (Zabell,
2016a). Hájek (1996, 2009) presents a total of 30 arguments against different
versions of frequentism, 15 against finite and 15 against hypothetical
frequentism, demanding that in order to rescue any kind of frequentist account
all these arguments need to be countered, where one counterargument would
still leave the other 29 unanswered. I won’t endeavor to reply to every single
argument, because not all counterarguments are in fact counterarguments but
rather characterize a frequentist’s account. Instead, I will first contrast
typicality frequentism with two most recent competitors, the range account and
the Humean Mentaculus. Then I will counter some recent arguments raised
against frequentism by La Caze (2016), who builds on Hájek’s papers.

###  3.1 The Range-Account of Probabilities

The work by von Kries (1886) was a rich source for further research. Henri
Poincaré and Eduard Hopf filled in a major gap by developing _the method of
arbitrary functions_ , which is also known as _the range-account of
probabilities_ , advocated and further refined in different versions by Abrams
(2012), Rosenthal (2010, 2016), and Strevens (2003, 2008, 2013). Here, the
probabilities, like in typicality frequentism, are related to some sort of
initial conditions, but, unlike typicality frequentism, regions of phase space
together with a probability density or a volume measure directly determine
probabilities. For example, for the coin toss a probability density over the
initial conditions for _every single toss_ is used (see Figure 3). The
physical state of a coin is completely described by its vertical velocity $v$
for the trajectory of the coin and the angular momentum $\omega$ for its
rotation, given a fixed height and further simplifying restrictions, like the
exclusion of bouncing (see Keller, 1986; Strzałko et al., 2008; Stefan and
Cheche, 2017, for detailed physical models). The phase space structure for the
coin toss has a regular structure, in which the size of the areas leading the
coin to land on the same side as it has started are approximately equal to the
size of the areas for which the coin changes faces (if $v$ or $\omega$ are not
too small). In the method of arbitrary functions, probabilities result when a
probability density is integrated over specific regions in this phase space.
The main two problems for the method of arbitrary functions is, first, to
justify the particular shape of the probability density and, second, to base
this justification on non-probabilistic facts in order not to explain
probabilities by probabilities. This is the main point in which Abrams,
Rosenthal, and Strevens disagree. They agree, however, that some measure must
be used to determine the sizes of phase space regions in terms of which
probabilities are defined.

The range account of probability is easily confused with typicality
frequentism. First, the range account does not define probabilities in terms
of frequencies. Nonetheless, Strevens’s account, for example, relies on a
close link to frequencies; he aims at explaining probabilities in long series
of trials and facts about frequencies determine facts about the (initial)
probability density (see also Strevens, 2011, section 4.2 and 4.3). Second,
typicality frequentism considers, like statistical mechanics, the initial
conditions _for the entire universe_ , where a measure of typicality is
imposed on. All these initial conditions are grouped into two main groups:
almost all initial conditions lead to typical behavior, whereas almost no
initial conditions lead to atypical behavior (there may be remaining sets that
do not fit in either category, but they are not important for our current
purposes). The typicality measure is only used to group the initial conditions
of the universe, from which probabilities are defined in terms of frequencies.

Figure 3: This shows the partition of phase space for the initial conditions
of a single coin, which determine how the coin will land after it is tossed.
The x-axis represents the initial conditions for the angular momentum around a
certain axis; the y-axis represents the vertical velocity of the entire coin.
Pink areas depict the initial conditions for which the coin lands on the same
face as it started, while the white areas stand for the initial conditions for
which the coin changes faces. In the method of arbitrary functions, one puts a
probability density on this phase space, which gives $\nicefrac{{1}}{{2}}$
once integrated over all the pink or all the white areas. (Picture from
Strzałko et al. (2008, p. 62) as an elaboration of Keller (1986, p. 193).)

There are several problems a range account faces, which Abrams, Rosenthal, and
Strevens are aware of and have reacted to. First, one needs to justify the
initial probability distribution. Where did it come from? Second, by
explaining the probabilities for a coin toss by a probability distribution
over the initial conditions, one would explain probabilities with
probabilities. It is a challenge to explain the properties of the initial
probability density from non-probabilistic facts in order not to make the
theory circular. Third, a probability distribution actually contains much more
information than is needed to get probabilities for frequencies. Typicality
frequentism, on the other hand, introduces something weaker than an initial
probability distribution that is more tailored to define probabilities as
special kinds of frequencies, and it does not suffer from a circular argument
(see the next section for a more detailed discussion of this point).

Fourth, typicality frequentism can explain the initial probability
distribution of the range-account (if it is the one used for frequencies). It
is known that the probabilities in repeatable processes are robust under many
changes of the initial probability distribution. Only very special
distributions (Rosenthal, 2016, calls them ‘eccentric’) would lead to
different probability assignments. In typicality frequentism these
distributions are, in fact, explained to arise from special initial conditions
of the universe yielding atypical behavior. Strevens (2011, p. 355–6) seems to
be aware of this when he says, “the typical actual pre-toss state, together
with the typical actual set of timings and the typical actual coin, usually
produce—because of such and such properties of the physiology of tossing—a
macroperiodic set of spin speeds.” But instead of embedding his theory into a
theory of typicality, Strevens borrows from Lewis’s possible-worlds semantics
to explain why we observe typical frequencies in our world.

###  3.2 The Humean Mentaculus

Albert (2000, 2015) and Loewer (2001, 2004, 2012) have been working on a
Humean account of probabilities.131313Hoefer (2007, 2011, 2019) developed a
more pragmatic account of Humean probabilities, which is closely linked to the
Albert–Loewer account. Similar to the range account, they postulate an initial
probability distribution, but this initial probability distribution is defined
on the phase space for the initial conditions of the _entire_ universe. More
precisely, the Albert–Loewer account of probabilities consists of three
postulates:

  1. 1.

The fundamental deterministic laws of physics.

  2. 2.

The existence of a special (low-entropy) macrostate (called the _past
hypothesis_).

  3. 3.

A probability distribution over the initial conditions of the universe (within
this macrostate).

These three postulates are embedded in a Humean interpretation of laws of
nature, so they are axioms in the best systematization of the Humean mosaic,
balancing simplicity, strength, and fit. The initial probability distribution
assigns a probability to all kinds of factual and counterfactual events. These
three postulates, the _Mentaculus_ , are said to form a probability map of the
history of the universe. Probabilities in this theory are defined, similarly
to the range account, as weighed regions of initial conditions _of the
universe_ (in phase space); in other words, one counts and weighs, according
to the initial probability distribution, all possible initial conditions of
the universe that would give rise to the relevant phenomenon. And again, as
the range-account, the Mentaculus needs to explain what it means for the
initial probability distribution to be a _probability_ distribution. So far,
the probability distribution axiomatically introduced by the Mentaculus is
merely a mathematical object that assigns numbers to certain sets.

The central feature and goal of the Albert–Loewer account is “to obtain a
definite numerical assignment of probability to every formulable proposition
about the physical history of the world” (Albert, 2015, p. 7-8). This
probability map assigns a probability not only to coin tosses but also to
events that may happen (or not) just once, like France defending the Soccer
World Cup title in 2022. There seems to be a shared intuition that these
single-case probabilities are meaningful and crucial to the notion of
probability—a point that has been raised against frequentism:

> The most famous problem for finite frequentism is _the problem of single
> case_. According to finite frequentism all single-case events automatically
> have the probability 0 or 1. Consider a coin that is only tossed once and
> comes up Heads. It appears that the probability of heads may be
> intermediate, but the finite frequentist is unable to say this. This goes
> against some strong intuitions about probability. A form of this problem
> remains in larger finite sequences. (La Caze, 2016, p. 343)

This criticism was raised early on against frequentism, to which von Mises
answered:

> ‘The probability of winning a battle’, for instance, has no place in our
> theory of probability, because we cannot think of a collective to which it
> belongs. The theory of probability cannot be applied to this problem any
> more than the physical concept of work can be applied to the calculation of
> the ‘work’ done by an actor in reciting his part in a play. (von Mises,
> quoted in Gillies, 2000, p. 98)

For many it was a shortcoming of frequentism that it does not assign
probabilities to single events, although it ought to do so (Hájek, 2009, p.
227–8). Von Mises argues, and I agree with him here, that scientific concepts
may not capture the full range of intuitive notions and it may not even be the
goal of science to form concepts that capture all the different meanings of an
intuitive notion. Scientific concepts are defined in a precise way for the
price of being less general. Probability, according to von Mises, is like the
word “work” in physics, which has a precise meaning in terms of an integral of
the forces along a certain path and which, thus, differs from the everyday
meaning of “work”. Von Mises was, therefore, open to a pluralistic account of
probability dependent on the field of application.

In contrast to von Mises, other frequentists tried to generalize probabilities
to single cases as a kind of fictitious value:

> Frequentists from Venn to Reichenbach have attempted to show how the
> frequency concept can be made to apply to the single case. According to
> Reichenbach, the probability concept is extended by giving probability a
> “fictitious” meaning in reference to single events. We find the probability
> associated with an infinite sequence and transfer that value to a given
> single member of it. (Salmon, 1966, p. 90)

Although one can formally or “fictitiously” assign these numbers from
frequencies to single events, their meaning is unclear, especially their
meaning as something objective or physical. This is not only a problem for
frequentism, but also for the Humean Mentaculus because it is unclear what a
probability in an objective or physical sense for a single event is in the
first place. A purely subjective account, on the other hand, would not have
this problem, as probabilities are an agent’s degree of belief, which are
meaningful for single events, because they capture how confident an agent is
to believe a proposition.

In the Mentaculus, probabilities are introduced by a probability density over
the initial conditions of the universe, but this probability density, it shall
be noted, merely axiomatically introduces numbers on the Humean mosaic. To
make this distribution of numbers a _probability_ distribution requires
further elaboration and an interpretation that turns these numbers into
probabilities. This is accomplished in two steps (Loewer, 2004). First, the
concept of “fit” is introduced. Every (probabilistic) proposition is said to
have a certain degree of fit, that is, how likely it is to be true, and this
is quantified by a probability. If a proposition with high probability matches
the actual facts, it has a better fit than a proposition with low
probability.141414The concept of fit leads to the _zero-fit_ problem; Elga
(2004) proposes a solution by invoking a certain notion of typicality. Second,
in order that fit in terms of probabilities is informative, an agent needs to
constrain her belief according to these probabilities (Loewer, 2004, p. 1122),
and this is done according to another axiom, the _Principal Principle_. It
roughly says that an agent ought to adjust her degree of belief or her
credence according to the probability of the proposition given by the Humean
best system.

It is not immediately clear what the physical meaning of single-case
probabilities is in this Humean theory. Let us say that there are two coins,
and the Mentaculus assigns a probability of landing heads of 0.4 to one coin
and 0.6 to the other. Each coin is just once tossed and then destroyed. What
can these numbers 0.4 and 0.6 mean? These probabilities indeed influence, by
the Principal Principle, an agent’s attitude and behavior toward the outcome
of the coin tosses. For example, an agent will bet differently on an event
with probability of 0.4 than on an event with a probability of 0.6. It seems,
however, that these single-case probabilities need also to say something about
the physical events themselves, whether their occurrence is in some way
constrained or not, which is then the basis for an agent to adjust her degree
of belief. Moreover, this example of two coins being tossed just once is in
principle repeatable, and so Humeans need to clarify the relationship between
these single-case probabilities and the frequencies of repeatable coin tosses.
Although the Albert–Loewer account of Humean probabilities explicitly
introduces and endorses single-case probabilities, it is, as of now, unclear
what their objective physical meaning is supposed to be.

###  3.3 Countering Standard Critique of Frequentism

Building on Hájek’s critique of frequentism, La Caze (2016) launched another
comprehensive attack. Here, I reply to four of La Caze’s arguments: (i) that
frequentism is a poor _analysis_ of probabilities, (ii) the problem of
ascertainability, (iii) the reference class problem, and (iv) that frequentism
is not completely objective.

####  3.3.1 It’s a Poor _Analysis_ of Probabilities

La Caze (2016) claims that hypothetical frequencies are not the right
description of probabilities because they provide a poor analysis of what
probabilities are:

> The hypothetical frequentist provides an answer to the question “What is
> probability?” with an analysis that has little relationship with what most
> people _mean_ by the probability statements they make. […] When stating that
> a specific coin has the probability of Heads of half, people are typically
> referring to their beliefs about the coin, their experience with this coin
> in a finite series of tosses, or their experience with similar-seeming
> coins. (La Caze, 2016, p. 350)

The aim of typicality frequentism is not to reduce all ways in which
probabilities are invoked to typical long-term frequencies. It, rather, aims
at showing how one can derive from fundamental physics physically meaningful
probabilities, and it is open to be complemented by other accounts of
probability outside its scope. Given the myriads of different cases in which
probabilities are used, it is plausible that all these cases are not unified
by one account. Typicality frequentism would be, in my view, one piece in a
pluralistic landscape of probabilities. Moreover, if typicality frequentism is
true, then people may need to re-think their intuitions they have about
probabilities of coins and other physical processes. I aim at giving an
account of objective probability, but I agree that we also need an account of
subjective probabilities, and I can envision that it may be possible, in
certain circumstances, to connect a particular interpretation of subjective
probabilities with typicality frequentism.

####  3.3.2 The Problem of Ascertainability

“[T]he problem of ascertainability is the most fundamental difficulty the
frequency interpretation faces,” says (Salmon, 1966, p. 89–90), and he defines
this problem in the following way:

> _Ascertainability._ This criterion requires that there be some method by
> which, _in principle at least_ , we can ascertain values of probabilities.
> It merely expresses the fact that a concept of probability will be useless
> if it is impossible _in principle_ to find out what the probabilities are.
> (Salmon, 1966, p. 64, my emphasis)

Actually, all interpretations of probability face in one form or other the
problem of ascertainability, that is, how to assign probabilities in practice.
A meaningful definition is not enough, because it may lack the instructions
for how to pick the right probabilities. Salmon stresses that these
instructions, however, are supposed to be applicable only _in principle_ , and
not necessarily in actual practice. Applied to hypothetical frequencies, they
are said to be unascertainable for the following reasons (see also Hájek,
2009, pp. 214–5):

> To ascertain a hypothetical frequency with certainty we would need to
> observe an infinite number of trials. Assuming that a specific sequence of
> observations will converge to a limiting relative frequency, there is no
> guarantee that it will do so within the number of trials that will be
> observed. And if a relative frequency appears to have converged in a finite
> number of trials, it is always possible that the relative frequency diverges
> from this value in subsequent trials. These points are direct consequences
> of the mathematics of infinite sequences. The task for the frequentist is to
> justify inferring a (frequentist) probability from a relative frequency
> observed in a finite number of trials, and there is no deductively valid way
> to do this. (La Caze, 2016, p. 353)

The argument amounts to the correct observation that we cannot figure out the
true probability (as a hypothetical frequency) by observing finite
frequencies. This objection is particularly damaging to von Mises and
Reichenbach because they defined the probabilities in the spirit of logical
empiricism based on obersvation. The only means that they had to reach the
hypothetical frequencies is by means of observable finite frequencies. In
order to mitigate this problem, von Mises introduced two principles
(Rowbottom, 2015, p. 100–1):

  1. 1.

_Law of Stability_ : The relative frequencies of attributes in collectives
become increasingly stable as observations increase.

  2. 2.

_Law of Randomness_ : Collectives involve random sequences, in the sense that
they contain no predictable patterns of attributes.

These laws are arguably ad hoc in von Mises theory, but at least they may be
justified by induction.

Similarly to von Mises, Reichenbach (1949/1971) bridged the gap between finite
and infinite sequences by induction; Reichenbach called his law the _Rule of
Induction by Enumeration_. Starting with an infinite sequence of events $A$,
we are interested in the relative frequency that some feature $B$ occurs in
this sequence. We can only observe a finite sequence of events of length $n$,
for example. The frequency of feature $B$ among the first $n$ members of $A$
is, say, $F^{n}(A,B)=\nicefrac{{m}}{{n}}$. In order to infer the limiting
frequency, the _Rule of Induction by Enumeration_ needs to be applied: Given
$F^{n}(A,B)=\nicefrac{{m}}{{n}}$, to infer that
$\lim\limits_{n\rightarrow\infty}F^{n}(A,B)=\nicefrac{{m}}{{n}}$ (Salmon,
1966, pp. 85–6).

La Caze, on the other hand, demands a deductive way to get to the hypothetical
frequencies, and this can be, in principle, accomplished by typicality
frequentism, as the hypothetical frequencies are predictions of the laws of
physics about typical behavior. By applying a physical theory, probably by
building a model as is standard in many cases (Cartwright, 1983; Morgan and
Morrison, 1999; Giere, 2004), the probabilities fall out of the theory as any
other empirical prediction. This move was not possible for von Mises and
Reichenbach, as they based their probabilities on observable behavior of the
physical processes. Typicality frequentism adheres to Salmon’s requirement for
solving the problem of ascertainability, because we can access the information
of a physical theory _in principle_ ; in practice, there might be strong
limitations on how to access all this information, but these obstacles are not
of a different nature than we normally encounter in other kinds of empirical
predictions.

One may argue that the “in principle” in typicality frequentism does a lot of
work.151515Thanks to an anonymous referee for raising this point. If we have a
powerful enough physical theory that also makes it easy to extract empirical
predictions, then we would be able to solve the problem of ascertainability.
But what if we cannot extract this information from a physical theory (for
whatever reason)? Then either we need to extract the right frequencies from
observations, or we need to apply further metaphysical or physical
assumptions. Both paths are problematic: the first because we would fall back
to the (empirical) problem of hypothetical frequentism, the second because
further theoretical assumptions need to be justified. I grant that this is
argument poses a challenge to the epistemology of typicality frequentism, that
is, how to ascertain the probabilities in practice. It is in general very
hard, and mostly impossible, to extract precise empirical information from a
physical theory for sufficiently complex systems—we cannot even analytically
solve the three-body problem in classical physics. Therefore, for practical
purposes we rely on other means to make empirical predictions: for example, by
making certain idealizations and approximations. In the case of probabilities,
we may need to rely on past incomplete empirical observations for future
predictions, or we may use theoretical assumptions, like symmetry arguments
(Zabell, 2016a).

####  3.3.3 The Reference Class Problem

The reference class problem is generally regarded to sound another death knell
to frequentism (Hájek, 2009, pp. 219), although it was originally raised _by
frequentists_ , like Venn (1888) and Reichenbach (1949/1971), _against single-
case probabilities_ :

> If we are asked to find the probability holding for an individual future
> event, we must first incorporate the case in a suitable reference class. An
> individual thing or event may be incorporated in many reference classes,
> from which different probabilities will result. This ambiguity has been
> called the _problem of the reference class_. (Reichenbach, 1949/1971, p.
> 374)

The reference class problem for single-case probabilities is a problem of how
to get the probability of one event when it can be part of many collections.
Venn’s example is the single-case probability of a man called John Smith, aged
50, to die at age 61. In order to make a qualified prediction of Smith’s life
in the future eleven years, one needs to compare Smith with other people
similar to Smith. In order to extract single-case probabilities from
frequentism, one would need to find a set of people similar to Smith and who
live until 61 and compare this number with all the people of this age. The
problem is, however, that it is not clear which properties the reference
class, that is, the people similar to Smith, need to have in order to count as
”similar to Smith,” (also because Smith himself has so many different
properties).

This example can be transferred into a reference class problem for frequentism
in general (La Caze, 2016, section 16.4.4); we just need to add to John Smith
any finite number of people of the same age and ask about the probability of
their life expectancy until age 61. What is the correct infinite collection of
people that give rise to the right probability? More precisely, given a finite
sequence of events $(x_{1},\dots,x_{n})$, what is the appropriate infinite
sequence $(y_{1},y_{2},\dots)$ that we shall associate with
$(x_{1},\dots,x_{n})$ in order to assign the probability
$p=\lim\limits_{m\rightarrow\infty}\frac{1}{m}\sum_{i=1}^{m}y_{i}$ for some
feature of $(x_{1},\dots,x_{n})$?

Furthermore, having found a suitable (or even the “correct’?) reference class,
the order of the members of the reference class may change the probability,
and there may be even an ordering where the sequence does not converge and no
probability can be assigned in the first place. Hájek (2007, p. 567) calls
this subcategory of the reference class problem the _reference sequence
problem_. Von Mises dealt with the reference sequence problem by restricting
the admissible sequences to give a unique ordering; these sequences, he called
_collectives_ , and they are defined by means of his two laws of probability,
the law of stability and the law of randomness. With this move, von Mises
could only solve, or propose a solution to, one aspect of the reference class
problem, namely, what Hájek (2007, p. 565 ) calls the _metaphysical_ reference
class problem. Given the two laws of probability, there is (hopefully) a fact
what the correct reference class is and what accordingly the probability is.
Still, this information may be practically inaccessible for an agent, which
amounts to an _epistemological_ reference class problem.

Does the reference class problem only arise in frequentist interpretations of
probabilities? Hájek (2007) argues that basically _all_ interpretations of
probability face their version of the reference class problem, and the best we
can hope for is to solve the metaphysical problem—the epistemological problem
will always remain. And theories that do not face a reference class problem in
the first place, like radical subjectivists à la de Finetti or certain
versions of the propensity interpretation, are, according to Hájek, no-theory
theories of probability, because they do not sufficiently specify what
probabilities are and how they are to be used to guide agents’ beliefs and
actions.

Typicality frequentism indeed solves the metaphysical reference class problem
by means of a physical theory, something Salmon also mentioned as a way out:

> When a sequence is generated by a physical process that is well understood
> in terms of accepted physical theory, we may be able to make theoretical
> inferences concerning convergence properties. For instance, our present
> knowledge of mechanics enables us to infer the frequency behavior of many
> kinds of gambling mechanisms. Our theory of probability must allow room for
> inferences of this kind. The basic problem, however, concerns sequences of
> events for which we are lacking such physical knowledge. (Salmon, 1966, p.
> 84)

The Theory of Everything ultimately determines the underlying physical
processes of a random sequence, and thus determines the limit of a finite
sequence if one were to repeat it infinitely. The reference class problem is
solved in typicality frequentism, because the reference class is the finite
sequence itself which gets extrapolated into an infinite sequence by means of
the Theory of Everything. Since we do not yet have a Theory of Everything, any
candidate for a fundamental physical theory determines the behavior of the
reference class. In other words, the truthmaker for singling out the reference
class and the corresponding behavior is the Theory of Everything, and for the
current situation we can replace the Theory of Everything by an appropriate
candidate for a fundamental physical theory or by a model of the physical
theory (given certain idealizations). So the gap in Reichenbach’s Rule of
Induction by Enumeration is closed not by induction from the observable
sequence itself, but by a physical theory describing the physical processes
underlying the sequence.

In a similar vein, the reference sequence problem is tackled. Intricate
orderings that yield different limits or no limit at all are physically
possible but atypical, given the initial conditions of the universe, which
determine the physical conditions of the physical processes governing the
sequence.161616Something similar has been proposed in certain versions of the
propensity interpretation. Miller (1994, p. 56) says that propensities are
determined by “the complete situation of the universe (or the light-cone) at
the time, and, for Fetzer (1982, p. 195), they are determined by “a complete
set of (nomically and/or causally) relevant conditions […] which happens to be
instantiated in that world at that time.” These solutions, however, are not
satisfactory for Hájek (2007, p. 576) because the propensities, such defined,
are not accessible to an agent to assign probabilities in practice. Therefore,
he subsumes these proposals under no-theory theories of probabilities. More
precisely, there is a physically distinguished “natural” ordering of the
sequence, namely, the temporal ordering as determined or predicted by physics.
Rowbottom (2015, p. 111) presents an argument that physics is not able to
single out a natural order for sequences , because, according to special
relativity, the order of, say, coin flips depends on the state of motion of an
observer. So two observers on two different trajectories may disagree on the
order of the same coin flips that they observe. But this would be only correct
when the observers would see _two_ different sequences of coin flips that are
space-like separated. If Rowbottom refers to one sequence of coin flips, and I
assume he does because this is the relevant case at issue, then the coin flips
are time-like separated, and, according to special relativity, the temporal
order of time-like separated events are objective, that is, independent of the
state of motion of observers.

####  3.3.4 Frequentism Is Not Completely Objective

Von Mises (1928/1957, p. 14) makes a strong assertion about frequentistic
probabilities when he says, ”The probability of a 6 is a physical property of
a given die and is a property analogous to its mass, specific heat, or
electrical resistance.” I agree with La Caze (2016, section 16.4.5) that
frequentism does not oblige upon us this strong metaphysical interpretation of
probabilities, but I disagree that frequentistic probabilities are not
objective. For La Caze, “[h]ypothetical frequencies are not divorced from
consideration of personal factors (including beliefs).”

His argument goes like this. Since the main advantage proclaimed of
frequentism is that it introduces objective probabilities, any subjective
trace in frequentistic probabilities would undermine the entire project. The
subjectivity that frequentism relies on comes from how the particular physical
process that gives rise to frequencies is modeled. The probabilities for a
coin toss, for example, depend on how the properties of the coin and the
tossing mechanism are modeled. That some particular physical model is suitable
for giving rise to the proper frequencies needs to be judged by an agent. And
this judgement is unequivocally subjective, as La Caze (2016, p. 358) says,
“Scientists employing frequentists probabilities need to make a judgement that
the data-generating processes providing the measured outcomes of the study are
adequately modeled by one or more of these approaches to specifying the
requirements on the expected sequence of outcomes.” The bar raised by this
requirement is so high that basically all our physical predictions are deemed
to be subjective, because they depend on certain idealizations to be made by
an agent. The practice of physics has for practical matters this “subjective”
ingredient but it does not make physics a subjective science. Therefore, I do
not see that frequentistic probabilities are less objective than other
predictions of physics.

Hájek (2009, pp. 215–7) also criticizes the idealizations made in hypothetical
frequentism, but he approaches this problem from a different direction:

> Consider the coin toss. We are supposed to imagine infinitely many results
> of tossing the coin: that is, a world in which coins are ‘immortal’, lasting
> forever, coin-tossers are immortal and never tire of tossing (or something
> similar anyway), or else in which coin tosses can be completed in ever
> shorter intervals of time… In short, we are supposed to imagine _utterly
> bizarre_ worlds […]. (Hájek, 2009, pp. 215–6)

For Hájek, the problem of hypothetical frequentism lies in the definition of
probabilities: in order to define hypothetical frequencies “utterly bizarre”
counterfactual scenarios need to be set up that “would have to be _very_
different from the actual world” (Hájek, 2009, pp. 215). I think this problem
can be remedied by a physical theory and the laws of nature in such a theory.
We know that laws of nature ground facts beyond the actual regularities (e.g.,
Maudlin, 2007b). The counterfactual idealizations that need to be made for
hypothetical frequentism—and also for typicality frequentism—may be more
radical or more detached from the actual world than in other applications,
like in the normal way of model building (Morgan and Morrison, 1999), but they
can be still grounded and made true by the laws in a physical theory.

###  3.4 Objections and Replies

  1. 1.

_Typicality seems to be too vague. How can it be meaningful?_

Typicality is intentionally a vague term. Not all notions need to be precise
to be meaningful. We know when someone is tall or when someone is bald. Of
course, there may be borderline cases when we may debate is this person really
tall or really bald, but for all practical purposes there is no ambiguity. The
same we encounter in physics. The initial macrostate the universe evolved from
according to statistical mechanics is also vague, because the boundaries are
fuzzy and not precisely specified. But when we reason about the evolution of
the universe we talk about microstates that do not lie on the boundary, so
this vagueness is harmless. It is a strength of the notion of typicality to be
vague, because we don’t need to cope with unnecessary details in our
explanation and we can use typicality in many different areas.

  2. 2.

_What is a formal definition of typicality?_

In many cases, typicality does not need a formal definition. It is basically a
technical term for _most_ or _almost all_. Maudlin (2018) and Wilhelm (2019)
propose two different approaches to formalize typicality. Maudlin interprets
typicality as a second-order predicate, that is, a predicate of a predicate.
We formally write $F(X)$ symbolizing that $X$ has property $F$. Typicality
would be a further qualification between $X$ and $F$. $T(F,X)$ would symbolize
that it is typical for $X$ to have $F$. One may even consider typicality as
another quantifier. Wilhelm, on the other hand, focuses on the explanatory
scheme of typicality explanations and points out that it resembles Hempel’s
deductive–nomological model.

  3. 3.

_What is the relationship between a probability measure and a typicality
measure?_

Mathematically, a typicality measure is usually represented as a probability
measure, but a probability measure contains more information than is actually
needed:

> While typicality is usually defined – as it was here – in terms of a
> probability measure, the basic concept is not genuinely probabilistic, but
> rather a less detailed concept. A measure $\mu$ of typicality need not be
> countably additive, nor even finitely additive. Moreover, for any event $E$,
> if $\mu$ is merely a measure of typicality, there is no point worrying
> about, nor any sense to, the question as to the real meaning of say
> ‘$\mu(E)=\nicefrac{{1}}{{2}}$‘. Distinctions such as between
> ‘$\mu(E)=\nicefrac{{1}}{{2}}$‘ and ‘$\mu(E)=\nicefrac{{3}}{{4}}$‘ are
> distinctions without a difference.
>
> The only thing that matters for a measure $\mu$ of typicality is ‘$\mu(E)\ll
> 1$´: a measure of typicality plays solely the role of informing us when a
> set $E$ of exceptions is sufficiently small that we may in effect ignore it
> and regard the phenomenon in question occurring of the set $E$, as having
> been explained. (Goldstein, 2001, p. 15)

  4. 4.

_What is the difference between typicality and probability? Is _typical_ just
another word for _highly probable_ and _atypical_ for _highly improbable_?_

Historically typicality evolved from abstracting from highly probable cases.
Boltzmann, for example, said that the second law of thermodynamics makes it
highly probable that a gas in a box equilibrates. But I think that typicality
is a more primitive notion than and different from probability, and this paper
showed how one can reduce probabilities to typicality. Typicality is a much
less-fine grained and more general concept than probability.

  5. 5.

_There are cases where something typical is highly improbable. For example, a
long, well-mixed sequence of heads is typical but improbable, e.g.
HTHTTHHHTTTHTHHT. Or a very specific event may be typical but improbable, e.g.
the probability of randomly selecting from the US population a man of height
exactly 175.4 cm is very low, even though this is the average height, and in a
good sense typical. How can one reconcile that?_ 171717Thanks to an anonymous
referee for raising this issue, who I quote almost verbatim.

The difference between typicality and probability have been addressed in more
detail in Wilhelm (2019). Typicality is not a categorical property. So it
doesn’t make sense to say that something is typical by itself. There needs to
be always a reference: “typical with respect to what?” It is typical for
clovers to have three leaves, because in the class of all clovers most of them
have three leaves. If we zoom in too much, for instance, when comparing the
particular shapes of the leaves, every leaf may be unique, and we may not be
able to find any “typical shape”. Applied to the coin toss, if we zoom in the
particular pattern of a series of tosses, and ask, “Is HTHTTHHHTTTHTHHT
typical?”. The right answer is, “Typical with respect to what?” Typical with
respect to the number of heads and tails. Then yes, because approximately 50%
are heads and tails (I ignore that the series needs to be much longer to make
such a statement). But what about the particular pattern HTHTTHHHTTTHTHHT? It
is very unlikely to repeat this particular pattern in an actual coin toss. But
this is the case for any particular pattern. The same is true in statistical
mechanics: every particular trajectory in phase space has measure zero and is
therefore very unlikely (or atypical, although it is meaningless to talk about
atypicality per se too). This point has been raised against typicality by
Uffink (2007), and I think it has been rightly answered by Lazarovici and
Reichert (2015, section 5.2). I agree on the example of the height, which is
similar to the way I define probabilities. The actual height 175.4 cm is
rarely found in a person but most people are close to the average.

  6. 6.

_If you reduce probabilities to typical longterm frequencies, then you cannot
account for all the uses of probability. Especially single-case probabilities
lack an explanation._

That is correct, but I claim that single-case probabilities are not
meaningfully interpreted as some kind of frequency. They may be properly
construed as purely subjective degrees of belief, as a kind of tool in
Bayesian updating, but not in an ontological sense. Therefore, I endorse a
pluralistic account of probabilities tailored to different applications.

##  4 Conclusion

If our world is correctly described by a deterministic physical theory, then
every event is determined by the initial conditions of the universe.
Typicality frequentism builds on this insight and singles out physical
processes that give rise to stable long-term frequencies. If these frequencies
are typical they define probabilities. As I have shown, the essential idea
behind this approach comes from how Boltzmann explained the thermodynamic
arrow of time and how he reduced thermodynamics to statistical mechanics. The
main advantage I see with typicality frequentism is that it carves objective
probabilities at the right joint by specifying those kinds of probabilities
that are meaningful within physics. In this way, typicality frequentism does
not face the same problems as traditional empiricist accounts of frequentism
do. Other applications of probability beyond physics may be properly described
by subjective approaches that would complement to a pluralistic picture of
probabilities.

## Acknowledgements

I wish to thank Frederick Eberhardt, Christopher Hitchcock, and Charles Sebens
for their helpful and detailed comments on previous drafts of this paper. I
also wish to thank David Albert, Jeffrey Barrett, Detlef Dürr, Sheldon
Goldstein, Dustin Lazarovici, Barry Loewer, Tim Maudlin, and Isaac Wilhelm for
many invaluable hours of discussions. I also thank the members of the _Caltech
Philosophy of Physics Reading Group_ , in particular Joshua Eisentahl and
James Woodward. I want to thank two anonymous reviewers for their helpful
comments, which significantly improved the paper. Especially one of the
anonymous reviewers spent considerable time and effort in the review process;
I particularly thank this reviewer.

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