What is the Philosophy of Statistics? (and how I was drawn to it)

What is the Philosophy of Statistics?
(and how I was drawn to it)
Deborah G Mayo
Dept of Philosophy, Virginia Tech
April 30, 2025
1

In a conversation with Sir David Cox
COX: Deborah, in some fields foundations do not
seem very important, but we both think foundations of
statistical inference are important; why do you think?
MAYO: …in statistics …we invariably cross into
philosophical questions about empirical knowledge
and inductive inference.
(“A Statistical Scientist Meets a Philosopher of Science” 2011) 2

3
At one level, statisticians and
philosophers of science ask
similar questions
• What should be observed and what may
justifiably be inferred from data?
• How well do data confirm or test a model?

Two-way street
• Statistics is a kind of “applied philosophy of
science” (Kempthorne, 1976).
4

5
“The Statistics Wars”
Statistics has long been the subject of
philosophical debate marked by unusual heights
of passion and controversy

6
Philosophy → Statistics
• A central job for philosophers of science:
minister to conceptual, epistemological, and
logical problems of sciences

7
Statistics → Philosophy
Statistical ideas are used in philosophy to:
1) Model Scientific Inference—ways to arrive at
evidence and inference
2) Solve (or reconstruct) Philosophical
Problems about inference and evidence (e.g.,
problem of induction, should we prefer novel
predictions?)

8
Logics of Confirmation
• Failure to justify (enumerative) induction led to
building logics of confirmation C(H,e), like
deductive logic: Carnap
• Evidence e is given, calculating C(H,e) is formal.
• Carnap: Scientists could come to the inductive
logician for rational degree of confirmation

9
“According to modern logical empiricist orthodoxy...It is
quite irrelevant whether e was known first and h
proposed to explain it, or whether e resulted from
testing predictions drawn from h”.
(Alan Musgrave 1974, p. 2).
The search for C(H,e) did not succeed, and the
program is challenged by post-positivists (Popper,
Lakatos, Laudan, Kuhn)

10
Post-Positivists
Evidence e is theory-laden, not given:
Collected with difficulty, must start with a problem
Popper: Falsification, not Confirmation:
Corroborate claims that survive severe tests

11
Popper never adequately cashed out
severity—tried adding requirements like novel
predictive success to C(h, e)
(due partly to his unfamiliarity with statistical
methods)

Neyman and Pearson (N-P):
Methodological falsification
“We may search for rules to govern our behavior…in
following which we insure that in the long run of
experience, we shall not be too often wrong” (Neyman
and Pearson 1933, 141-2)
• Probability is assigned, not to hypotheses, but to
methods 12

Series of models
It is a fundamental contribution of modern mathematical
statistics to have recognized the explicit need of a
model in analyzing the significance of experimental
data. (Patrick Suppes 1969, 33)
13

“I might never have sauntered into that first class on
mathematical statistics had the Department of
Statistics not been situated so closely to the
Department of Philosophy [at U Penn]. ...I
suspected that understanding how these statistical
methods worked would offer up solutions to the
vexing problem of how we learn about the world in
the face of error.” (Mayo 1996, Error and the
Growth of Experimental Knowledge: EGEK)
14

15
What really influenced me in the years before EGEK
(here at Virginia Tech):
In statistics: famous Bayesian I.J. Good:
Why wasn’t I a Bayesian?
In philosophy, leading Popperian Laudan:
Can I make progress on their problems about
large scale units: paradigms and research programs

16
What I found then holds true:
• Insights from grappling with foundational
problems of statistics (and data science)
provide a gold mine for making progress on
philosophical problems
• Turn now to controversies of PhilStat

17
Long-standing philosophical
controversy on roles of probability
Frequentist (performance): to control and assess
the relative frequency of misinterpretations of
data—error probabilities
(e.g., P-values, confidence intervals,
randomization, resampling)
Bayesian (and other probabilisms): to assign
degrees of belief or support in claims
(e.g., Bayes factors, Bayesian posterior
probabilities)

Beyond performance and
probabilism
• Long-standing battles simmer below the surface
in today’s ”statistical (replication) crisis” in science
• Biggest source of handwringing? High powered
methods make it easy to find impressive-looking
but spurious effects
18

I set sail with a minimal principle
of evidence (severity principle)
• We don’t have evidence for a claim C if little if
anything has been done that would have
found C flawed, even if it is
19

Statistical inference as
severe testing
Probability arises (in statistical inference) to
assess and control how capable methods are at
uncovering and avoiding erroneous
interpretations of data
20

I invite the reader: “Envision yourself embarking on
a special interest cruise” (Excursions and Tours)
The severity principle is an excavation tool: to get
beyond the “statistics wars” and for appraising
reforms
21

Replication crisis leads to
“reforms”
Several are welcome:
• preregistration of protocol, replication
checks, avoid cookbook statistics
Others are radical
• and even lead to violating our minimal
requirement for evidence
22

Being an outside philosopher helps
To unearth assumptions, combat paradoxical,
“reforms” requires taking on strong ideological
leaders (philosophical observer)
23

What is the replication crisis?
• Results that had been found statistically
significant are not found significant when an
independent group seeks to replicate them.
• What are statistical significance tests?
24

25
Simple significance (Fisherian) tests
“…to test the conformity of the particular data
under analysis with H0 in some respect….” (Mayo
and Cox 2006, p. 81)
…the P-value: the probability the test would yield
an even larger (or more extreme) value of a test
statistic T assuming H0 chance variability or noise
NOT Pr(data|H0 )

26
Testing reasoning:
statistical modus tollens
• Small P-values indicate* some underlying
discrepancy from H0 because very probably
(1- P) you would have seen a less impressive
difference were H0 true.
• Still not evidence of a genuine statistical effect
H1 yet alone a scientific conclusion H*
*(until an audit is conducted testing assumptions)

27
Neyman and Pearson tests (1933) put
Fisherian tests on firmer ground:
Introduce alternative hypotheses H0, H1
e.g., H0: no benefit vs. H1: some benefit
Controls Type I errors (erroneous rejections) and
Type II errors (erroneously failing to reject); ensures
that as effect sizes increase, so does test’s power

Hypotheses vs Events
Statistical hypotheses assign probabilities to data or
events Pr(x0;H0)
• Correctly guessing tea or milk first is like a coin tossing
trial Pr(success) on each trial = .5 (lady tasting tea)
• A drug does not improve lung function in patients
• The deflection of light due to the sun is 1.75 degrees
(an estimate)
It’s rare to assign frequentist probabilities to stat
hypotheses (not random variables); inference is
qualified by probabilistic properties of the test method28

Error control is lost by biasing
selection effects: fooled by
randomness
• Sufficient finagling—cherry-picking,
significance seeking, multiple testing, post-
data subgroups, trying and trying again—
may practically guarantee an impressive-
looking effect, even if it’s unwarranted by
evidence
• Violates error control and severity
29

30
Data Dredging (Torturing): P-hacking
The case of the Drug CEO: Finding no statistically
significant benefit on the primary endpoint (lung benefit),
nor on 10 other secondaries....ransacks the data
Harkonen vs. United States (2013)—guilty of wire fraud
Trump pardons

31
Violates: Severity Requirement
We have evidence for a claim C only to the extent C
has been subjected to and passes a test that would
probably have found C flawed, just if it is.
• This probability is the stringency or severity with
which C has passed the test.
• applicable to any error-prone problem

Severity gives an evidential twist
• What is the epistemic value of good performance
relevant for inference in the case at hand?
• What bothers you with selective reporting, cherry
picking, stopping when the data look good, P-hacking
• We cannot say the test has done its job in the case at
hand in avoiding sources of misinterpreting data
32

A claim C is not warranted _______
• Probabilism: unless C is true or probable (gets
a probability boost, made comparatively firmer,
more believable)
• Performance: unless it stems from a method
with low long-run error
• Probativism (severe testing) unless
something (a fair amount) has been done to
probe (& rule out) ways we can be wrong
about C
33

Informal Example: Severe Tests
To test if I’ve gained weight between the start and
end of the pandemic, I use a series of well-calibrated
scales, at the start and after.
All show an over 4 lb gain, none shows a difference
in weighing EGEK, I’m forced to infer:
H: I’ve gained at least 4 pounds
We argue about the source of the readings from the
high capacity to reveal if any scales were wrong
34

35
The severe tester is assumed to be
in a context of wanting to find
things out
• I could insist all the scales are wrong—they
work fine with weighing known objects—but
this would prevent correctly finding out about
weight….. (rigged alternative)
• What sort of extraordinary circumstance could
cause them all to go astray just when we do
not know the weight of the test object?

36
Aside:
Severity led to reformulating significance tests,
after 1999, I was going to focus on problems in
philosophy of science and epistemology:
e.g., when can proportions in populations give
probabilities for the single case? (almost never)
(Laudan shifts to work on probability: error in the
law)

37
Prof. Aris Spanos Wilson Schmidt Professor
of Economics at Virginia Tech

In 2003 I invited Sir David Cox
(Oxford) to be part of a session on
Phil Stat
• “Our goal is to identify a key principle of
evidence by which hypothetical error
probabilities may be used for inductive
inference.” (Mayo and Cox 2006)
• Fisher tried fiducial probability
38

A rival (and more philosophically
popular) view of stat evidence
underlies ‘probabilism’
• Philosopher Ian Hacking (1965) “Law of
Likelihood”:
Pr(x0;H0)/Pr(x0;H1)
x0 supports H0 less well than H1 if H0 is less likely
than H1 in this technical sense.
(Note: the likelihood of a hypothesis is not its
probability)
39

Problem
Any hypothesis that perfectly fits the data is
maximally likely (even if data-dredged)
• “there always is such a rival hypothesis viz.,
that things just had to turn out the way they
actually did” (Barnard 1972, 129)
40

Error probabilities are
“one level above” a fit measure:
Pr(H0 is found less well supported than H1; H0)
is high for some H1 or other
41

“There is No Such Thing as a Logic
of Statistical Inference”
• Hacking changes his mind (1972, 1980)
“I now believe that Neyman, Peirce, and
Braithwaite were on the right lines to follow in
the analysis of inductive arguments”
(Hacking 1980, 141)
The analogous debate Mill (1888). Keynes
(1921) vs. Whewell (1847), Peirce (1931-5);
Popper (1959)
42

Likelihood Principle (LP)
A pervasive view remains: all the evidence
from x is contained in the ratio of likelihoods*:
Pr(x;H0) / Pr(x;H1)
• Follows from inference by Bayes theorem
*for inference in a model
43

On the LP, error probabilities
appeal to something irrelevant
“Sampling distributions, significance levels,
power, all depend on something more [than the
likelihood function]–something that is irrelevant
in Bayesian inference–namely the sample
space” (Lindley 1971, 436)
44

Some alternatives offered to replace
significance tests obey the LP
• “Bayes factors can be used in the complete absence
of a sampling plan…” (Bayarri, Benjamin, Berger,
Sellke 2016, 100)
• It seems very strange that a frequentist could not
analyze a given set of data…if the stopping rule is
not given. (Berger and Wolpert, The Likelihood
Principle 1988, 78)
• Stopping rules refer to a second kind of multiplicity 45

Optional Stopping:
H0: no effect vs. H1: some effect
Instead of fixing the sample size n in advance:
Keep sampling until H0 is rejected at (“nominal”)
0.05 level
46

In testing the mean of a standard
normal distribution
47

Optional Stopping
48
• “if an experimenter uses this [optional stopping]
procedure, then with probability 1 he will
eventually reject any sharp null hypothesis,
even though it be true”
(Edwards, Lindman, and Savage 1963, 239)
• Understandably, they observe, the significance
tester deems this cheating, requiring an
adjustment of the P-values

Nevertheless, on their view:
• “the import of the...data actually observed will
be exactly the same as it would had you
planned to take n observations.” (ibid)
• What counts as cheating depends on statistical
philosophy
49

Contrast this with reforms from
replication research
• Simmons, Nelson, and Simonsohn (2011):
“Authors must decide the rule for terminating
data collection before data collection begins and
report this rule in the articles” (ibid. 1362).
50

Replication Paradox
• Significance test critic: It’s too easy to obtain low
P-values
• You: Why is it so hard to replicate low P-values with
preregistered protocols?
• Significance test critic: Initial studies were guilty of
P-hacking, cherry-picking, data-dredging (QRPs)
• You: So, replication researchers want methods that
pick up on these biasing selection effects.
• Significance test critic: Actually, reforms
recommend methods insensitive to gambits that
alter error probabilities
51

52
A question
If a method is insensitive to error probabilities, does
it escape inferential consequences of gambits that
inflate error rates?
It depends on a critical understanding of “escape
inferential consequences”

53
It would not escape consequences for a
severe tester:
• What alters error probabilities alters the
method’s error probing capability
• Alters the severity of what’s inferred

54
Notes/qualifications:
• Bayesians and frequentists use sequential trials—
difference is whether/how to adjust
• Post-data selective inference is a major research
area of its own in AI/ML
• Data-dredging need not be pejorative: It’s a biasing
selection effect only when it injures severity
• It can even increase severity (e.g.,searching for a
match with DNA testing in a complete data base)

Bayesian Probabilists may (indirectly)
block intuitively unwarranted
inferences
(without error probabilities)
• Likelihoods + prior probabilities
• Give high prior probability to “no effect”
(spike prior)
55

Problems
• It doesn’t show what researchers had done
wrong—battle of beliefs
• The believability of data-dredged hypotheses is
what makes them so seductive
• Additional source of flexibility
56

No help with the severe tester’s
key problem
• How to distinguish the warrant for a single
hypothesis H with different methods
(e.g., one with selection effects, another, pre-
registered results)?
• Since there’s a single H, its prior would be the
same
57

Most Bayesians (last 15 years) use
“objective” or “conventional” priors
• ‘Eliciting’ subjective priors too difficult:
“[V]irtually never would different experts give prior
distributions that even overlapped” (J. Berger 2006,
392)
• Conventional priors are supposed to prevent prior
beliefs from influencing the posteriors–data
dominant (ideally, some performance guarantees)
58

How should we interpret them?
• “The priors are not to be considered expressions
of uncertainty, ignorance, or degree of belief.
Conventional priors may not even be
probabilities…” (Cox and Mayo 2010, 299)
• No agreement on rival systems for default/non-
subjective priors
(maximum entropy, invariance, maximizing the
missing information, coverage matching.)
59

60
Severity Reformulates tests to
avoid classic fallacies
in terms of discrepancies (effect sizes) that are and
are not severely-tested
SEV(Test T, data x, claim C)
• In a nutshell: one tests several discrepancies
from a test hypothesis and infers those well or
poorly warranted
Mayo 1991-; Mayo and Spanos (2006, 2011); Mayo and
Cox (2006); Mayo and Hand (2022)

61
SEV(𝛍 > 𝛍𝟏), 𝛍𝟏 = 𝛍𝟎+ 
to avoid misinterpreting low P-values
(SE =1)

Some Bayesians reject probabilism
(Gelman and Shalizi (2013)):
Falsificationist Bayesian
“[M]ost of [the] received view of Bayesian inference is
wrong. ...[C]rucial parts of Bayesian data analysis, …
can be understood as ‘error probes’ in Mayo’s sense.”
(10)
“[W]hat we are advocating, then, is what Cox and
Hinkley (1974) call ‘pure significance testing.” (10).
• Last part of SIST: (Probabilist) Foundations Lost,
(Probative/ Error Statistical) Foundations Found 62

63
• A silver lining to distinguishing highly
probable and highly probed–can use
different methods for different contexts
• Attempts to unify is behind much of the
confusion about stat concepts today in
medicine, economics, law, psychology,
climate science, social science
• Philosophical insight is needed

Aside: there’s a famous argument
(Birnbaum, 1962) that frequentists
principles entail LP
• I claim the argument is unsound—using logic alone
• In a book out a few weeks ago (Rod Little)—Little
doesn’t agree with me
64

Thank you!
If there’s time after questions, I’ll
say more about the severity
reinterpretation
65

66
SEV(𝛍 > 𝛍𝟏), 𝛍𝟏 = 𝛍𝟎+ 
to avoid misinterpreting low P-values
(SE =1)

67
Severity for 𝛍 > 𝛍𝟏 vs Power

In the same way, severity avoids
the “large n” problem
• Fixing the P-value, increasing sample
size n, the 2SE cut-off gets smaller
68

Severity tells us:
• A difference just significant at level α indicates less
of a discrepancy from the null if it results from larger
(n1) rather than a smaller (n2) sample size (n1 > n2 )
• What’s more indicative of a large effect (fire), a fire
alarm that goes off with burnt toast or one that
doesn’t go off unless the house is fully ablaze?
• [The larger sample size is like the one that goes off
with burnt toast] 69

70
What about fallacies of
non-significant results?
• Not evidence of no discrepancy, but not
uninformative
• Minimally: Test was incapable of
distinguishing the effect from noise
• Can also use severity reasoning to rule out
discrepancies

71
SEV(𝛍 < 𝛍𝟏), to set upper bounds

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77
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Jimmy Savage on the LP:
“According to Bayes' theorem,…. if y is the
datum of some other experiment, and if it
happens that P(x|µ) and P(y|µ) are
proportional functions of µ (that is,
constant multiples of each other), then
each of the two data x and y have exactly
the same thing to say about the values of
µ…” (Savage 1962, 17)
78

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