Meeting #1 Slides Phil 6334/Econ 6614 SP2019

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PHIL 6334/ ECON 6614: Spring 2019
Current Debates on Statistical Inference and
Modeling
Wed. 4-6:30 Derring 1076
Deborah G. Mayo Aris Spanos

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What is the Philosophy of
Statistics (PhilStat)?
At one level of analysis, statisticians and
philosophers of science ask many of the same
questions:
What should be observed and what may justifiably
be inferred from the resulting data?
How well do data confirm or fit a model?
What is a good test?

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Must predictions be “novel” in some sense? (selection
effects, double counting, data mining)
How can spurious relationships be distinguished from
genuine regularities? from causal regularities?
How can we infer more accurate and reliable
observations from less accurate ones?
When does a fitted model account for regularities in
the data?

• These very general questions are entwined with long
standing debates in philosophy of science
• No wonder the field of statistics tends to cross over
so often into philosophical territory.
• Statistics is a kind of “applied philosophy of science”
(Kempthorne, 1976).
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Statistics  Philosophy
3 ways statistical accounts are used in philosophy
of science
(1) Model Scientific Inference—capture the actual or
rational ways to arrive at evidence and inference
(2) Resolve Philosophical Problems about scientific
inference, observation, experiment;
(problem of induction, objectivity, underdetermination).
(3) Perform a Metamethodological Critique
-scrutinize methodological rules (e.g., novel data)

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Philosophy  Statistics
job is to help resolve the conceptual, logical, and
methodological discomforts of scientists
In tackling the statistics wars one also makes
progress on:
the philosopher's problems of induction, objective
evidence, underdetermination.
“2-way street”

Start from the preface:
1. The Statistics Wars
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2. Role of Probability: performance
or probabilism?
(Frequentist vs. Bayesian)
• End of foundations? (Unifications and
Eclecticism)
• Long-standing battles still simmer below the
surface (agreement on numbers)
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Let’s brush the dust off the pivotal debates,
walk into the museums to hear the founders:
Fisher, Neyman, Pearson, Savage and
many others in relation to today’s statistical
crisis in science (xi)
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Stat Sci/Phil Sci

3. Statistical inference as severe
testing
• Main source of the statistical crisis in science?
• We set sail with a simple tool: if little or nothing
has been done to rule out flaws in inferring a
claim, it has not passed a severe test
• Sufficiently general to apply to any methods
now in use
• You needn’t accept this philosophy to use it to
excavate the statistics wars
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4. A philosophical excursion
“Taking the severity principle, along with the aim
that we desire to find things out… let’s set sail on a
philosophical excursion to illuminate statistical
inference.” (8)
“…and engage with a host of tribes marked by
family quarrels, peace treaties, and shifting
alliances” (xiv)
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5. Getting Beyond the Statistics
Wars
Revisit some taboos: problems of induction &
falsification, science vs. pseudoscience
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Excursion 1 How to Tell What’s
True About Statistical inference
Tour I: Beyond Probabilism and
Performance
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Most findings are false?
“Several methodologists have pointed out that the
high rate of nonreplication of research discoveries
is a consequence of the convenient, yet ill-
founded strategy of claiming conclusive research
findings solely on the basis of a single study
assessed by formal statistical significance,
typically for a p-value less than 0.05. …
It can be proven that most claimed research
findings are false.” (John Ioannidis 2005, 0696)
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R.A. Fisher
“[W]e need, not an isolated record, but a reliable
method of procedure. In relation to the test of
significance, we may say that a phenomenon is
experimentally demonstrable when we know how to
conduct an experiment which will rarely fail to give
us a statistically significant result.” (Fisher 1947, 14)
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Simple significance tests (Fisher)
p-value. …to test the conformity of the particular data
under analysis with H0 in some respect:
…we find a function d(X) of the data, the test statistic,
such that
• the larger the value of d(X) the more inconsistent
are the data with H0;
• d(X) has a known probability distribution when H0 is
true.
…the p-value corresponding to any d(x) (or d0bs)
p = p(t) = Pr(d(X) ≥ d(x); H0)
(Mayo and Cox 2006, 81; d for t, x for y) 17

Testing Reasoning
• If even larger differences than d0bs occur fairly
frequently under H0 (i.e., P-value is not small),
there’s scarcely evidence of incompatibility
with H0
• Small P-value indicates some underlying
discrepancy from H0 because very probably
you would have seen a less impressive
difference than d0bs were H0 true.
• This still isn’t evidence of a genuine statistical
effect H1, let alone a scientific conclusion H*
Stat-Sub fallacy H => H*
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Fallacy of rejection
• H* makes claims that haven’t been probed by the
statistical test
• The moves from experimental interventions to H*
don’t get enough attention–but your statistical
account should block them
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Neyman-Pearson (N-P) tests:
A null and alternative hypotheses H0, H1
that are exhaustive*
H0: μ ≤ 0 vs. H1: μ > 0
“no effect” vs. “some positive effect”
• So this fallacy of rejection H1H* is blocked
• Rejecting H0 only indicates statistical alternatives
H1 (how discrepant from null)
*(introduces Type 2 error)
20

Not an incompatible hybrid
• They both fall under tools for “appraising and
bounding the probabilities (under respective
hypotheses) of seriously misleading
interpretations of data” (Birnbaum 1970, 1033)–
error probabilities
• Can place all under the rubric of error statistics
• Confidence intervals, N-P and Fisherian tests,
resampling, randomization
• Details of tests do not enter until Excursion 3 21

We must get beyond N-P/ Fisher
incompatibilist caricature
• Beyond “inconsistent hybrid”: Fisher–inferential; N-P–
long run performance
• “Incompatibilism” results in P-value users robbed of
features from N-P tests they need (power)
• Wrongly supposes N-P testers have only fixed error
probabilities & are not allowed to report P-values
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Both Fisher and N-P methods: it’s
easy to lie with statistics
by selective reporting
• Sufficient finagling—cherry-picking, significance
seeking, multiple testing, post-data subgroups,
trying and trying again—may practically
guarantee a preferred claim H gets support,
even if it’s unwarranted by evidence
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Severity Requirement:
If the test had little or no capability of finding
flaws with H (even if H is incorrect), then
agreement between data x0 and H provides
poor (or no) evidence for H
• Such a test fails a minimal requirement for
a stringent or severe test
• N-P and Fisher did not put it in these terms
but our severe tester does
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Need to Reformulate Tests
Severity function: SEV(Test T, data x, claim C)
• Tests are reformulated in terms of a discrepancy γ
from H0
• Instead of a binary cut-off (significant or not) the
particular outcome is used to infer discrepancies
that are or are not warranted
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This alters the role of probability:
Probabilism. To assign a degree of probability,
confirmation, support or belief in a hypothesis,
given data x0 (absolute or comparative)
(e.g., Bayesian, likelihoodist, Fisher (at times))
Performance. Ensure long-run reliability of
methods, coverage probabilities (frequentist,
behavioristic Neyman-Pearson, Fisher (at times))
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What happened to using
probability to assess error probing
capacity?
• Neither “probabilism” nor “performance” directly
captures it
• Good long-run performance is a necessary, not
a sufficient, condition for severity
27

Key to revising the role of error
probabilities
• What bothers you with selective reporting,
cherry picking, stopping when the data look
good, P-hacking
• Not problems about long-runs—
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We cannot say the case at hand has done a
good job of avoiding the sources of
misinterpreting data

A claim C is not warranted _______
• Probabilism: unless C is true or probable (gets
a probability boost, made comparatively firmer)
• 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 ways we
can be wrong about C
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Spurious P-Value
He reports: Such results would be difficult to
achieve under the assumption of H0
When in fact such results are common under the
assumption of H0
There are many more ways to be wrong with
hunting (different sample space)
(Formally):
He says Pr(P-value ≤ Pobs; H0) ~ Pobs = small
But in fact it is high
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Informal Examples
Bad Evidence No Test (BENT): Texas marksman
(Having drawn a bull’s eye around tightly clustered
shots, claims marksman ability)
Strong Severity (argument from coincidence to
weight gain):
Lift-off (p. 14)
arguing from error (p. 16)
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Jump to Tour II: Error Probing Tools
vs. Logics of Evidence (p. 30)
To understand the stat wars, start with the holy
grail–a purely formal (syntactical) logic of
evidence
It should be like deductive logic but with
probabilities
Engine behind probabilisms (e.g., Carnapian
confirmation theories Likelihood accounts,
Bayesian posteriors)
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Likelihood Principle (LP)
In probabilisms, the import of the data is via the
ratios of likelihoods of hypotheses
Pr(x0;H0)/Pr(x0;H1)
The data x0 are fixed, while the hypotheses vary
A pivotal disagreement in the philosophy of
statistics battles
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A note on Likelihoods
• To see the difference between the likelihood of
a hypothesis and its probability, any hypothesis
that perfectly fits the data gets maximal
likelihood (1)
• Two rival hypotheses can both have likelihood 1
Lik(H0) = Pr(x0;H0)
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Logic of Support
• Ian Hacking (1965) “Law of Likelihood”: x
support hypothesis H0 less well than H1 if,
Pr(x;H0) < Pr(x;H1)
(rejects in 1980)
• “there always is such a rival hypothesis viz., that
things just had to turn out the way they actually
did” (Barnard 1972, 129).
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Error Probability:
• Pr(H0 is less well supported than H1 ;H0 ) is high
for some H1 or other
“In order to fix a limit between ‘small’ and ‘large’
values of [the likelihood ratio] we must know how
often such values appear when we deal with a
true hypothesis.” (Pearson and Neyman 1967,
106)
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All error probabilities
violate the LP
(even without selection effects):
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)
The LP implies…the irrelevance of predesignation,
of whether a hypothesis was thought of before
hand or was introduced to explain known effects
(Rosenkrantz 1977, 122) 38

Optional Stopping:
Error probing capacities are altered not just by data
dredging, but also via data dependent stopping rules:
H0: no effect vs. H1: some effect
Xi ~ N(μ, σ2), 2-sided H0: μ = 0 vs. H1: μ ≠ 0.
Instead of fixing the sample size n in advance, in some
tests, n is determined by a stopping rule:
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• Keep sampling until H0 is rejected at (“nominal”)
0.05 level
Keep sampling until sample mean M > 1.96 SE
• Trying and trying again: Having failed to rack up a
1.96SE difference after 10 trials, go to 20, 30 and
so on until obtaining a 1.96 SE difference
SE = s/√n
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Nominal vs. Actual
significance levels again:
• With n fixed the Type 1 error probability is 0.05
• With this stopping rule the actual significance
level differs from, and will be greater than 0.05
(proper stopping rule)
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Optional Stopping
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• “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 frowns on this, or at least requires
adjustment of the P-values

• “Imagine instead if an account advertised itself as
ignoring stopping rules” (43)
• “[the] irrelevance of stopping rules to statistical
inference restores a simplicity and freedom to
experimental design that had been lost by
classical emphasis on significance levels (in the
sense of Neyman and Pearson).” (Edwards,
Lindman, and Savage 1963, 239)
• “…these same authors who warn that to ignore
stopping rules is to guarantee rejecting the null
hypothesis even if it’s true” (43) declare it
irrelevant.
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What counts as cheating depends
on statistical philosophy
• Are they contradicting themselves?
• “No. It is just that what looks to be, and
indeed is, cheating from the significance
testing perspective is not cheating from
[their] Bayesian perspective.” (43)
44

21 Word Solution
• Replication researchers (re)discovered that data-
dependent hypotheses and stopping are a major
source of spurious significance levels.
• Statistical critics, Simmons, Nelson, and
Simonsohn (2011) place at the top of their list the
need to block flexible stopping
“Authors must decide the rule for terminating data
collection before data collection begins and report
this rule in the articles” (Simmons, Nelson, and
Simonsohn 2011, 1362).
45

Berger and Wolpert: “Likelihood
Principle”
• It seems very strange that a frequentist could
not analyze a given set of data…if the stopping
rule is not given….Data should be able to speak
for itself. (Berger and Wolpert 1988, 78)
• “intentions” should be irrelevant
• Inference by Bayes Theorem satisfies this:
“How Stopping Rules Drop Out” (ibid., 45) is
derived in 1 page.
46

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)
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Probabilists can still block intuitively
unwarranted inferences
(without error probabilities)?
• Supplement with subjective beliefs: What do I
believe? As opposed to What is the evidence?
(Royall)
• Likelihoodists + prior probabilities
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Richard Royall
1. What do I believe, given x
2. What should I do, given x
3. How should I interpret this observation x as
evidence? (comparing 2 hypotheses)
For #1–degrees of belief, Bayesian posteriors
For #2–frequentist performance
For #3–LL
p. 33 49

Bayesian accounts
• Statistical hypotheses assign probabilities to data
Pr(x0;H1), but it’s rare to assign frequentist
probabilities to hypotheses
Hypotheses
• The deflection of light due to the sun is 1.75 degrees
• Deficiency in vitamin D increases chances of
fractures
• HRT decreases age-related dementia
• IQ is more variable in men than women
You might assign degrees of belief: betting
50

Bayes Rule
Pr(H|x) = Pr(x|H)Pr(H)
Pr(x|H)Pr(H) + Pr(x|~H)Pr(~H)
(p. 24)
• Requires an exhaustive set of
hypotheses
51

Problems with appealing to priors to
block inferences based on
selection effects
• Could work in some cases, but still wouldn’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, priors and biasing
selection effects
52

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 has biasing selection effects,
another, pre-registered results and
precautions)?
• Since there’s a single H, its prior would be the
same
53

Current State of Play in
Bayesian-Frequentist Wars
1.3 View from a Hot-Air Balloon (p. 23)
How can a discipline, central to science and to critical
thinking, have two methodologies, two logics, two
approaches that frequently give substantively different
answers to the same problems? Is complacency in the
face of contradiction acceptable for a central discipline
of science? (Donald Fraser 2011, p. 329)
54

Bayesian-Frequentist Wars
The Goal of Frequentist Inference: Construct
procedures with frequentist guarantees
good long-run performance
The Goal of Bayesian Inference: Quantify and
manipulate your degrees of beliefs
belief probabilism
Larry Wasserman (p. 24)
But now we have marriages and reconciliations (pp.
25-8) 55

Most Bayesians (last decade) use
“default” priors: unification
• ‘Eliciting’ subjective priors too difficult, scientists
reluctant for subjective beliefs to overshadow data
“[V]irtually never would different experts give prior
distributions that even overlapped” (J. Berger 2006,
392)
• Default priors are supposed to prevent prior beliefs
from influencing the posteriors–data dominant
56

Marriages of Convenience
Why?
For subjective Bayesians, to be less
subjective
For frequentists, to have an inferential or
epistemic interpretation of error probabilities
57

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
58

Some Bayesians reject probabilism
(Falsificationist Bayesians)
• “[C]rucial parts of Bayesian data analysis, such
as model checking, can be understood as ‘error
probes’ in Mayo’s sense” which might be seen as
using modern statistics to implement the
Popperian criteria of severe tests.
(Andrew Gelman and Cosma Shalizi 2013, 10).
59

Decoupling
Break off stat methods from their traditional
philosophies
Can Bayesian methods find a new foundation in
error statistical ideas? (p. 27)
Excursion 6: (probabilist) foundations lost;
(probative) foundations found
60

Choose Your Stat War: Old or New
• Replication crisis: should replication be
required? What kind?
• Does science need fixing?
• Change “perverse incentives” (publishing, merit
awarding, open science), badges? Less
competition?
• Should P-values be redefined? (lower p-value)
• Does data dredging, selective reporting alter the
import of data (always? Sometimes?)
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• What is the role of probability in inference (probabilism,
performance, probativism, or something else?)
• Different roles for different contexts? Or a unification?
• Issues in the philosophy and history of statistics,
confirmation, experimentation, replication
• Neyman-Fisher wars: what were they really about?
• Frequentist vs Bayesian wars: then and now?
• What’s missing from current stat wars?
Underlying philosophical disputes
Need to test assumptions of statistical models
Relating statistical to substantive claims, problems of
measurement
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• Should tests be replaced by confidence
intervals?
• How to interpret non-significant results?
• New Eclecticism: model selection, Bayes
Factors, empirical Bayes posterior predictive
checking, AI, ML
• Is experimental philosophy a better way to do
philosophy?
• What about experimental economics?
• What about the ethics of research & Big Data
Modeling?
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Meeting #1 Slides Phil 6334/Econ 6614 SP2019

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