Transcript Fiducial – trust, reliability, belief

Case Study of Faith-Academic
Discipline Integration:
Statistical Inference
Andrew M. Hartley, PhD
Associate Statistical Science Director,
PPD
All of Life Redeemed
• Proverbs 9:10: “The fear of the LORD is the
beginning of wisdom, and knowledge of the
Holy One is understanding.”
• Kuyper: “There is not a square inch in the
whole domain of our human existence over
which Christ, who is Sovereign over all, does
not cry, ‘Mine!'”
• Christ laid down His life for us; how can we fail
to surrender all to Him?
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Medicine to be Redeemed
Kuyper: questions for Medical Doctors that
depend on religion
1. Tell a dying person they are dying?
2. Urge self-control or indulgence for passionate
youth?
3. Counsel the psychologically distressed, or
drug them?
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Statistics, too, to be Redeemed
Statistical Qns that depend on religion
1.
2.
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4.
5.
Can the data alone tell us what to believe about scientific
hypotheses?
Is every individual free to interpret data the way s/he sees fit?
If most people believe in alpha=0.05, does that make it a good rule?
Is the human conscience a tabla rasa that can interpret data
completely without bias? Should it be so?
Is it true that data are data (concept of the “blind statistician”) so
that we should analyze them the same regardless of their
implications?
Connections often missed between religious beliefs &
science, because philosophy not recognized as the link.
Religion sets bounds for philosophy…
philosophy sets bounds for science.
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Parallels - Sphere Sovereignty & the
Philosophy of the Law Idea (PLI)
Note the similarities
• Kuyper’s SS: “This perfect Sovereignty of the sinless
Messiah at the same time directly denies and
challenges all absolute Sovereignty among sinful men
on earth, and does so by dividing life into separate
spheres, each with its own sovereignty.”
• Dooyeweerd’s PLI: Creation reveals itself to us in many
distinct yet inter-connected aspects, or kinds of laws &
properties. No aspect is more important or real than
any other, & to treat it as such is idolatry.
More, now, on “kinds of laws & properties…”
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Some of the 14 Aspects
Aspect = kind of property & law
1. Quantitative – how much, how many
2. Sensitive
3. Biological
4. Logical
5. Social
6. Economic
7. Fiducial – trust, reliability, belief
Each science investigates a small number of aspects.
Statistical inference – investigates quantitative &
fiducial aspects
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PLI’s Claims about the Aspects
(Notice parallels with Sphere Sovereignty)
• All aspects equally real & important (“nonreduction”) – Avoid idolizing aspects
• Pre-scientific vs Scientific Experience
– Pre-scientific (Everyday) experience – multiaspectual, coherent, non-abstract
– Each science enhances everyday experience, but
does not replace it (“primacy of the prescientific”)
Statistical paradigms might respect or violate
the above principles
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Driven to Worship
Humans will worship (trust, believe in) either
God or something God created, for happiness,
fulfillment, security, sure knowledge…
• Romans 1 - “For although they knew God, they neither
glorified him as God nor gave thanks to him, but their thinking
became futile and their foolish hearts were darkened…They
exchanged the truth about God for a lie, and worshiped and
served created things rather than the Creator…”
• Richard Lovelace - “It is the paradox of earthly blessing that
because of our own wayward hearts we can worship the gift
rather than the giver.”
• Bob Dylan: “You’re gonna have to serve somebody. Well, it
may be the devil or it may be the Lord.”
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Idolatry Possible on Many Levels
Idolatry: Worshiping something other than God
• Personal, e.g., materialism
• Cultural , e.g., statism
• Scientific, e.g.,
– Economism – “Fix the economy & all will be well”
– Logicism – “Only logic is trustworthy”
– Chemicism – “Chemical reactions explain
everything”
• At all levels, recognizing all aspects as equally
valid & important helps avoid idolatry
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Aims of Statistical Inference & Select
Aspects Involved
• Given
– Interesting scientific hypothesis(es)
– Sample data
• Conclude
– We can (or should) be C% certain that the hypothesis is
true
• Example: 75% certain that Vitamin D deficiency
increases risk of rickets
• In PLI terms
– C% = a quantitative property
– Certain = a fiducial property
(However, “statistics” > “statistical inference”)
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Proposed 21st Century Christian
Principles of Statistical Inference
1. From the PLI’s Non-reduction Principle: Humans do
not set the standards for what to believe; rather,
inference constrains permissible beliefs.
2. From the PLI’s Primacy of Pre-scientific Experience:
Quantitative data alone are insufficient to indicate what
to believe (& what to do); rather, data, and properties
and laws from all the aspects are needed.
A statistical inference paradigm that respects these
principles…
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Proposed Christian Process Flow of
Statistical Inference
1. Applied scientist forms hypothesis (H) to investigate
2. Scientist & statistician state initial strength of beliefs
about H quantitatively (recognizing primacy of prescientific. Enhance, not replace.)
3. Statistician
a) models Data (x), measures evidence
b) combines evidence mathematically with initial beliefs
(identifying lawful constraints on beliefs)
c) derives post-analytic certainties about H
(Flow is being discussed among Christian statisticians)
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Proposed Christian Process Flow of
Statistical Inference (con’t)
Modeled
Data (x,
Evidence)
Fiducial
Specific Sci
Synthesis
of Priors &
Evidence
P(H|x)
Quantitative
Natural & intuitive? Yes & Yes.
However, most statistical inference
does not follow this pattern.
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Humanism as Analyzed by PLI
• A religion, a faith (=trust) in humans
• Central tenets: Humans are
– Self-sufficient
– Free from external authority
– Able to discern (or even decide!) right/wrong
Examples
1. “Medical care? Food? Water? Education?
We believe the answer is: All the above.”
2. “I can follow the rules that I set for myself.”
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Humanism’s Opposing Poles
• Personality Pole
– Humans are happiest & most fulfilled without
external constraints
– Humanity = Self-expression, emotion, freedom
• Control Pole
– Humans demonstrate dominance through mastery
of the universe
– Humanity = exercise of scientific (esp. logical &
mathematical) reasoning
• Both poles place complete faith in humans’
abilities
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Standard (Frequentist)
Statistical Testing
1. Identify a hypothesis (H) to test
2. Fix an arbitrary critical region (C) such that
P(x will fall in C|H) is small
[P(x will fall in C|H) = α, “alpha”]
3. Decision:
If x does not fall into C, “accept” H
If x falls into C, “reject” H at significance level α
Qn: Once we accept or reject H, how strongly should
we believe in H?
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Example: Frequentist Testing
• Consumer advocate suspects average mass of “1 kg”
packages of candy < 1 kg. Decides to test
H: µ ≥1
• C: {sample mean < 0.83}
• Pr(C | H) = 0.05
• Assess 20 packages
• Sample mean = 0.81, H “rejected”
• Qns:
– How strongly should we believe µ<1?
– Should we act as if µ<1?
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“Frequentist Inference:”
An Oxymoron?
• Inference = statements about hypotheses, given data
(inductive, outward-looking)
• Frequentist results = statements about data, given
hypotheses (deductive, inward-looking). Examples:
– Type 1 error rate = probability of rejecting the hypothesis,
if the hypothesis is true
– P-value = probability of results as extreme as those
observed, given the tested hypothesis.
• Frequentist results are silent about strengths of
belief
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Humanist Control over Interpreting
Frequentist “Inferential” Results
• Humanist “Personality” Pole
– “Probabilities about x given H constitute
‘evidence’ that people are free to interpret as they
wish”
– Leads to “indirect” frequentism
• Humanist “Control” Pole
– “Probabilities about x given H convey direct
meanings about H itself”
– Leads to “direct” frequentism
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Indirect Frequentism: Examples
• McClean: “Statistics is about judgment.”
• RA Fisher:
– “We have the duty of…communicating our conclusions…in
recognition of the right of other free minds to utilize them
in making their own decisions.”
– “The deviation [might be] in the direction expected for
certain influences which seemed to me not improbable.”
• Motulsky: “What conclusion [from p] should you
reach? That’s up to you. Statistical calculations
provide [p]. You have to interpret it.”
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Direct Frequentism: Statements to Consider
Given a p-value of 0.01, which of the following are true (all, none or
some)?
1. You have absolutely disproved the null hypothesis (i.e., there is no
difference between the population means).
2. You have found the probability of the null hypothesis being true.
3. You have absolutely proved your experimental hypothesis (that
there is a difference between the population means).
4. You can deduce the probability of the experimental hypothesis
being true.
5. You know, if you decide to reject the null hypothesis, the probability
that you are making the wrong decision.
6. You have a reliable experimental finding in the sense that if,
hypothetically, the experiment were repeated a great number of
times, you would obtain a significant result on 99% of occasions.
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Ref: Gigerenzer et al., 2004
Usefulness of Frequentist Results
• How can frequentist results aid the applied sciences?
• Under some conditions,
– Frequentist p-value approximates P(H|x)
– Frequentist 95% confidence interval approximates an
interval with 95% probability of containing the unknown
quantity
–…
• Correspondences do not hold when, e.g.,
– 2-sided Testing
– Strong prior information exists about H
Hence, consider discriminating use of results
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The Missing Link, Supplied
• Many Christians attempt to integrate faith &
discipline, to show what obedience to God
means in their profession
• These attempts often fail; relations between
faith & discipline are not revealed
• Missing link = the philosophical framework
• Often, efforts fall short because the
philosophy is not examined & made Biblically
consistent
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Summary
• PLI = a philosophical framework for
– Analyzing relations between religion & science
– Identifying biblically consistent science
• Much of science in Western culture coheres with
Humanist philosophies
– Personality pole: The individual subject is the law-giver
– Control pole: Only logical & quantitative reasoning are
trustworthy
• When humanity fell, all of creation fell
• Christ calls His disciples to re-claim creation
for Himself
Andrew M Hartley, PhD
[email protected]
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