Mayo: Day #2 slides

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Day #2: 6334

Today we’ll try to cover a number of things:

1. Learning philosophy/philosophy of statistics

2. Situating the broad issues within philosophy of science

3. Little bit of logic

4. Probability and random variables

(need a full list to get participants in Scholar!)

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Learning to do philosophy is very different from learning

philosophy as a museum of different views, what so and so

said, various “isms”

How to teach “doing philosophy,” and how the student is to

recognize acquiring the skill, is not cut and dried—(no one can

really say, but I can tell when you’re “getting philosophical”)

To teach you to be philosophers—even if it’s only in this

class—we’ve got to teach something that is unfamiliar and

maybe even painful for some. (I know we have some philo-

sufferers here, who should know what I mean.)

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We won’t typically come out and “give the answer” in

wrestling with a philosophical issue, but may deliberately hold

back to encourage you to wrestle with it.

On the other hand, Professor Spanos and I have been working

in this general area for a long time, and developing positions,

responses to challenges, etc.

You also learn how to do philosophy by witnessing the

strongest arguments, and the effort that goes into carefully

engaging them.

So there’s a balance here wherein we want to both give and

hold back, feed but make you struggle…

So expect that.

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Everyone can see our publications, no secret there, and people

participating in this course (listed or added to Scholar) will also

get bits from my still-being-written book (on which I definitely

want your feedback).

So you will be getting our strongest arguments, but also the

strongest arguments out there, and we’ll want you to work

through the arguments on your own.

The most important thing—admittedly, also the most

unusual—is that anyone who wants to make progress in the

area of philosophy of statistics must be prepared to question

and challenge everything you read (even by the highest of the

high priests).

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We know it feels strange. (This does not typically hold for

other subjects.)

In that first chapter of How To Tell What’s True About

Statistical Inference, I note that “a certain trepidation and

groupthink take over when it comes to philosophically tinged

notions such as evidence, inductive reasoning, objectivity,

rationality, truth”.

“The general area of philosophy that deals with knowledge,

evidence, inference, and rationality is called epistemology.

The epistemological standpoints of leaders, be they

philosophers or scientists, are too readily taken on faith as

canon by others, including researchers who would otherwise

cut through unclarity, ambiguity, and the fuzziness and

handwaving that often surround these claims”.

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Progress in mathematical statistics doesn’t require philosophy,

but too often “we don’t need to be philosophical about these

concepts or methods” means “I don’t want to examine them

very closely”.

Writing philosophy (I gave you some pointers already on the

blog):

o Here, too, a slow examination of the question involved is

required.

o Less (much less!) is more.

o We guarantee you will find this kind of analytical writing

of value in your other fields.

o As a start, the first assignment will be make-believe for

next time. (e-mail)

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PHILOSOPHY:

Understand and Justify Human Knowledge About the World

PHILOSOPHY OF SCIENCE:

How Do We Learn About the World In The Face of Uncertainty

and Error?

Is there a scientific method?

How do we obtain good evidence?

How do we make reliable inferences from evidence?

What makes an inquiry “scientific” or rational?

Is there scientific progress?

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Problems with answers from logic empiricist philosophers of

science:


The “logics” for science are oversimple, open to paradoxes

Standard canons are violated in actual science

Scientific methods change with changing aims, values,

technologies, societies


Empirical data are uncertain, finite, probabilistic

Data are not just “given”, they have to be interpreted

introducing biases, theory-ladenness, value-ladenness

(scientific, social, ethical, economic)

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Inductivist: cannot justify “induction”

logics of induction failed

Falsificationist: deduction won’t teach anything new; cannot

pinpoint blame for a failed prediction


None of the philosophical attempts to erect a demarcation for

science seem to work.


no account of cumulative growth of knowledge

old “paradigms” are swept away by new ones which are

“incommensurable” with the old

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PHILOSOPHY:

Understand and Justify Human Knowledge About the World

PHILOSOPHY OF SCIENCE:

How Do We Learn About the World In The Face of Uncertainty and Error?

LOGICAL EMPIRICIST ATTEMPTS: 1930-60

Carnap Popper

Inductive Logics: C(h,e) Logic of Falsification

“Armchair Philosophy of Science”

Problems, Paradoxes

KUHN (1962)

CRISIS IN PHILOSOPHY OF SCIENCE

POST-POSITIVISM

“Historicism in Philosophy of

Science”

STS/ HPS

pessimistic optimistic

THEORY CHANGE

MOVEMENT-1970s, 80s

Kuhn, Lakatos, Laudan

New Models of Rationality

“naturalistic turn” Search for More Adequate

Theories of Induction, Testing,

& Decision-Making

“Rational Reconstruction”

BAYESIANS

Inference as updating degrees of belief Sc

New Experimentalist Turn 1980s,1990s

Error Statistical Philosophy: Mayo 1996?

Relativism

Post-

modernism

anarchy

dadaism

irrationality

social

constructivism

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Little Bit of Logic

(Double purpose: both for arguing philosophically and for

understanding inductive/deductive methods)

Argument:

A group of statements, one of which (the conclusion) is claimed to

follow from one or more others (the premises), which are regarded

as supplying evidence for the truth of that one.

This is written:

P1, P2,…Pn/ ∴ C.

In a 2-value logic, any statement A is regarded as true or false.

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A deductively valid argument: if the premises are all true then,

necessarily, the conclusion is true.

To use the “⊨” (double turnstile) symbol:

P1, P2,…Pn ⊨ C.

Note: Deductive validity is a matter of form—any argument with

the same form or pattern as a valid argument is also a valid

argument.

(Simple truth tables serve to determine validity)

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EXAMPLES (listing premises followed by the conclusion)

Modus Ponens Modus Tollens

If H then E If H then E

H_______ Not-E____

E Not-H

If (H) GTR, then (E) deflection effect.

(not-E) No light deflection observed.

(not-H) GTR is false

(falsification)

These results depend on the English meaning of “if then” and of

“not.” In context, sentence meanings aren’t always so clear.

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Disjunctive syllogism:

(1) Either the (A) experiment is flawed or (B) GTR

is false

(2) GTR is true (i.e., not-B).

Conclusion: Therefore, (A) experiment is flawed.

If either A or B are true, and not-B, then infer A.

Since if A were not true, you can’t also hold the two

premises true—without contradiction.

(e.g., soup or salad)

Either A or B. (disjunction)

Not-B

Therefore, A

So we have 3 valid forms.

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Deductively Valid Argument (argument form):

Three equivalent definitions:

An argument where if the premises are all true, then

necessarily, the conclusion is true. (i.e., if the conclusion is

false, then (necessarily) one of the premises is false.)

An argument where it’s (logically) impossible for the premises

to be all true and the conclusion false. (i.e., to have the

conclusion false with the premises true leads to a logical

contradiction: A & not-A.)

An argument that maps true premises into a true conclusion

with 100% reliability. (i.e., if the premises are all true, then

100% of the time the conclusion is true).

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True or False?

If an argument is deductively valid, then its conclusion must be

true.

To detach the conclusion of a deductively valid argument as true,

the premises must be true.

Here’s an instance of the valid form modus tollens:

Example: Let A be: Bayesian methods are acceptable. and let B stand for: Bayesian methods always give the same answer as frequentist methods. If A then B Not-B Not A.

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So you may criticize a valid argument by showing one of its

premises is false.

(Deductively) Sound argument: deductively valid + premises are

true/approximately true.

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Exercise: showing deductively valid arguments can have false conclusions. (1) (I did this above) Give an example of an argument that follows the valid pattern of modus tollens that has a false conclusion. (2) Exercise (YOU do this one) Show that an argument following the form of a disjunctive syllogism can have a false conclusion. Either A or B Not-B Therefore A.

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Invalid Argument: Consider this argument:

If H then E

E________

H

(H) If everyone watched the SOTU speech last night, then (E)

you did.

(E)You did._________________________

So, (H) everyone did.

Affirming the consequent.

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Invalid argument: An argument where it’s possible to have all

true premises and a false conclusion without contradiction.

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Inductive argument. With an inductive argument, the conclusion

goes beyond the premises. So it’s logically possible for all the

premises to be true and the conclusion false. So an inductive

argument is invalid.

The premises might be experimental outcomes or data points:

E1, E2,…En (e.g., light deflection observations, drug reactions,

radiation levels in fish)

Even if all observed cases had a property, or followed a law,

there’s no logical contradiction in the falsity of the generalization:

H: All E’s are F

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This is also true for a statistical generalization:

60% in this class watched SOTU

Thus, H: 60% of all people watched SOTU

H agrees with the data, but it’s possible to have such good

agreement even if H is false.

The problem of induction: how to justify such inductive

inferences.

It’s not clear how to characterize it much less to justify the move

(if it should be justified).

(Some would say we only use deduction, and we’ll talk about this

two classes from now.)

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Inductively “good” argument (how should we define it?)

Can we try to parallel the definition of deduction validity?

(i)A deductively valid argument, if the premises are all true then,

necessarily, the conclusion is true.

E1, E2,…En ⊨ H.

(i)* An inductively good argument is one where if the premises are

true, then the conclusion is probable (?)

E1, E2,…En ⊨ probably H.

(i)* probabilism

Does this mean infer H is probably true: Pr(H) = high?

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Try a second definition of deductively valid:

(ii) Deductive. The argument leads from true premises to a true

conclusion 100% of the time.

(ii)* Inductive. The argument leads from true (or approximately

true) premises to a true conclusion (1 – α)% of the time. (?)

(ii)* highly reliable method (reliabilism?)

(Differs from highly reliable conclusion)

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High “reliability” of the method, low long-run error isn’t enough.

Remember the BP Deepwater Horizon oil spill of 2010?

The BP representatives claimed to have good (inductive) evidence

that

H: the cement seal is adequate (no gas should leak)

In fact they kept decreasing the pressure until H passed, rather

than performing a more severe test (as they were supposed to)

called "a cement bond log” (using acoustics)

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Oil Exec: Our inference to H: the cement seal is fine

is highly reliable

Senator: But didn’t you just say no cement bond log was

performed, when it initially failed…?

Oil Exec: That’s what we did on April 20, but usually we do---I’m

giving the average.

(Imaginary): We use a randomizer that most of the time directs us

to run the gold-standard check on pressure, but with small

probability tells us to assume the pressure is fine, or keep

decreasing the pressure of the test till it passes….

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Overall, this test might rarely err but that is irrelevant in appraising

the inference from the data on April 20.

The oil rep gives a highly misleading of the stringency of the

actual test that H managed to “pass”.

Passing this overall “test” made it too easy for H for pass, even if

false.

The long-run reliability of the rule is a necessary but not a

sufficient condition to infer H (with severity)

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Severity demands considering, for the context at hand, what

error(s)* must be (statistically) ruled out in order the have

evidence for some claim H

(*construed as any erroneous claim or misunderstanding

empirical and theoretical)

And being able to evaluate a tools probative capacity to have

discerned the flaw, were it present.

An inductive argument is good if its conclusion has passed a

severe test (with the premises and background).

H agrees with the data, but it’s highly improbable to have such

good agreement even if H is false.

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Error (Probability) Statistics

Error probabilities may be used to quantify probativeness or

severity of tests (for a given inference)

The logic of severe testing (or corroboration) is not probability

logic

It may well be that there’s a role for both, but that’s not to unify

them, or claim the differences don’t matter…

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I began with a set of questions, and hope to show how they are

tackled in the error statistical philosophy.

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Problems with answers from logic empiricist philosophers of

science:


The “logics” for science are oversimple, open to paradoxes

Standard canons are violated in actual science

Scientific methods change with changing aims, values,

technologies, societies


Empirical data are uncertain, finite, probabilistic

Data are not just “given”, they have to be interpreted introducing

biases, theory-ladenness, value-ladenness

(scientific, social, ethical, economic)

32


Inductivist: cannot justify “induction”

logics of induction failed

Falsificationist: deduction won’t teach anything new; cannot

pinpoint blame for a failed prediction


None of the philosophical attempts to erect a demarcation for

science seem to work.


no account of cumulative growth of knowledge

old “paradigms” are swept away by new ones which are

“incommensurable” with the old

Education

Mayo: Day #2 slides