Central Topics in the Philosophy of Science Basic Probability … · 2019. 11. 26. · 2 / 30 Pardon the Interruption Ask, challenge, clarify, and so on. I might be vague, unclear,

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Central Topics in the Philosophy of ScienceBasic Probability Theory – Session 1

Jürgen Landes

Munich Center for Mathematical PhilosophyLMU Munich

15.10.2019

Jürgen Landes Central Topics in the Philosophy of Science Basic Probability Theory – Session 1

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This Module in General Terms

This course is taught alternately every winter semester:In one year, it provides a somewhat old-fashionedsystematic overview over the main problems discussed ingeneral philosophy of science.In the other year (this time), it focuses on the Bayesianapproach to all these problems and introduces probabilitytheory and the theory of Bayesian networks, which are ourmain modelling tool, in a systematic way.

The seminar is accompanied by a tutorial in which you willsolve mathematical problems from probability theory andlearn how to use the theory of Bayesian Networks to solvephilosophical problems.This session provides some background, it introduces thebasics of probability theory, and it addressesorganisational issues.


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Pardon the Interruption

Ask,challenge,clarify,and so on.

I might be vague, unclear, false on purpose – or just plainwrong.I do ask questions, too!


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Philosophy of Science – Problems and Questions

Philosophy of science addresses philosophical questionsabout science or raised by science. Which?These questions are either

metaphysical (e.g., what is a law of nature? what is thenature of space and time?),epistemological (e.g., what distinguishes scientificknowledge from other kinds of knowledge?, how aretheories confirmed?), ormethodological (e.g., how do theories relate?, what is ascientific explanation?).

Philosophy of science also addresses specific foundationalquestions within a specific science (general philosophy ofscience vs. philosophy of the special sciences).This course focuses on epistemological andmethodological questions of general philosophy of scienceand it addresses them from a Bayesian point of view.


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Some Steps towards Modern Philosophy of Science

Predecessors: P. Duhem and H. Poincaré (around 1900)The beginnings: the Vienna Circle (R. Carnap, O. Neurath,M. Schlick,. . . ), the Berlin group (H. Reichenbach, C.Hempel), and K. Popper (around 1930)Institutionalization: Logical Empiricism in the US (R.Carnap, C. Hempel, E. Nagel, P. Suppes,. . . ) (1950-1960)The naturalistic/Wittgensteinian turn: T.S. Kuhn, P.Feyerabend, N. R. Hanson,. . . (1960ies)Different schools in Germany: Critical Rationalism (H.Albert), Analytic Philosophy (W. Stegmüller),Erlangen-Konstanz School (P. Lorenzen, J. Mittelstrass)(1970ies)Today: There is a plurality of approaches, e.g. casestudies, social studies of science, and formal approaches(logic, probability).


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Probability Theory

Don’t be scared!


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The Axioms of Kolmogorov

Let us fix a finite set of possible worlds (see [Lowe, 2002,§ 5]), call this set Ω.The propositions of interest are identified with the set ofpossible worlds in which they hold, F ⊆ Ω.A function P : PΩ→ [0,∞] is a probability function, if andonly if it satisfies these axioms:

1 P(Ω) = 1 and2 P(F ∪ E) = P(E) + P(F ) if F ∩ E = ∅.PΩ is the power set of Ω, it consists of all the subsets of Ω(including the empty set [∅] and Ω).


https://en.wikipedia.org/wiki/Possible_world

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An Intuitive Picture

A bucket full of paint.


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Some Consequences of the Axioms – Proofs?

P(F ) = 1− P(F ) for all F ⊆ Ω.P(F ) ≤ 1 for all F ⊆ Ω.P(F ) ∈ [0,1] for all F ⊆ Ω.P(∅) = 0.P(F ) =

∑ω∈F P(ω).

P(F ) =∑n

i=1 P(F ∩ Ei), if Ei ∩ Ek = ∅ [i 6= k ] andΩ =

⋃ni=1 Ei .

The Ei are mutually exclusive and exhaustive.The Ei are said to form a partition of Ω.Law of Total Probability.Use bucket of paint!


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Conditional Probabilities I

What are the probabilities, if F is true?Turn to picture?...Normalise!Keep ratios fixed.We write P(E |F ), probability of E ⊆ Ω given that F is true.Read as: the conditional probability of E given F .We also sometimes write P(·|F ).The conditional probability function.Supposing that F is true.


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Conditional Probabilities II

P(E |F ) := P(E∩F )P(F ) ,

if P(F ) > 0!If P(F ) = 0? (Popper Functions)What’s it all good for?

Dynamism!Prior and posterior probabilities.


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Bayes’ Theorem

Theorem

P(H|E) =P(E ∩ H)

P(E)=

P(H)

P(E)· P(E |H) (1)

=P(H) · P(E |H)∑N

i=1 P(E ∩ Hi)

=P(H) · P(E |H)∑N

i=1 P(Hi) · P(E |Hi).

If the Hi form a partition – and P(Hi) > 0 for all i .


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The Importance of Bayes’ Theorem

Reminder: P(H|E) = P(H)·P(E |H)∑Ni=1 P(Hi )·P(E |Hi )

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Posterior of H, a scientific hypothesis of interest is equalto....Some rather complicated looking term,using only simple ingredientsPrior probabilities, the P(H) and P(Hi)

likelihoods, P(E |Hi). Statistics.


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Probabilistic Independence

Learning that A is the case does not change the probabilityof B.Example!A Trump will be re-elected, B: It will rain in Munichtomorrow.A ⊆ Ω and B ⊆ Ω are probabilistically independent, if andonly if P(A ∩ B) = P(A) · P(B).Equivalently: P(A|B) = P(A) or P(B|A) = P(B), ifP(A),P(B) > 0.


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Conditional Independence

A is conditionally independent of B given C, if and only ifP(A|B,C) = P(A|C).Example: A: yellow fingers, B: lung cancer, C: smokingA and B are positively correlated, i.e., learning that aperson has A raises the probability of B. Yet, if we know C,A leaves the probability of B unchanged.C can be a common cause of A and B.


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Some Probabilistic Fallacies and Puzzles

Probability theory has a number of not obviousconsequences (e.g., Simpson’s Paradox) and lay peopleas well as scientists make serious mistakes.Here are three examples.N.B.: For more on this, read [Kahneman, 2012].


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1. The Base Rate Fallacy (Tversky and Kahneman)

There is a new test for a rare (and, indeed, rather horrible)disease. Only 2% of the patients who are tested for thedisease actually have it. Unfortunately, like all medicaltests, the new test is not fully reliable. 5% of the testsreport that the patient has the disease although she doesnot have it (“false positives”). And 1% of the tests reportthat the patient does not have the disease although shedoes has it (“false negatives”). A patient is tested and theresult is positive. What is the probability that she has thedisease?What do you think and why?What do people think and why?Answer: 28.8%.H,E ....


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2. The Conjunction Fallacy (Tversky and Kahneman)

Linda is 31 years old, single, outspoken, and verybright. She majored in philosophy. As a student, shewas deeply concerned with issues of discriminationand social justice, and also participated in anti-nucleardemonstrations. Which is more probable?

1 Linda is a bank teller.2 Linda is a bank teller and is active in the feminist

movement.3 What do you think?4 What do people think?5 In experiments, 85% of the participants consider

the second option more likely. They committedthe conjunction fallacy.


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3. The Monty Hall Problem

As a contestant on a game show you are given the choicebetween three doors: Behind one door is a car; behind theother two doors is a goat.You’d quite like to have the car, and have no use for goats.You pick a door, say No. 1.The host, who knows what’s behind the doors, opensanother door, say No. 3, which has a goat.He then says to you, “Do you want to pick door No. 2?”What should you do? Is it to your advantage to switchdoors?


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Organisational Matters


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Plan of the Course

Week 1 (15.10. 2019): Basic Probability TheoryWeek 2 (22.10. 2019): Bayesian EpistemologyWeek 3 (29.11. 2019): Interpretations of ProbabilityWeek 4 (05.11. 2019): Bayesian NetworksWeek 5 (12.11. 2019): ConfirmationWeek 6 (19.11. 2019): The Value of InformationWeek 7 (26.11. 2019): The Variety of Evidence ThesisWeek 8 (03.12. 2019): The No Alternatives ArgumentWeek 9 (10.12. 2019): The Problem of Old EvidenceWeek 10 (17.12. 2019): Midterm-ExamWeek 11 (07.01. 2020): Objective Bayesian EpistemologyWeek 12 (14.01. 2020): Epistemic Utility TheoryWeek 13 (21.01. 2020): TBAWeek 14 (28.01. 2020): Essay Session 1Week 15 (04.02. 2020): Essay Session 2


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Coursework

Attend all sessions,carefully study the reading assignments (published on mywebsite https://jlandes.wordpress.com/teaching),submit one discussion question about the readings everyweek,this week: browse GPS volumes and find one topic youwant to use probability theory on andparticipate in the classroom discussion.


https://jlandes.wordpress.com/teaching

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Taking the Course for Credit

1 Attend the weekly tutorials (Monday 18:15-19:45, in thisroom) and hand in solutions of the weekly problem sets atthe beginning of each tutorial,

it’s good to work in groups! – write up must be individual!one point for every typo you and only you spot in the formalparts of the lecture notes.

2 pass the Midterm-Exam on 17 December 2019 and3 hand in a 500-words outline of the term paper (what is the

problem? how does the argument proceed?) and aliterature list by31 January 2020, register for this course on LSF by 13-24January and submit a term paper of about 3000 words by28 March 2020.

4 No tutorials after the exam.


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Assessment: Final Mark

the weekly problem sets (25%),the Midterm-Exam (35%), andthe term paper (40%).


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Intended Learning Outcomes

Familiarity with central topics in philosophy of science.Mastery of probabilistic modelling to address problems inthe philosophy of science.And elsewhere!


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Keeping Updated

Syllabus, lecture notes and problem sets are available onmy website:https://jlandes.wordpress.com/teachingI know how to use email: [email protected] teaching assistants, too:

Gašper Stukelj ([email protected]) andTimo Freiesleben ([email protected]).

Please, send an email to Maria Csauscher-Wuck([email protected]) so that she canset up a mailing list (urgent cancellations, etc.).


https://jlandes.wordpress.com/teaching

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Useful Resources

Harvard Writing Guide.Uni online library.Google Scholar.Phil Papers.Stanford Encyclopedia of Philosophy.Teachers, teaching assistants, peers, student union,student bodies, student representatives, PhD students,friends & family, ....SPEAK UPCourse rep.


https://philosophy.fas.harvard.edu/files/phildept/files/brief_guide_to_writing_philosophy_paper.pdf

https://login.emedien.ub.uni-muenchen.de/login

https://scholar.google.com/

https://philpapers.org

https://plato.stanford.edu/

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Enjoy the course!


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References I

Curd, M. and Cover, J. A., editors (1998).Philosophy of Science: The Central Issues.W. W. Norton & Company.

Hempel, C. (1966).Philosophy of Natural Science.Prentice Hall.

Kahneman, D. (2012).Thinking, Fast and Slow.Penguin.

Ladyman, J. (2012).Understanding Philosophy of Science.Taylor & Francis, London.


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References II

Lowe, J. E. (2002).A Survey of Metaphysics.Oxford University Press.


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See you next week – One Application of ProbabilityTheory!


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Central Topics in the Philosophy of Science Basic Probability … · 2019. 11. 26. · 2 / 30 Pardon the Interruption Ask, challenge, clarify, and so on. I might be vague, unclear,