Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes...
Transcript of Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes...
![Page 1: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/1.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Maximum Entropy and Inductive Logic I
Jürgen Landes
Spring School on Inductive Logic
Canterbury, 20.04.2015 - 21.04.2015
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 2: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/2.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Outline
1 Motivation & ProblemMotivationProblem
2 FormalitiesLanguageUncertaintyKnowledge
3 Inference ProcessesThe GistThe PointRight!
4 Desiderata for Inference Processes
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 3: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/3.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Outline
1 Motivation & ProblemMotivationProblem
2 FormalitiesLanguageUncertaintyKnowledge
3 Inference ProcessesThe GistThe PointRight!
4 Desiderata for Inference Processes
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 4: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/4.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Background
Ballpark: Rational BeliefsWhat does an agent rationally belief?As opposed to: When do we say that an agent beliefs X?Imagine: You are the agent.Normative formal approach
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 5: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/5.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Background
Ballpark: Rational BeliefsWhat does an agent rationally belief?As opposed to: When do we say that an agent beliefs X?Imagine: You are the agent.Normative formal approach
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 6: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/6.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Background
Ballpark: Rational BeliefsWhat does an agent rationally belief?As opposed to: When do we say that an agent beliefs X?Imagine: You are the agent.Normative formal approach
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 7: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/7.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Background
Ballpark: Rational BeliefsWhat does an agent rationally belief?As opposed to: When do we say that an agent beliefs X?Imagine: You are the agent.Normative formal approach
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 8: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/8.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Background
Ballpark: Rational BeliefsWhat does an agent rationally belief?As opposed to: When do we say that an agent beliefs X?Imagine: You are the agent.Normative formal approach
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 9: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/9.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Outline
1 Motivation & ProblemMotivationProblem
2 FormalitiesLanguageUncertaintyKnowledge
3 Inference ProcessesThe GistThe PointRight!
4 Desiderata for Inference Processes
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 10: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/10.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Ouch
Patient, migraines, doctorGiven the doctor’s background knowledge,patient complaining of certain symptoms,exhibiting certain traits:what should the doctor believe?Eventually, what should the doctor do?On the basis of imperfect information.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 11: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/11.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Ouch
Patient, migraines, doctorGiven the doctor’s background knowledge,patient complaining of certain symptoms,exhibiting certain traits:what should the doctor believe?Eventually, what should the doctor do?On the basis of imperfect information.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 12: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/12.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Ouch
Patient, migraines, doctorGiven the doctor’s background knowledge,patient complaining of certain symptoms,exhibiting certain traits:what should the doctor believe?Eventually, what should the doctor do?On the basis of imperfect information.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 13: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/13.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Ouch
Patient, migraines, doctorGiven the doctor’s background knowledge,patient complaining of certain symptoms,exhibiting certain traits:what should the doctor believe?Eventually, what should the doctor do?On the basis of imperfect information.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 14: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/14.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Ouch
Patient, migraines, doctorGiven the doctor’s background knowledge,patient complaining of certain symptoms,exhibiting certain traits:what should the doctor believe?Eventually, what should the doctor do?On the basis of imperfect information.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 15: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/15.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Ouch
Patient, migraines, doctorGiven the doctor’s background knowledge,patient complaining of certain symptoms,exhibiting certain traits:what should the doctor believe?Eventually, what should the doctor do?On the basis of imperfect information.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 16: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/16.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Ouch
Patient, migraines, doctorGiven the doctor’s background knowledge,patient complaining of certain symptoms,exhibiting certain traits:what should the doctor believe?Eventually, what should the doctor do?On the basis of imperfect information.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 17: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/17.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Ouch
Patient, migraines, doctorGiven the doctor’s background knowledge,patient complaining of certain symptoms,exhibiting certain traits:what should the doctor believe?Eventually, what should the doctor do?On the basis of imperfect information.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 18: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/18.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Ouch
Patient, migraines, doctorGiven the doctor’s background knowledge,patient complaining of certain symptoms,exhibiting certain traits:what should the doctor believe?Eventually, what should the doctor do?On the basis of imperfect information.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 19: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/19.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Ouch
Patient, migraines, doctorGiven the doctor’s background knowledge,patient complaining of certain symptoms,exhibiting certain traits:what should the doctor believe?Eventually, what should the doctor do?On the basis of imperfect information.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 20: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/20.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Carnap’s Principle
Carnap1947: A rational agent ought to take all availableevidence into account when forming beliefs.Reasonable.So, the doctor has to take all background knowledge andthe patient’s individual properties into account.Ignoring some information is a “No no”....Reasonable, ... really?
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 21: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/21.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Carnap’s Principle
Carnap1947: A rational agent ought to take all availableevidence into account when forming beliefs.Reasonable.So, the doctor has to take all background knowledge andthe patient’s individual properties into account.Ignoring some information is a “No no”....Reasonable, ... really?
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 22: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/22.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Carnap’s Principle
Carnap1947: A rational agent ought to take all availableevidence into account when forming beliefs.Reasonable.So, the doctor has to take all background knowledge andthe patient’s individual properties into account.Ignoring some information is a “No no”....Reasonable, ... really?
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 23: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/23.jpg)
...
6 / 36
Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Carnap’s Principle
Carnap1947: A rational agent ought to take all availableevidence into account when forming beliefs.Reasonable.So, the doctor has to take all background knowledge andthe patient’s individual properties into account.Ignoring some information is a “No no”....Reasonable, ... really?
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 24: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/24.jpg)
...
6 / 36
Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Carnap’s Principle
Carnap1947: A rational agent ought to take all availableevidence into account when forming beliefs.Reasonable.So, the doctor has to take all background knowledge andthe patient’s individual properties into account.Ignoring some information is a “No no”....Reasonable, ... really?
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 25: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/25.jpg)
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6 / 36
Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
MotivationProblem
Carnap’s Principle
Carnap1947: A rational agent ought to take all availableevidence into account when forming beliefs.Reasonable.So, the doctor has to take all background knowledge andthe patient’s individual properties into account.Ignoring some information is a “No no”....Reasonable, ... really?
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 26: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/26.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Outline
1 Motivation & ProblemMotivationProblem
2 FormalitiesLanguageUncertaintyKnowledge
3 Inference ProcessesThe GistThe PointRight!
4 Desiderata for Inference Processes
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 27: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/27.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Outline
1 Motivation & ProblemMotivationProblem
2 FormalitiesLanguageUncertaintyKnowledge
3 Inference ProcessesThe GistThe PointRight!
4 Desiderata for Inference Processes
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 28: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/28.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Our Language
Finite propositional language LVariables v1, . . . , vn
Connectives ∧,∨,¬,→,←,↔Sentences of L, SL
No funny business, self-reference, truth predicates,self-fulfilling
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 29: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/29.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Our Language
Finite propositional language LVariables v1, . . . , vn
Connectives ∧,∨,¬,→,←,↔Sentences of L, SL
No funny business, self-reference, truth predicates,self-fulfilling
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 30: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/30.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Our Language
Finite propositional language LVariables v1, . . . , vn
Connectives ∧,∨,¬,→,←,↔Sentences of L, SL
No funny business, self-reference, truth predicates,self-fulfilling
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 31: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/31.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Our Language
Finite propositional language LVariables v1, . . . , vn
Connectives ∧,∨,¬,→,←,↔Sentences of L, SL
No funny business, self-reference, truth predicates,self-fulfilling
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 32: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/32.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Our Language
Finite propositional language LVariables v1, . . . , vn
Connectives ∧,∨,¬,→,←,↔Sentences of L, SL
No funny business, self-reference, truth predicates,self-fulfilling
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 33: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/33.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Our Language
Finite propositional language LVariables v1, . . . , vn
Connectives ∧,∨,¬,→,←,↔Sentences of L, SL
No funny business, self-reference, truth predicates,self-fulfilling
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 34: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/34.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Outline
1 Motivation & ProblemMotivationProblem
2 FormalitiesLanguageUncertaintyKnowledge
3 Inference ProcessesThe GistThe PointRight!
4 Desiderata for Inference Processes
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 35: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/35.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Probabilities
Probabilistic frameworkProbability function, PSubjective degrees of belief – Dutch BookP : SL→ [0,1]
P(τ) = 1 for all tautologies τ ∈ SL.P(ϕ ∨ θ) = P(ϕ) + P(θ), if |= ¬(ϕ ∧ θ).ϕ |= θ implies P(ϕ) ≤ P(θ).P(ϕ ∨ θ) = P(ϕ) + P(θ)− P(ϕ ∧ θ).Set of probability functions P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 36: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/36.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Probabilities
Probabilistic frameworkProbability function, PSubjective degrees of belief – Dutch BookP : SL→ [0,1]
P(τ) = 1 for all tautologies τ ∈ SL.P(ϕ ∨ θ) = P(ϕ) + P(θ), if |= ¬(ϕ ∧ θ).ϕ |= θ implies P(ϕ) ≤ P(θ).P(ϕ ∨ θ) = P(ϕ) + P(θ)− P(ϕ ∧ θ).Set of probability functions P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 37: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/37.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Probabilities
Probabilistic frameworkProbability function, PSubjective degrees of belief – Dutch BookP : SL→ [0,1]
P(τ) = 1 for all tautologies τ ∈ SL.P(ϕ ∨ θ) = P(ϕ) + P(θ), if |= ¬(ϕ ∧ θ).ϕ |= θ implies P(ϕ) ≤ P(θ).P(ϕ ∨ θ) = P(ϕ) + P(θ)− P(ϕ ∧ θ).Set of probability functions P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 38: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/38.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Probabilities
Probabilistic frameworkProbability function, PSubjective degrees of belief – Dutch BookP : SL→ [0,1]
P(τ) = 1 for all tautologies τ ∈ SL.P(ϕ ∨ θ) = P(ϕ) + P(θ), if |= ¬(ϕ ∧ θ).ϕ |= θ implies P(ϕ) ≤ P(θ).P(ϕ ∨ θ) = P(ϕ) + P(θ)− P(ϕ ∧ θ).Set of probability functions P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 39: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/39.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Probabilities
Probabilistic frameworkProbability function, PSubjective degrees of belief – Dutch BookP : SL→ [0,1]
P(τ) = 1 for all tautologies τ ∈ SL.P(ϕ ∨ θ) = P(ϕ) + P(θ), if |= ¬(ϕ ∧ θ).ϕ |= θ implies P(ϕ) ≤ P(θ).P(ϕ ∨ θ) = P(ϕ) + P(θ)− P(ϕ ∧ θ).Set of probability functions P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 40: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/40.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Probabilities
Probabilistic frameworkProbability function, PSubjective degrees of belief – Dutch BookP : SL→ [0,1]
P(τ) = 1 for all tautologies τ ∈ SL.P(ϕ ∨ θ) = P(ϕ) + P(θ), if |= ¬(ϕ ∧ θ).ϕ |= θ implies P(ϕ) ≤ P(θ).P(ϕ ∨ θ) = P(ϕ) + P(θ)− P(ϕ ∧ θ).Set of probability functions P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 41: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/41.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Probabilities
Probabilistic frameworkProbability function, PSubjective degrees of belief – Dutch BookP : SL→ [0,1]
P(τ) = 1 for all tautologies τ ∈ SL.P(ϕ ∨ θ) = P(ϕ) + P(θ), if |= ¬(ϕ ∧ θ).ϕ |= θ implies P(ϕ) ≤ P(θ).P(ϕ ∨ θ) = P(ϕ) + P(θ)− P(ϕ ∧ θ).Set of probability functions P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 42: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/42.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Probabilities
Probabilistic frameworkProbability function, PSubjective degrees of belief – Dutch BookP : SL→ [0,1]
P(τ) = 1 for all tautologies τ ∈ SL.P(ϕ ∨ θ) = P(ϕ) + P(θ), if |= ¬(ϕ ∧ θ).ϕ |= θ implies P(ϕ) ≤ P(θ).P(ϕ ∨ θ) = P(ϕ) + P(θ)− P(ϕ ∧ θ).Set of probability functions P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 43: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/43.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Probabilities
Probabilistic frameworkProbability function, PSubjective degrees of belief – Dutch BookP : SL→ [0,1]
P(τ) = 1 for all tautologies τ ∈ SL.P(ϕ ∨ θ) = P(ϕ) + P(θ), if |= ¬(ϕ ∧ θ).ϕ |= θ implies P(ϕ) ≤ P(θ).P(ϕ ∨ θ) = P(ϕ) + P(θ)− P(ϕ ∧ θ).Set of probability functions P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 44: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/44.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Worlds
Possible world, states
ω = v1 ∧ v2 ∧ ¬v3 ∧ . . . ∧ vn1 ∧ ¬vn
Set of possible worlds Ω.Proposition F ⊆ Ω.
P(ϕ) =∑ω∈Ωω|=ϕ
P(ω).
A probability function P ∈ P is uniquely determined by itsvalues on possible worlds, 〈P(ω) : ω ∈ Ω〉.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 45: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/45.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Worlds
Possible world, states
ω = v1 ∧ v2 ∧ ¬v3 ∧ . . . ∧ vn1 ∧ ¬vn
Set of possible worlds Ω.Proposition F ⊆ Ω.
P(ϕ) =∑ω∈Ωω|=ϕ
P(ω).
A probability function P ∈ P is uniquely determined by itsvalues on possible worlds, 〈P(ω) : ω ∈ Ω〉.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 46: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/46.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Worlds
Possible world, states
ω = v1 ∧ v2 ∧ ¬v3 ∧ . . . ∧ vn1 ∧ ¬vn
Set of possible worlds Ω.Proposition F ⊆ Ω.
P(ϕ) =∑ω∈Ωω|=ϕ
P(ω).
A probability function P ∈ P is uniquely determined by itsvalues on possible worlds, 〈P(ω) : ω ∈ Ω〉.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 47: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/47.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Worlds
Possible world, states
ω = v1 ∧ v2 ∧ ¬v3 ∧ . . . ∧ vn1 ∧ ¬vn
Set of possible worlds Ω.Proposition F ⊆ Ω.
P(ϕ) =∑ω∈Ωω|=ϕ
P(ω).
A probability function P ∈ P is uniquely determined by itsvalues on possible worlds, 〈P(ω) : ω ∈ Ω〉.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 48: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/48.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Worlds
Possible world, states
ω = v1 ∧ v2 ∧ ¬v3 ∧ . . . ∧ vn1 ∧ ¬vn
Set of possible worlds Ω.Proposition F ⊆ Ω.
P(ϕ) =∑ω∈Ωω|=ϕ
P(ω).
A probability function P ∈ P is uniquely determined by itsvalues on possible worlds, 〈P(ω) : ω ∈ Ω〉.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 49: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/49.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Outline
1 Motivation & ProblemMotivationProblem
2 FormalitiesLanguageUncertaintyKnowledge
3 Inference ProcessesThe GistThe PointRight!
4 Desiderata for Inference Processes
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 50: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/50.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Knowledge
Think of P as some set of functions.Spanned by the possible worlds ω.It can be represented by a simplex.Dimension equal to the number of possibleworlds/states.Doctor’s background knowledge KIdentify K with a set E ⊆ P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 51: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/51.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Knowledge
Think of P as some set of functions.Spanned by the possible worlds ω.It can be represented by a simplex.Dimension equal to the number of possibleworlds/states.Doctor’s background knowledge KIdentify K with a set E ⊆ P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 52: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/52.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Knowledge
Think of P as some set of functions.Spanned by the possible worlds ω.It can be represented by a simplex.Dimension equal to the number of possibleworlds/states.Doctor’s background knowledge KIdentify K with a set E ⊆ P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 53: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/53.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Knowledge
Think of P as some set of functions.Spanned by the possible worlds ω.It can be represented by a simplex.Dimension equal to the number of possibleworlds/states.Doctor’s background knowledge KIdentify K with a set E ⊆ P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 54: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/54.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Knowledge
Think of P as some set of functions.Spanned by the possible worlds ω.It can be represented by a simplex.Dimension equal to the number of possibleworlds/states.Doctor’s background knowledge KIdentify K with a set E ⊆ P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 55: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/55.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Knowledge
Think of P as some set of functions.Spanned by the possible worlds ω.It can be represented by a simplex.Dimension equal to the number of possibleworlds/states.Doctor’s background knowledge KIdentify K with a set E ⊆ P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 56: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/56.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Knowledge
Think of P as some set of functions.Spanned by the possible worlds ω.It can be represented by a simplex.Dimension equal to the number of possibleworlds/states.Doctor’s background knowledge KIdentify K with a set E ⊆ P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 57: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/57.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Knowledge
Think of P as some set of functions.Spanned by the possible worlds ω.It can be represented by a simplex.Dimension equal to the number of possibleworlds/states.Doctor’s background knowledge KIdentify K with a set E ⊆ P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 58: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/58.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Knowledge
Think of P as some set of functions.Spanned by the possible worlds ω.It can be represented by a simplex.Dimension equal to the number of possibleworlds/states.Doctor’s background knowledge KIdentify K with a set E ⊆ P
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 59: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/59.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Watt’s Assumption
The set E contains all the doctor’s knowledge.If you don’t formalise all knowledge, then you should not besurprised by our answer.Garbage in, garbage out.This course is not about the highly relevant and non-trivialproblem of obtaining E from K .We will assume that E 6= ∅.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 60: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/60.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Watt’s Assumption
The set E contains all the doctor’s knowledge.If you don’t formalise all knowledge, then you should not besurprised by our answer.Garbage in, garbage out.This course is not about the highly relevant and non-trivialproblem of obtaining E from K .We will assume that E 6= ∅.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 61: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/61.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Watt’s Assumption
The set E contains all the doctor’s knowledge.If you don’t formalise all knowledge, then you should not besurprised by our answer.Garbage in, garbage out.This course is not about the highly relevant and non-trivialproblem of obtaining E from K .We will assume that E 6= ∅.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 62: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/62.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Watt’s Assumption
The set E contains all the doctor’s knowledge.If you don’t formalise all knowledge, then you should not besurprised by our answer.Garbage in, garbage out.This course is not about the highly relevant and non-trivialproblem of obtaining E from K .We will assume that E 6= ∅.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 63: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/63.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Watt’s Assumption
The set E contains all the doctor’s knowledge.If you don’t formalise all knowledge, then you should not besurprised by our answer.Garbage in, garbage out.This course is not about the highly relevant and non-trivialproblem of obtaining E from K .We will assume that E 6= ∅.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 64: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/64.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Example Knowledge
Symptom S1 is a very good indicator of condition C.10% of migraines are triggered by stress.In 23% of cases in which patients complain about symptomS2, migraines will spontaneously cease within half a day.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 65: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/65.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Example Knowledge
Symptom S1 is a very good indicator of condition C.10% of migraines are triggered by stress.In 23% of cases in which patients complain about symptomS2, migraines will spontaneously cease within half a day.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 66: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/66.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
LanguageUncertaintyKnowledge
Example Knowledge
Symptom S1 is a very good indicator of condition C.10% of migraines are triggered by stress.In 23% of cases in which patients complain about symptomS2, migraines will spontaneously cease within half a day.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 67: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/67.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Outline
1 Motivation & ProblemMotivationProblem
2 FormalitiesLanguageUncertaintyKnowledge
3 Inference ProcessesThe GistThe PointRight!
4 Desiderata for Inference Processes
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 68: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/68.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Outline
1 Motivation & ProblemMotivationProblem
2 FormalitiesLanguageUncertaintyKnowledge
3 Inference ProcessesThe GistThe PointRight!
4 Desiderata for Inference Processes
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 69: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/69.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Condensing Knowledge
Idea:Input knowledge KOutput belief function P+ ∈ PAdopt this function P+ for decision making.P+ reflects the doctor’s knowledge K .We are not fabricating knowledge out of thin air.We do the best we can, given limited information.Some information is lost in this process.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 70: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/70.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Condensing Knowledge
Idea:Input knowledge KOutput belief function P+ ∈ PAdopt this function P+ for decision making.P+ reflects the doctor’s knowledge K .We are not fabricating knowledge out of thin air.We do the best we can, given limited information.Some information is lost in this process.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 71: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/71.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Condensing Knowledge
Idea:Input knowledge KOutput belief function P+ ∈ PAdopt this function P+ for decision making.P+ reflects the doctor’s knowledge K .We are not fabricating knowledge out of thin air.We do the best we can, given limited information.Some information is lost in this process.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 72: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/72.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Condensing Knowledge
Idea:Input knowledge KOutput belief function P+ ∈ PAdopt this function P+ for decision making.P+ reflects the doctor’s knowledge K .We are not fabricating knowledge out of thin air.We do the best we can, given limited information.Some information is lost in this process.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 73: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/73.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Condensing Knowledge
Idea:Input knowledge KOutput belief function P+ ∈ PAdopt this function P+ for decision making.P+ reflects the doctor’s knowledge K .We are not fabricating knowledge out of thin air.We do the best we can, given limited information.Some information is lost in this process.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 74: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/74.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Condensing Knowledge
Idea:Input knowledge KOutput belief function P+ ∈ PAdopt this function P+ for decision making.P+ reflects the doctor’s knowledge K .We are not fabricating knowledge out of thin air.We do the best we can, given limited information.Some information is lost in this process.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 75: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/75.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Condensing Knowledge
Idea:Input knowledge KOutput belief function P+ ∈ PAdopt this function P+ for decision making.P+ reflects the doctor’s knowledge K .We are not fabricating knowledge out of thin air.We do the best we can, given limited information.Some information is lost in this process.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 76: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/76.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Condensing Knowledge
Idea:Input knowledge KOutput belief function P+ ∈ PAdopt this function P+ for decision making.P+ reflects the doctor’s knowledge K .We are not fabricating knowledge out of thin air.We do the best we can, given limited information.Some information is lost in this process.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 77: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/77.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Conditionalisation
What about the properties of the patient?Yes, what about them??I have not forgotten about them.Properties ψ1, . . . , ψr
ψ := ψ1 ∧ . . . ∧ ψr
Consider conditional probability (P+(ψ) > 0)
P+(ϕ|ψ) :=P+(ϕ ∧ ψ)
P+(ψ).
Roughly, assume that ψ is true.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 78: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/78.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Conditionalisation
What about the properties of the patient?Yes, what about them??I have not forgotten about them.Properties ψ1, . . . , ψr
ψ := ψ1 ∧ . . . ∧ ψr
Consider conditional probability (P+(ψ) > 0)
P+(ϕ|ψ) :=P+(ϕ ∧ ψ)
P+(ψ).
Roughly, assume that ψ is true.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 79: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/79.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Conditionalisation
What about the properties of the patient?Yes, what about them??I have not forgotten about them.Properties ψ1, . . . , ψr
ψ := ψ1 ∧ . . . ∧ ψr
Consider conditional probability (P+(ψ) > 0)
P+(ϕ|ψ) :=P+(ϕ ∧ ψ)
P+(ψ).
Roughly, assume that ψ is true.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 80: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/80.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Conditionalisation
What about the properties of the patient?Yes, what about them??I have not forgotten about them.Properties ψ1, . . . , ψr
ψ := ψ1 ∧ . . . ∧ ψr
Consider conditional probability (P+(ψ) > 0)
P+(ϕ|ψ) :=P+(ϕ ∧ ψ)
P+(ψ).
Roughly, assume that ψ is true.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 81: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/81.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Conditionalisation
What about the properties of the patient?Yes, what about them??I have not forgotten about them.Properties ψ1, . . . , ψr
ψ := ψ1 ∧ . . . ∧ ψr
Consider conditional probability (P+(ψ) > 0)
P+(ϕ|ψ) :=P+(ϕ ∧ ψ)
P+(ψ).
Roughly, assume that ψ is true.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 82: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/82.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Conditionalisation
What about the properties of the patient?Yes, what about them??I have not forgotten about them.Properties ψ1, . . . , ψr
ψ := ψ1 ∧ . . . ∧ ψr
Consider conditional probability (P+(ψ) > 0)
P+(ϕ|ψ) :=P+(ϕ ∧ ψ)
P+(ψ).
Roughly, assume that ψ is true.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 83: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/83.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Conditionalisation
What about the properties of the patient?Yes, what about them??I have not forgotten about them.Properties ψ1, . . . , ψr
ψ := ψ1 ∧ . . . ∧ ψr
Consider conditional probability (P+(ψ) > 0)
P+(ϕ|ψ) :=P+(ϕ ∧ ψ)
P+(ψ).
Roughly, assume that ψ is true.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 84: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/84.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
The General Picture
Formally, an inference process is a map from a set of prob-ability functions (here E) to the set of probability functions.An inference process is a map (or function)Input: EOutput: P+.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 85: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/85.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
The General Picture
Formally, an inference process is a map from a set of prob-ability functions (here E) to the set of probability functions.An inference process is a map (or function)Input: EOutput: P+.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 86: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/86.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
The General Picture
Formally, an inference process is a map from a set of prob-ability functions (here E) to the set of probability functions.An inference process is a map (or function)Input: EOutput: P+.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 87: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/87.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
The General Picture
Formally, an inference process is a map from a set of prob-ability functions (here E) to the set of probability functions.An inference process is a map (or function)Input: EOutput: P+.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 88: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/88.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Outline
1 Motivation & ProblemMotivationProblem
2 FormalitiesLanguageUncertaintyKnowledge
3 Inference ProcessesThe GistThe PointRight!
4 Desiderata for Inference Processes
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 89: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/89.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Inductive Logic
How strongly do uncertain premises entail a conclusion?Uncertain premises ...conclusion ...entail ...
P∗(ϕ1) ∈ I1,P∗(ϕ2) ∈ I2, . . . ,P∗(ϕk ) ∈ Ik︸ ︷︷ ︸Knowledge
|≈︸︷︷︸Entailment
ψ?︸︷︷︸Conclusion
? = P+(ψ), a single real number.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 90: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/90.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Inductive Logic
How strongly do uncertain premises entail a conclusion?Uncertain premises ...conclusion ...entail ...
P∗(ϕ1) ∈ I1,P∗(ϕ2) ∈ I2, . . . ,P∗(ϕk ) ∈ Ik︸ ︷︷ ︸Knowledge
|≈︸︷︷︸Entailment
ψ?︸︷︷︸Conclusion
? = P+(ψ), a single real number.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 91: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/91.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Inductive Logic
How strongly do uncertain premises entail a conclusion?Uncertain premises ...conclusion ...entail ...
P∗(ϕ1) ∈ I1,P∗(ϕ2) ∈ I2, . . . ,P∗(ϕk ) ∈ Ik︸ ︷︷ ︸Knowledge
|≈︸︷︷︸Entailment
ψ?︸︷︷︸Conclusion
? = P+(ψ), a single real number.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 92: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/92.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Inductive Logic
How strongly do uncertain premises entail a conclusion?Uncertain premises ...conclusion ...entail ...
P∗(ϕ1) ∈ I1,P∗(ϕ2) ∈ I2, . . . ,P∗(ϕk ) ∈ Ik︸ ︷︷ ︸Knowledge
|≈︸︷︷︸Entailment
ψ?︸︷︷︸Conclusion
? = P+(ψ), a single real number.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 93: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/93.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Inductive Logic
How strongly do uncertain premises entail a conclusion?Uncertain premises ...conclusion ...entail ...
P∗(ϕ1) ∈ I1,P∗(ϕ2) ∈ I2, . . . ,P∗(ϕk ) ∈ Ik︸ ︷︷ ︸Knowledge
|≈︸︷︷︸Entailment
ψ?︸︷︷︸Conclusion
? = P+(ψ), a single real number.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 94: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/94.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Inductive Logic
How strongly do uncertain premises entail a conclusion?Uncertain premises ...conclusion ...entail ...
P∗(ϕ1) ∈ I1,P∗(ϕ2) ∈ I2, . . . ,P∗(ϕk ) ∈ Ik︸ ︷︷ ︸Knowledge
|≈︸︷︷︸Entailment
ψ?︸︷︷︸Conclusion
? = P+(ψ), a single real number.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 95: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/95.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Inductive Logic
How strongly do uncertain premises entail a conclusion?Uncertain premises ...conclusion ...entail ...
P∗(ϕ1) ∈ I1,P∗(ϕ2) ∈ I2, . . . ,P∗(ϕk ) ∈ Ik︸ ︷︷ ︸Knowledge
|≈︸︷︷︸Entailment
ψ?︸︷︷︸Conclusion
? = P+(ψ), a single real number.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 96: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/96.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Outline
1 Motivation & ProblemMotivationProblem
2 FormalitiesLanguageUncertaintyKnowledge
3 Inference ProcessesThe GistThe PointRight!
4 Desiderata for Inference Processes
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 97: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/97.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Obviously right – Tomorrow!
Adopt the function, P† = P+, which solve this optimisationproblem
maximise: −∑ω∈Ω
P(ω) log(P(ω))
subject to: P ∈ E .
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 98: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/98.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Intuitively right
Pick a “middling” function in E.How formalise “middling”?
Pick the centre of E!Okay, yes, but how does one work out the centre?
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 99: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/99.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Intuitively right
Pick a “middling” function in E.How formalise “middling”?
Pick the centre of E!Okay, yes, but how does one work out the centre?
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 100: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/100.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Intuitively right
Pick a “middling” function in E.How formalise “middling”?
Pick the centre of E!Okay, yes, but how does one work out the centre?
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 101: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/101.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Intuitively right
Pick a “middling” function in E.How formalise “middling”?
Pick the centre of E!Okay, yes, but how does one work out the centre?
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 102: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/102.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
Intuitively right
Pick a “middling” function in E.How formalise “middling”?
Pick the centre of E!Okay, yes, but how does one work out the centre?
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 103: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/103.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
At the tip of a finger
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 104: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/104.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
The centre
Centre of MassPoint of Balance
P+CoM :=
∫P∈E P dp∫P∈E dp
.
Centre of Mass inference process.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 105: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/105.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
The centre
Centre of MassPoint of Balance
P+CoM :=
∫P∈E P dp∫P∈E dp
.
Centre of Mass inference process.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 106: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/106.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
The centre
Centre of MassPoint of Balance
P+CoM :=
∫P∈E P dp∫P∈E dp
.
Centre of Mass inference process.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 107: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/107.jpg)
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28 / 36
Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
The GistThe PointRight!
The centre
Centre of MassPoint of Balance
P+CoM :=
∫P∈E P dp∫P∈E dp
.
Centre of Mass inference process.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 108: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/108.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Outline
1 Motivation & ProblemMotivationProblem
2 FormalitiesLanguageUncertaintyKnowledge
3 Inference ProcessesThe GistThe PointRight!
4 Desiderata for Inference Processes
Jürgen Landes Maximum Entropy and Inductive Logic I
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
You
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Jürgen Landes Maximum Entropy and Inductive Logic I
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Internal
P+ ∈ E......if E is not emptyconvexand closed.
Jürgen Landes Maximum Entropy and Inductive Logic I
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Internal
P+ ∈ E......if E is not emptyconvexand closed.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 112: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/112.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Internal
P+ ∈ E......if E is not emptyconvexand closed.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 113: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/113.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Internal
P+ ∈ E......if E is not emptyconvexand closed.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 114: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/114.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Internal
P+ ∈ E......if E is not emptyconvexand closed.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 115: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/115.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Internal
P+ ∈ E......if E is not emptyconvexand closed.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 116: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/116.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Careful, careful
Open-mindednessDo not rule anything out, which you consider possible.If there exists a probability function P ∈ E with P(ω) > 0,then P+(ω) > 0.
Jürgen Landes Maximum Entropy and Inductive Logic I
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Careful, careful
Open-mindednessDo not rule anything out, which you consider possible.If there exists a probability function P ∈ E with P(ω) > 0,then P+(ω) > 0.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 118: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/118.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Careful, careful
Open-mindednessDo not rule anything out, which you consider possible.If there exists a probability function P ∈ E with P(ω) > 0,then P+(ω) > 0.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 119: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/119.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Open-mindedness is Possible
For all such ω, pick an Pω ∈ Pand also pick an rω ∈ (0,1) such that
∑rω = 1.
Then the convex combination∑
rωPω is a probability func-tion. (P is convex.)It is possible to satisfy open-mindedness.If E is convex, then
∑rωPω ∈ E.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 120: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/120.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Open-mindedness is Possible
For all such ω, pick an Pω ∈ Pand also pick an rω ∈ (0,1) such that
∑rω = 1.
Then the convex combination∑
rωPω is a probability func-tion. (P is convex.)It is possible to satisfy open-mindedness.If E is convex, then
∑rωPω ∈ E.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 121: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/121.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Open-mindedness is Possible
For all such ω, pick an Pω ∈ Pand also pick an rω ∈ (0,1) such that
∑rω = 1.
Then the convex combination∑
rωPω is a probability func-tion. (P is convex.)It is possible to satisfy open-mindedness.If E is convex, then
∑rωPω ∈ E.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 122: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/122.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Open-mindedness is Possible
For all such ω, pick an Pω ∈ Pand also pick an rω ∈ (0,1) such that
∑rω = 1.
Then the convex combination∑
rωPω is a probability func-tion. (P is convex.)It is possible to satisfy open-mindedness.If E is convex, then
∑rωPω ∈ E.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 123: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/123.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Open-mindedness is Possible
For all such ω, pick an Pω ∈ Pand also pick an rω ∈ (0,1) such that
∑rω = 1.
Then the convex combination∑
rωPω is a probability func-tion. (P is convex.)It is possible to satisfy open-mindedness.If E is convex, then
∑rωPω ∈ E.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 124: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/124.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Language Invariance
What if we started with v1, v2, . . . , vn, vn+1
the same knowledge and the same patient?Apply the above recipe and obtain P ′.For a sentence in the original language ϕ ∈ SL, pleasecomplete the below
P ′(ϕ) = ?
Shocker: Centre of Mass is not language invariant!
Jürgen Landes Maximum Entropy and Inductive Logic I
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Language Invariance
What if we started with v1, v2, . . . , vn, vn+1
the same knowledge and the same patient?Apply the above recipe and obtain P ′.For a sentence in the original language ϕ ∈ SL, pleasecomplete the below
P ′(ϕ) = ?
Shocker: Centre of Mass is not language invariant!
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 126: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/126.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Language Invariance
What if we started with v1, v2, . . . , vn, vn+1
the same knowledge and the same patient?Apply the above recipe and obtain P ′.For a sentence in the original language ϕ ∈ SL, pleasecomplete the below
P ′(ϕ) = ?
Shocker: Centre of Mass is not language invariant!
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 127: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/127.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Language Invariance
What if we started with v1, v2, . . . , vn, vn+1
the same knowledge and the same patient?Apply the above recipe and obtain P ′.For a sentence in the original language ϕ ∈ SL, pleasecomplete the below
P ′(ϕ) = ?
Shocker: Centre of Mass is not language invariant!
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 128: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/128.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
Language Invariance
What if we started with v1, v2, . . . , vn, vn+1
the same knowledge and the same patient?Apply the above recipe and obtain P ′.For a sentence in the original language ϕ ∈ SL, pleasecomplete the below
P ′(ϕ) = ?
Shocker: Centre of Mass is not language invariant!
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 129: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/129.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
References I
Paris, J. B. (2006).The Uncertain Reasoner’s Companion: A MathematicalPerspective, volume 39 of Cambridge Tracts in TheoreticalComputer Science.Cambridge University Press, 2 edition.
Jürgen Landes Maximum Entropy and Inductive Logic I
![Page 130: Maximum Entropy and Inductive Logic I - Blogs at Kent · Desiderata for Inference Processes Motivation Problem Maximum Entropy and Inductive Logic I Jürgen Landes Spring School on](https://reader033.fdocuments.in/reader033/viewer/2022041504/5e23c0575322212e6b7d8d16/html5/thumbnails/130.jpg)
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Motivation & ProblemFormalities
Inference ProcessesDesiderata for Inference Processes
That’s it. Thank you! Questions?
Jürgen Landes Maximum Entropy and Inductive Logic I