general formulations for structures markov logic
play

General Formulations for Structures: Markov Logic CS 6355: - PowerPoint PPT Presentation

Aug 22, 2022 •326 likes •767 views

General Formulations for Structures: Markov Logic CS 6355: Structured Prediction 1 This lecture Graphical models Bayesian Networks Markov Random Fields (MRFs) Formulations of structured output Joint models Markov Logic

  1. General Formulations for Structures: Markov Logic CS 6355: Structured Prediction 1

  2. This lecture • Graphical models – Bayesian Networks – Markov Random Fields (MRFs) • Formulations of structured output – Joint models • Markov Logic Network – Conditional models • Conditional Random Fields (again) • Constrained Conditional Models 2

  3. Representing and reasoning about knowledge Consider the following statements – Smoking causes cancer – If two people are friends and one smokes, so does the other Questions to think about – How do we represent this knowledge? – How do we answer questions like: “If Anna is friends with Bob, and Bob smokes, can Anna get cancer?” 3

  4. Representing and reasoning about knowledge Consider the following statements – Smoking causes cancer – If two people are friends and one smokes, so does the other Questions to think about – How do we represent this knowledge? – How do we answer questions like: “If Anna is friends with Bob, and Bob smokes, can Anna get cancer?” Logic is a natural language for declaratively stating knowledge and making inferences. 4

  5. Representing knowledge “Smoking causes cancer.” ∀𝑦, Smokes 𝑦 ⇒ Cancer 𝑦 We have predicates Smokes and Cancer here in this universally quantified statement. 5

  6. Representing knowledge “Smoking causes cancer.” ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) “If two people are friends and one smokes, so does the other.” ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) 6

  7. Reasoning about knowledge “Smoking causes cancer.” ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) “If two people are friends and one smokes, so does the other.” ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) Suppose we have two friends Anna and Bob, and Bob smokes. What can we infer about Anna? 1. Anna and Bob are friends: Friends Anna, Bob 2. Bob smokes: Smokes Bob 3. We know that: Friends Anna, Bob ∧ Smokes Bob ⇒ Smokes(Anna) 4. And we also know that: Smokes Anna ⇒ Cancer(Anna) 7

  8. Reasoning about knowledge “Smoking causes cancer.” ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) “If two people are friends and one smokes, so does the other.” ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) Suppose we have two friends Anna and Bob, and Bob smokes. What can we infer about Anna? 1. Anna and Bob are friends: Friends Anna, Bob 2. Bob smokes: Smokes Bob 3. We know that: Friends Anna, Bob ∧ Smokes Bob ⇒ Smokes(Anna) 4. And we also know that: Smokes Anna ⇒ Cancer(Anna) 8

  9. Reasoning about knowledge “Smoking causes cancer.” ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) “If two people are friends and one smokes, so does the other.” ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) Suppose we have two friends Anna and Bob, and Bob smokes. What can we infer about Anna? 1. Anna and Bob are friends: Friends Anna, Bob 2. Bob smokes: Smokes Bob 3. We know that: Friends Anna, Bob ∧ Smokes Bob ⇒ Smokes(Anna) 4. And we also know that: Smokes Anna ⇒ Cancer(Anna) 9

  10. Reasoning about knowledge “Smoking causes cancer.” ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) “If two people are friends and one smokes, so does the other.” ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) Suppose we have two friends Anna and Bob, and Bob smokes. What can we infer about Anna? 1. Anna and Bob are friends: Friends Anna, Bob 2. Bob smokes: Smokes Bob 3. We know that: Friends Anna, Bob ∧ Smokes Bob ⇒ Smokes(Anna) 4. And we also know that: Smokes Anna ⇒ Cancer(Anna) 10

  11. Reasoning about knowledge “Smoking causes cancer.” ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) “If two people are friends and one smokes, so does the other.” ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) Suppose we have two friends Anna and Bob, and Bob smokes. What can we infer about Anna? 1. Anna and Bob are friends: Friends Anna, Bob 2. Bob smokes: Smokes Bob 3. We know that: Friends Anna, Bob ∧ Smokes Bob ⇒ Smokes(Anna) 4. And we also know that: Smokes Anna ⇒ Cancer(Anna) 11

  12. Reasoning about knowledge “Smoking causes cancer.” ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) “If two people are friends and one smokes, so does the other.” ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) Suppose we have two friends Anna and Bob, and Bob smokes. What can we infer about Anna? 1. Anna and Bob are friends: Friends Anna, Bob 2. Bob smokes: Smokes Bob 3. We know that: Friends Anna, Bob ∧ Smokes Bob ⇒ Smokes(Anna) 4. And we also know that: Smokes Anna ⇒ Cancer(Anna) Logic is an expressive language, but how do we deal with uncertainty? 12

  13. From logic to Markov networks Consider the following statements – Smoking causes cancer – If two people are friends and one smokes, so does the other In logic: • ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) 13 [Example from Domingos and Lowd 2009]

  14. From logic to Markov networks Consider the following statements – Smoking causes cancer – If two people are friends and one smokes, so does the other In logic: • ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) The statements are not necessarily absolutely true • – How do we associate degrees of belief to statements? 14 [Example from Domingos and Lowd 2009]

  15. Markov Logic Networks From rules to graphical models • Convert to clauses ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) • Associate a potential function for each clause – Think of each formula as a factor – Typically, log-linear in all the variables involved • Ground the logical expressions to all x, y that you care about 15 [Example from Domingos and Lowd 2009]

  16. Markov Logic Networks From rules to graphical models • Convert to clauses ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) • Associate a potential function for each clause Recall: – Think of each formula as a factor – Typically, log-linear in all the variables involved • A literal is predicate or its negation • A clause is a disjunction of literals • Any implication 𝐵 ⇒ 𝐶 is equivalent to ¬𝐵 ∨ 𝐶 • Ground the logical expressions to all x, y that you care about 16 [Example from Domingos and Lowd 2009]

  17. Markov Logic Networks From rules to graphical models • Convert to clauses ∀𝑦, ¬Smokes 𝑦 ∨ Cancer 𝑦 ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) ∀𝑦, 𝑧, ¬Friends 𝑦, 𝑧 ∨ ¬Smokes 𝑧 ∨ Cancer(𝑧) ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) • Associate a potential function for each clause Recall: – Think of each formula as a factor – Typically, log-linear in all the variables involved • A literal is predicate or its negation • A clause is a disjunction of literals • Any implication 𝐵 ⇒ 𝐶 is equivalent to ¬𝐵 ∨ 𝐶 • Ground the logical expressions to all x, y that you care about 17 [Example from Domingos and Lowd 2009]

  18. Markov Logic Networks From rules to graphical models • Convert to clauses ∀𝑦, ¬Smokes 𝑦 ∨ Cancer 𝑦 ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) ∀𝑦, 𝑧, ¬Friends 𝑦, 𝑧 ∨ ¬Smokes 𝑧 ∨ Cancer(𝑧) ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) • Associate a potential function for each clause – Think of each formula as a factor – Could be log-linear in all the variables involved • Ground the logical expressions to all x, y that you care about 18 [Example from Domingos and Lowd 2009]

  19. Markov Logic Networks From rules to graphical models • Convert to clauses ∀𝑦, ¬Smokes 𝑦 ∨ Cancer 𝑦 ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) ∀𝑦, 𝑧, ¬Friends 𝑦, 𝑧 ∨ ¬Smokes 𝑧 ∨ Cancer(𝑧) ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) • Associate a potential function for each clause – Think of each formula as a factor – Could be log-linear in all the variables involved • Ground the logical expressions to all x, y that you care about 19 [Example from Domingos and Lowd 2009]

  20. Example of a ground network 1.5 ∀𝑦, ¬Smokes 𝑦 ∨ Cancer 𝑦 ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) ∀𝑦, 𝑧, ¬Friends 𝑦, 𝑧 ∨ ¬Smokes 𝑧 ∨ Cancer(𝑧) 1.0 ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) Each rule is associated with a weight 20 [Example from Domingos and Lowd 2009]

  21. Weighted formulas → ground network ∀𝑦, ¬Smokes 𝑦 ∨ Cancer 𝑦 ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) ∀𝑦, 𝑧, ¬Friends 𝑦, 𝑧 ∨ ¬Smokes 𝑧 ∨ Cancer(𝑧) ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) 21 [Example from Domingos and Lowd 2009]

  22. Weighted formulas → ground network 1.5 ∀𝑦, ¬Smokes 𝑦 ∨ Cancer 𝑦 ∀𝑦, Smokes 𝑦 ⇒ Cancer(𝑦) ∀𝑦, 𝑧, ¬Friends 𝑦, 𝑧 ∨ ¬Smokes 𝑧 ∨ Cancer(𝑧) 1.0 ∀𝑦, 𝑧 Friends 𝑦, 𝑧 ∧ Smokes 𝑦 ⇒ Smokes(𝑧) Suppose there are two people in the world: Anna (A), Bob (B) 22 [Example from Domingos and Lowd 2009]

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Markov Logic Markov Logic Probability First-Order Logic Propositional Logic Markov Logic

Markov Logic Markov Logic Probability First-Order Logic Propositional Logic Markov Logic

Putting the Pieces Together Markov Logic Markov Logic Probability First-Order Logic Propositional Logic Markov Logic Definition A Markov Logic Network (MLN) is a set of A logical KB is a set of hard constraints pairs (F, w) where on

350 views • 6 slides
Markov Logic Networks Matt Richardson and Pedro Domingos (2006), Markov Logic Networks, Machine

Markov Logic Networks Matt Richardson and Pedro Domingos (2006), Markov Logic Networks, Machine

Markov Logic Networks Matt Richardson and Pedro Domingos (2006), Markov Logic Networks, Machine Learning , 62, 107-136, 2006. CS 786 University of Waterloo Lecture 23: July 19, 2012 Outline Markov Logic Networks Alchemy 2 CS786

344 views • 15 slides
Markov Logic Networks November 18, 2008 CS 486/686 University of Waterloo Outline Markov

Markov Logic Networks November 18, 2008 CS 486/686 University of Waterloo Outline Markov

Markov Logic Networks November 18, 2008 CS 486/686 University of Waterloo Outline Markov Logic Networks Alchemy Readings: Matt Richardson and Pedro Domingos (2006), Markov Logic Networks, Machine Learning , 62, 107-136, 2006.

351 views • 21 slides
Markov Logic Networks Matt Richardson and Pedro Domingos (2006), Markov Logic Networks, Machine

Markov Logic Networks Matt Richardson and Pedro Domingos (2006), Markov Logic Networks, Machine

Markov Logic Networks Matt Richardson and Pedro Domingos (2006), Markov Logic Networks, Machine Learning , 62, 107-136, 2006. CS 486/686 University of Waterloo Lecture 21: Nov 20, 2012 Outline Markov Logic Networks Alchemy 2

618 views • 13 slides
Markov Logic Networks Andrea Passerini passerini@disi.unitn.it Statistical relational learning

Markov Logic Networks Andrea Passerini passerini@disi.unitn.it Statistical relational learning

Markov Logic Networks Andrea Passerini passerini@disi.unitn.it Statistical relational learning Markov Logic Networks Combining logic with probability Motivation First-order logic is a powerful language to represent complex relational

721 views • 54 slides
Integrating Logical Representations with Probabilistic Information using Markov Logic Dan

Integrating Logical Representations with Probabilistic Information using Markov Logic Dan

Integrating Logical Representations with Probabilistic Information using Markov Logic Dan Garrette, Katrin Erk, and Raymond Mooney The University of Texas at Austin 1 Overview Some phenomena best modeled through logic , others statistically

711 views • 51 slides
Logic as a Tool Chapter 3: Understanding First-order Logic 3.1 First-order structures and

Logic as a Tool Chapter 3: Understanding First-order Logic 3.1 First-order structures and

Logic as a Tool Chapter 3: Understanding First-order Logic 3.1 First-order structures and languages. Terms and formulae of first-order logic. Valentin Goranko Stockholm University October 2016 Goranko Propositional logic is too weak

188 views • 14 slides
Seminar Statistics for structures A graphical perspective on Gauss-Markov process priors

Seminar Statistics for structures A graphical perspective on Gauss-Markov process priors

Seminar Statistics for structures A graphical perspective on Gauss-Markov process priors Moritz Schauer University of Amsterdam 1 / 26 Outline Midpoint displacement construction of a Brownian motion Corresponding Gaussian Markov

481 views • 26 slides
General Considerations for Age-Appropriate Formulations: FDA Clinical Perspective Erica Radden,

General Considerations for Age-Appropriate Formulations: FDA Clinical Perspective Erica Radden,

General Considerations for Age-Appropriate Formulations: FDA Clinical Perspective Erica Radden, M.D. Medical Officer, Division of Pediatric and Maternal Health Office of New Drugs, FDA 1 Disclosure Statement I have no financial

413 views • 18 slides
Logic as a Tool Chapter 3: Understanding First-order Logic 3.1 First-order structures and

Logic as a Tool Chapter 3: Understanding First-order Logic 3.1 First-order structures and

Logic as a Tool Chapter 3: Understanding First-order Logic 3.1 First-order structures and languages. Terms and formulae of first-order logic. Valentin Goranko Stockholm University October 2016 Goranko Propositional logic is too weak Goranko

1.69k views • 109 slides
Learning and Inference in Markov Logic Networks CS 486/686 University of Waterloo Lecture 23:

Learning and Inference in Markov Logic Networks CS 486/686 University of Waterloo Lecture 23:

Learning and Inference in Markov Logic Networks CS 486/686 University of Waterloo Lecture 23: November 27, 2012 Outline Markov Logic Networks Parameter learning Lifted inference 2 CS486/686 Lecture Slides (c) 2012 P. Poupart 1

223 views • 10 slides
+ Decision Structures & Boolean Logic CSCI-UA.002 + Sequence Structures n What we have been

+ Decision Structures & Boolean Logic CSCI-UA.002 + Sequence Structures n What we have been

+ Decision Structures & Boolean Logic CSCI-UA.002 + Sequence Structures n What we have been programming so far is known as a sequence structure n Sequence structures are sets of statements that execute in the order in which they

882 views • 74 slides
Scalable Uncertainty Management 06 Markov Logic Rainer Gemulla July 13, 2012 Overview In

Scalable Uncertainty Management 06 Markov Logic Rainer Gemulla July 13, 2012 Overview In

Scalable Uncertainty Management 06 Markov Logic Rainer Gemulla July 13, 2012 Overview In this lecture Statistical relational learning (SRL) Introduction to probabilistic graphical models (PGM) Basics of undirected models (called Markov

1.14k views • 78 slides
Continue Data Structures Grand Tour FixedLengthQueue Markov Orientation and kickoff Turn in

Continue Data Structures Grand Tour FixedLengthQueue Markov Orientation and kickoff Turn in

Continue Data Structures Grand Tour FixedLengthQueue Markov Orientation and kickoff Turn in the written problem from HW 18. Before tomorrow Reading assignment from Weiss Read and understand the Markov assignment Take the short

228 views • 19 slides
Learning and Inference in Markov Logic Networks CS 786 University of Waterloo Lecture 24: July

Learning and Inference in Markov Logic Networks CS 786 University of Waterloo Lecture 24: July

Learning and Inference in Markov Logic Networks CS 786 University of Waterloo Lecture 24: July 24, 2012 Outline Markov Logic Networks Parameter learning Lifted inference 2 CS786 Lecture Slides (c) 2012 P. Poupart 1 Parameter

98 views • 6 slides
Probabilistic Event Calculus based on Markov Logic Networks Anastasios Skarlatidis 1 , 2 ,

Probabilistic Event Calculus based on Markov Logic Networks Anastasios Skarlatidis 1 , 2 ,

Probabilistic Event Calculus based on Markov Logic Networks Anastasios Skarlatidis 1 , 2 , Georgios Paliouras 1 , George Vouros 2 and Alexander Artikis 1 1 Institute of Informatics and Telecommunications NCSR Demokritos, Athens 15310, Greece

504 views • 29 slides
Motivation Beyond local representation of language Information Extraction Reason

Motivation Beyond local representation of language Information Extraction Reason

Probabilistic First Order Models for Coreference Aron Culotta Information Extraction & Synthesis Lab University of Massachusetts joint work with advisor Andrew McCallum Motivation Beyond local representation of language

361 views • 17 slides
!"#$%$&'&()&*+,"%-.&*%'+/#01'(+234-5)+6789:

!"#$%$&'&()&*+,"%-.&*%'+/#01'(+234-5)+6789:

Cmput 651 - Learning Undirected Models {14,17}/11/2008 !"#$%$&'&()&*+,"%-.&*%'+/#01'(+234-5)+6789: ;1%"<&<=+><0&"1*)10+/#01'( /%)).1?+@"#?< A8BC8DEF88FGHHI

529 views • 29 slides
Model inference s e from l b a v observed data r e s b o time dynamics underlying

Model inference s e from l b a v observed data r e s b o time dynamics underlying

Model inference s e from l b a v observed data r e s b o time dynamics underlying mechanisms predictive model (correlations) (interactions) Many issues : limitation over temporal and spatial sampling, noise (measurement,

554 views • 41 slides
GMN GMNN: Gr Graph Ma Mark rkov Neur Neural al Ne Networks Meng Qu 1 2 , Yoshua Bengio 1 2 4

GMN GMNN: Gr Graph Ma Mark rkov Neur Neural al Ne Networks Meng Qu 1 2 , Yoshua Bengio 1 2 4

GMN GMNN: Gr Graph Ma Mark rkov Neur Neural al Ne Networks Meng Qu 1 2 , Yoshua Bengio 1 2 4 , Jian Tang 1 3 4 1 Quebec AI Institute (Mila) 2 University of Montreal 3 HEC Montreal 4 Canadian Institute for Advanced Research (CIFAR) Se Semi

204 views • 7 slides
Estimating Recall the general mean-variance specification E( Y | x ) = f ( x , ) , var( Y |

Estimating Recall the general mean-variance specification E( Y | x ) = f ( x , ) , var( Y |

ST 762 Nonlinear Statistical Models for Univariate and Multivariate Response Estimating Recall the general mean-variance specification E( Y | x ) = f ( x , ) , var( Y | x ) = 2 g ( , , x ) 2 . We have, so far, considered as a

498 views • 21 slides
Learning Deep Structured Models for Semantic Segmentation Guosheng Lin Semantic Segmentation

Learning Deep Structured Models for Semantic Segmentation Guosheng Lin Semantic Segmentation

Learning Deep Structured Models for Semantic Segmentation Guosheng Lin Semantic Segmentation Outline Exploring Context with Deep Structured Models Guosheng Lin, Chunhua Shen, Ian Reid, Anton van dan Hengel; Efficient Piecewise Training

992 views • 45 slides
EBSpat an R package devoted to simulation and estimation around nearest-neighbour type Gibbs point

EBSpat an R package devoted to simulation and estimation around nearest-neighbour type Gibbs point

EBSpat an R package devoted to simulation and estimation around nearest-neighbour type Gibbs point processes R. Drouilhet LJK Grenoble R. Drouilhet (LJK Grenoble) EBSpat R package 1 / 40 Plan Motivation 1 The Delaunay and Vorono

710 views • 59 slides
KDE-HMMs New, Nonparametric Acoustic Models for Speech Synthesis Gustav Eje Henter Joint work

KDE-HMMs New, Nonparametric Acoustic Models for Speech Synthesis Gustav Eje Henter Joint work

KDE-HMMs New, Nonparametric Acoustic Models for Speech Synthesis Gustav Eje Henter Joint work with W. Bastiaan Kleijn and Arne Leijon at KTH CSTR internal presentation Monday 20 January, 2014 Gustav Eje Henter (CSTR) KDE-HMMs for Speech

886 views • 61 slides

More recommend