The document is an introduction to graphical models. It discusses that graphical models define probability distributions over random variables using graphs to encode conditional independence assumptions. It then describes popular classes of graphical models including directed Bayesian networks and undirected Markov random fields. Bayesian networks define a factorization of the joint distribution over parent variables, while Markov random fields factorize over potentials at cliques in the graph. An example Markov random field is also shown.
Robust Shape and Topology Optimization - Northwestern Altair
A robust shape and topology optimization (RSTO) approach with consideration of random field uncertainty in various sources such as loading, material properties, and geometry has been developed. The approach integrates the state-of-the-art level set methods for shape and topology optimization and the latest research development in design under uncertainty. To characterize the high-dimensional random-field uncertainty with a reduced set of random variables, the Karhunen-Loeve expansion is employed.
How can we apply machine learning techniques on graphs to obtain predictions in a variety of domains? Know more from Sami Abu-El-Haija, an AI Scientist with experience from both industry (Google Research) and academia (University of Southern California).
Aristidis Likas, Associate Professor and Christoforos Nikou, Assistant Professor, University of Ioannina, Department of Computer Science , Mixture Models for Image Analysis
Kernelization algorithms for graph and other structure modification problemsAnthony Perez
Thesis defense on November 14th, 2011, in Montpellier.
Jury:
Stéphane Bessy, Bruno Durand, Frédéric Havet, Rolf Niedermeier, Christophe Paul & Ioan Todinca.
A type system for the vectorial aspects of the linear-algebraic lambda-calculusAlejandro Díaz-Caro
We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms resulting from the reduction of programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We show that the resulting typed lambda-calculus is strongly normalizing and features a weak subject-reduction.
A discussion on sampling graphs to approximate network classification functionsLARCA UPC
The problem of network classification consists on assigning a finite set of labels to the nodes of the graphs; the underlying assumption is that nodes with the same label tend to be connected via strong paths in the graph. This is similar to the assumptions made by semi-supervised learning algorithms based on graphs, which build an artificial graph from vectorial data. Such semi-supervised algorithms are based on label propagation principles and their accuracy heavily relies on the structure (presence of edges) in the graph.
In this talk I will discuss ideas of how to perform sampling in the network graph, thus sparsifying the structure in order to apply semi-supervised algorithms and compute efficiently the classification function on the network. I will show very preliminary experiments indicating that the sampling technique has an important effect on the final results and discuss open theoretical and practical questions that are to be solved yet.
Robust Shape and Topology Optimization - Northwestern Altair
A robust shape and topology optimization (RSTO) approach with consideration of random field uncertainty in various sources such as loading, material properties, and geometry has been developed. The approach integrates the state-of-the-art level set methods for shape and topology optimization and the latest research development in design under uncertainty. To characterize the high-dimensional random-field uncertainty with a reduced set of random variables, the Karhunen-Loeve expansion is employed.
How can we apply machine learning techniques on graphs to obtain predictions in a variety of domains? Know more from Sami Abu-El-Haija, an AI Scientist with experience from both industry (Google Research) and academia (University of Southern California).
Aristidis Likas, Associate Professor and Christoforos Nikou, Assistant Professor, University of Ioannina, Department of Computer Science , Mixture Models for Image Analysis
Kernelization algorithms for graph and other structure modification problemsAnthony Perez
Thesis defense on November 14th, 2011, in Montpellier.
Jury:
Stéphane Bessy, Bruno Durand, Frédéric Havet, Rolf Niedermeier, Christophe Paul & Ioan Todinca.
A type system for the vectorial aspects of the linear-algebraic lambda-calculusAlejandro Díaz-Caro
We describe a type system for the linear-algebraic lambda-calculus. The type system accounts for the part of the language emulating linear operators and vectors, i.e. it is able to statically describe the linear combinations of terms resulting from the reduction of programs. This gives rise to an original type theory where types, in the same way as terms, can be superposed into linear combinations. We show that the resulting typed lambda-calculus is strongly normalizing and features a weak subject-reduction.
A discussion on sampling graphs to approximate network classification functionsLARCA UPC
The problem of network classification consists on assigning a finite set of labels to the nodes of the graphs; the underlying assumption is that nodes with the same label tend to be connected via strong paths in the graph. This is similar to the assumptions made by semi-supervised learning algorithms based on graphs, which build an artificial graph from vectorial data. Such semi-supervised algorithms are based on label propagation principles and their accuracy heavily relies on the structure (presence of edges) in the graph.
In this talk I will discuss ideas of how to perform sampling in the network graph, thus sparsifying the structure in order to apply semi-supervised algorithms and compute efficiently the classification function on the network. I will show very preliminary experiments indicating that the sampling technique has an important effect on the final results and discuss open theoretical and practical questions that are to be solved yet.
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 3.
More info at http://summerschool.ssa.org.ua
In many applications one observes rapid change of the solution in the boundary region. Accurate and numerically efficient resolution of the solution close to the moving boundaries is considered to be an important problem. We develop an approach to the optimization of the discretization grids for finite-difference scheme. Using the suggested approach we are able to achieve the exponential convergence of the boundary Neumann- to-Dirichlet maps. It increases the convergence order without increasing the stencil size of the finite-difference scheme and preserves stability.
Kernel based models for geo- and environmental sciences- Alexei Pozdnoukhov –...Beniamino Murgante
Kernel based models for geo- and environmental sciences- Alexei Pozdnoukhov – National Centre for Geocomputation, National University of Ireland , Maynooth (Ireland)
Intelligent Analysis of Environmental Data (S4 ENVISA Workshop 2009)
Basics of probability in statistical simulation and stochastic programmingSSA KPI
AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 2.
More info at http://summerschool.ssa.org.ua
GraphRAG is All You need? LLM & Knowledge GraphGuy Korland
Guy Korland, CEO and Co-founder of FalkorDB, will review two articles on the integration of language models with knowledge graphs.
1. Unifying Large Language Models and Knowledge Graphs: A Roadmap.
https://arxiv.org/abs/2306.08302
2. Microsoft Research's GraphRAG paper and a review paper on various uses of knowledge graphs:
https://www.microsoft.com/en-us/research/blog/graphrag-unlocking-llm-discovery-on-narrative-private-data/
Communications Mining Series - Zero to Hero - Session 1DianaGray10
This session provides introduction to UiPath Communication Mining, importance and platform overview. You will acquire a good understand of the phases in Communication Mining as we go over the platform with you. Topics covered:
• Communication Mining Overview
• Why is it important?
• How can it help today’s business and the benefits
• Phases in Communication Mining
• Demo on Platform overview
• Q/A
The Art of the Pitch: WordPress Relationships and SalesLaura Byrne
Clients don’t know what they don’t know. What web solutions are right for them? How does WordPress come into the picture? How do you make sure you understand scope and timeline? What do you do if sometime changes?
All these questions and more will be explored as we talk about matching clients’ needs with what your agency offers without pulling teeth or pulling your hair out. Practical tips, and strategies for successful relationship building that leads to closing the deal.
UiPath Test Automation using UiPath Test Suite series, part 4DianaGray10
Welcome to UiPath Test Automation using UiPath Test Suite series part 4. In this session, we will cover Test Manager overview along with SAP heatmap.
The UiPath Test Manager overview with SAP heatmap webinar offers a concise yet comprehensive exploration of the role of a Test Manager within SAP environments, coupled with the utilization of heatmaps for effective testing strategies.
Participants will gain insights into the responsibilities, challenges, and best practices associated with test management in SAP projects. Additionally, the webinar delves into the significance of heatmaps as a visual aid for identifying testing priorities, areas of risk, and resource allocation within SAP landscapes. Through this session, attendees can expect to enhance their understanding of test management principles while learning practical approaches to optimize testing processes in SAP environments using heatmap visualization techniques
What will you get from this session?
1. Insights into SAP testing best practices
2. Heatmap utilization for testing
3. Optimization of testing processes
4. Demo
Topics covered:
Execution from the test manager
Orchestrator execution result
Defect reporting
SAP heatmap example with demo
Speaker:
Deepak Rai, Automation Practice Lead, Boundaryless Group and UiPath MVP
Sudheer Mechineni, Head of Application Frameworks, Standard Chartered Bank
Discover how Standard Chartered Bank harnessed the power of Neo4j to transform complex data access challenges into a dynamic, scalable graph database solution. This keynote will cover their journey from initial adoption to deploying a fully automated, enterprise-grade causal cluster, highlighting key strategies for modelling organisational changes and ensuring robust disaster recovery. Learn how these innovations have not only enhanced Standard Chartered Bank’s data infrastructure but also positioned them as pioneers in the banking sector’s adoption of graph technology.
GDG Cloud Southlake #33: Boule & Rebala: Effective AppSec in SDLC using Deplo...James Anderson
Effective Application Security in Software Delivery lifecycle using Deployment Firewall and DBOM
The modern software delivery process (or the CI/CD process) includes many tools, distributed teams, open-source code, and cloud platforms. Constant focus on speed to release software to market, along with the traditional slow and manual security checks has caused gaps in continuous security as an important piece in the software supply chain. Today organizations feel more susceptible to external and internal cyber threats due to the vast attack surface in their applications supply chain and the lack of end-to-end governance and risk management.
The software team must secure its software delivery process to avoid vulnerability and security breaches. This needs to be achieved with existing tool chains and without extensive rework of the delivery processes. This talk will present strategies and techniques for providing visibility into the true risk of the existing vulnerabilities, preventing the introduction of security issues in the software, resolving vulnerabilities in production environments quickly, and capturing the deployment bill of materials (DBOM).
Speakers:
Bob Boule
Robert Boule is a technology enthusiast with PASSION for technology and making things work along with a knack for helping others understand how things work. He comes with around 20 years of solution engineering experience in application security, software continuous delivery, and SaaS platforms. He is known for his dynamic presentations in CI/CD and application security integrated in software delivery lifecycle.
Gopinath Rebala
Gopinath Rebala is the CTO of OpsMx, where he has overall responsibility for the machine learning and data processing architectures for Secure Software Delivery. Gopi also has a strong connection with our customers, leading design and architecture for strategic implementations. Gopi is a frequent speaker and well-known leader in continuous delivery and integrating security into software delivery.
Epistemic Interaction - tuning interfaces to provide information for AI supportAlan Dix
Paper presented at SYNERGY workshop at AVI 2024, Genoa, Italy. 3rd June 2024
https://alandix.com/academic/papers/synergy2024-epistemic/
As machine learning integrates deeper into human-computer interactions, the concept of epistemic interaction emerges, aiming to refine these interactions to enhance system adaptability. This approach encourages minor, intentional adjustments in user behaviour to enrich the data available for system learning. This paper introduces epistemic interaction within the context of human-system communication, illustrating how deliberate interaction design can improve system understanding and adaptation. Through concrete examples, we demonstrate the potential of epistemic interaction to significantly advance human-computer interaction by leveraging intuitive human communication strategies to inform system design and functionality, offering a novel pathway for enriching user-system engagements.
Dr. Sean Tan, Head of Data Science, Changi Airport Group
Discover how Changi Airport Group (CAG) leverages graph technologies and generative AI to revolutionize their search capabilities. This session delves into the unique search needs of CAG’s diverse passengers and customers, showcasing how graph data structures enhance the accuracy and relevance of AI-generated search results, mitigating the risk of “hallucinations” and improving the overall customer journey.
Why You Should Replace Windows 11 with Nitrux Linux 3.5.0 for enhanced perfor...SOFTTECHHUB
The choice of an operating system plays a pivotal role in shaping our computing experience. For decades, Microsoft's Windows has dominated the market, offering a familiar and widely adopted platform for personal and professional use. However, as technological advancements continue to push the boundaries of innovation, alternative operating systems have emerged, challenging the status quo and offering users a fresh perspective on computing.
One such alternative that has garnered significant attention and acclaim is Nitrux Linux 3.5.0, a sleek, powerful, and user-friendly Linux distribution that promises to redefine the way we interact with our devices. With its focus on performance, security, and customization, Nitrux Linux presents a compelling case for those seeking to break free from the constraints of proprietary software and embrace the freedom and flexibility of open-source computing.
A tale of scale & speed: How the US Navy is enabling software delivery from l...sonjaschweigert1
Rapid and secure feature delivery is a goal across every application team and every branch of the DoD. The Navy’s DevSecOps platform, Party Barge, has achieved:
- Reduction in onboarding time from 5 weeks to 1 day
- Improved developer experience and productivity through actionable findings and reduction of false positives
- Maintenance of superior security standards and inherent policy enforcement with Authorization to Operate (ATO)
Development teams can ship efficiently and ensure applications are cyber ready for Navy Authorizing Officials (AOs). In this webinar, Sigma Defense and Anchore will give attendees a look behind the scenes and demo secure pipeline automation and security artifacts that speed up application ATO and time to production.
We will cover:
- How to remove silos in DevSecOps
- How to build efficient development pipeline roles and component templates
- How to deliver security artifacts that matter for ATO’s (SBOMs, vulnerability reports, and policy evidence)
- How to streamline operations with automated policy checks on container images
Essentials of Automations: The Art of Triggers and Actions in FMESafe Software
In this second installment of our Essentials of Automations webinar series, we’ll explore the landscape of triggers and actions, guiding you through the nuances of authoring and adapting workspaces for seamless automations. Gain an understanding of the full spectrum of triggers and actions available in FME, empowering you to enhance your workspaces for efficient automation.
We’ll kick things off by showcasing the most commonly used event-based triggers, introducing you to various automation workflows like manual triggers, schedules, directory watchers, and more. Plus, see how these elements play out in real scenarios.
Whether you’re tweaking your current setup or building from the ground up, this session will arm you with the tools and insights needed to transform your FME usage into a powerhouse of productivity. Join us to discover effective strategies that simplify complex processes, enhancing your productivity and transforming your data management practices with FME. Let’s turn complexity into clarity and make your workspaces work wonders!
LF Energy Webinar: Electrical Grid Modelling and Simulation Through PowSyBl -...DanBrown980551
Do you want to learn how to model and simulate an electrical network from scratch in under an hour?
Then welcome to this PowSyBl workshop, hosted by Rte, the French Transmission System Operator (TSO)!
During the webinar, you will discover the PowSyBl ecosystem as well as handle and study an electrical network through an interactive Python notebook.
PowSyBl is an open source project hosted by LF Energy, which offers a comprehensive set of features for electrical grid modelling and simulation. Among other advanced features, PowSyBl provides:
- A fully editable and extendable library for grid component modelling;
- Visualization tools to display your network;
- Grid simulation tools, such as power flows, security analyses (with or without remedial actions) and sensitivity analyses;
The framework is mostly written in Java, with a Python binding so that Python developers can access PowSyBl functionalities as well.
What you will learn during the webinar:
- For beginners: discover PowSyBl's functionalities through a quick general presentation and the notebook, without needing any expert coding skills;
- For advanced developers: master the skills to efficiently apply PowSyBl functionalities to your real-world scenarios.
In his public lecture, Christian Timmerer provides insights into the fascinating history of video streaming, starting from its humble beginnings before YouTube to the groundbreaking technologies that now dominate platforms like Netflix and ORF ON. Timmerer also presents provocative contributions of his own that have significantly influenced the industry. He concludes by looking at future challenges and invites the audience to join in a discussion.
Threats to mobile devices are more prevalent and increasing in scope and complexity. Users of mobile devices desire to take full advantage of the features
available on those devices, but many of the features provide convenience and capability but sacrifice security. This best practices guide outlines steps the users can take to better protect personal devices and information.
Generative AI Deep Dive: Advancing from Proof of Concept to ProductionAggregage
Join Maher Hanafi, VP of Engineering at Betterworks, in this new session where he'll share a practical framework to transform Gen AI prototypes into impactful products! He'll delve into the complexities of data collection and management, model selection and optimization, and ensuring security, scalability, and responsible use.
Removing Uninteresting Bytes in Software FuzzingAftab Hussain
Imagine a world where software fuzzing, the process of mutating bytes in test seeds to uncover hidden and erroneous program behaviors, becomes faster and more effective. A lot depends on the initial seeds, which can significantly dictate the trajectory of a fuzzing campaign, particularly in terms of how long it takes to uncover interesting behaviour in your code. We introduce DIAR, a technique designed to speedup fuzzing campaigns by pinpointing and eliminating those uninteresting bytes in the seeds. Picture this: instead of wasting valuable resources on meaningless mutations in large, bloated seeds, DIAR removes the unnecessary bytes, streamlining the entire process.
In this work, we equipped AFL, a popular fuzzer, with DIAR and examined two critical Linux libraries -- Libxml's xmllint, a tool for parsing xml documents, and Binutil's readelf, an essential debugging and security analysis command-line tool used to display detailed information about ELF (Executable and Linkable Format). Our preliminary results show that AFL+DIAR does not only discover new paths more quickly but also achieves higher coverage overall. This work thus showcases how starting with lean and optimized seeds can lead to faster, more comprehensive fuzzing campaigns -- and DIAR helps you find such seeds.
- These are slides of the talk given at IEEE International Conference on Software Testing Verification and Validation Workshop, ICSTW 2022.
1. Graphical Models Factor Graphs Test-time Inference Training
Part 2: Introduction to Graphical Models
Sebastian Nowozin and Christoph H. Lampert
Colorado Springs, 25th June 2011
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
2. Graphical Models Factor Graphs Test-time Inference Training
Graphical Models
Introduction
Model: relating observations x to
quantities of interest y
f
Example 1: given RGB image x, infer
depth y for each pixel
Example 2: given RGB image x, infer X Y
presence and positions y of all objects f :X →Y
shown
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
3. Graphical Models Factor Graphs Test-time Inference Training
Graphical Models
Introduction
Model: relating observations x to
quantities of interest y
f
Example 1: given RGB image x, infer
depth y for each pixel
Example 2: given RGB image x, infer X Y
presence and positions y of all objects f :X →Y
shown
X : image, Y: object annotations
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
4. Graphical Models Factor Graphs Test-time Inference Training
Graphical Models
Introduction
General case: mapping x ∈ X to y ∈ Y
Graphical models are a concise
language to define this mapping x
Mapping can be ambiguous:
f (x)
measurement noise, lack of X Y
well-posedness (e.g. occlusions) f :X →Y
Probabilistic graphical models: define
form p(y |x) or p(x, y ) for all y ∈ Y
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
5. Graphical Models Factor Graphs Test-time Inference Training
Graphical Models
Introduction
General case: mapping x ∈ X to y ∈ Y
Graphical models are a concise ?
language to define this mapping x
Mapping can be ambiguous: ?
measurement noise, lack of X Y
well-posedness (e.g. occlusions) p(Y |X = x)
Probabilistic graphical models: define
form p(y |x) or p(x, y ) for all y ∈ Y
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
6. Graphical Models Factor Graphs Test-time Inference Training
Graphical Models
Graphical Models
A graphical model defines
a family of probability distributions over a set of random variables,
by means of a graph,
so that the random variables satisfy conditional independence
assumptions encoded in the graph.
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
7. Graphical Models Factor Graphs Test-time Inference Training
Graphical Models
Graphical Models
A graphical model defines
a family of probability distributions over a set of random variables,
by means of a graph,
so that the random variables satisfy conditional independence
assumptions encoded in the graph.
Popular classes of graphical models,
Undirected graphical models (Markov
random fields),
Directed graphical models (Bayesian
networks),
Factor graphs,
Others: chain graphs, influence
diagrams, etc.
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
8. Graphical Models Factor Graphs Test-time Inference Training
Graphical Models
Bayesian Networks
Graph: G = (V , E), E ⊂ V × V Yi Yj
directed
acyclic
Variable domains Yi Yk
Factorization
p(Y = y ) = p(yi |ypaG (i) )
Yl
i∈V
over distributions, by conditioning on parent A simple Bayes net
nodes.
Example
p(Y = y ) =p(Yl = yl |Yk = yk )p(Yk = yk |Yi = yi , Yj = yj )
p(Yi = yi )p(Yj = yj ).
Sebastian Nowozin and Christoph H. Lampert
Family of distributions
Part 2: Introduction to Graphical Models
9. Graphical Models Factor Graphs Test-time Inference Training
Graphical Models
Bayesian Networks
Graph: G = (V , E), E ⊂ V × V Yi Yj
directed
acyclic
Variable domains Yi Yk
Factorization
p(Y = y ) = p(yi |ypaG (i) )
Yl
i∈V
over distributions, by conditioning on parent A simple Bayes net
nodes.
Example
p(Y = y ) =p(Yl = yl |Yk = yk )p(Yk = yk |Yi = yi , Yj = yj )
p(Yi = yi )p(Yj = yj ).
Sebastian Nowozin and Christoph H. Lampert
Family of distributions
Part 2: Introduction to Graphical Models
10. Graphical Models Factor Graphs Test-time Inference Training
Graphical Models
Undirected Graphical Models
Yi Yj Yk
= Markov random field (MRF) = Markov
network A simple MRF
Graph: G = (V , E), E ⊂ V × V
undirected, no self-edges
Variable domains Yi
Factorization over potentials ψ at cliques,
1
p(y ) = ψC (yC )
Z
C ∈C(G )
Constant Z = y ∈Y C ∈C(G ) ψC (yC )
Example
1
p(y ) = ψi (yi )ψj (yj )ψl (yl )ψi,j (yi , yj )
Z
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
11. Graphical Models Factor Graphs Test-time Inference Training
Graphical Models
Undirected Graphical Models
Yi Yj Yk
= Markov random field (MRF) = Markov
network A simple MRF
Graph: G = (V , E), E ⊂ V × V
undirected, no self-edges
Variable domains Yi
Factorization over potentials ψ at cliques,
1
p(y ) = ψC (yC )
Z
C ∈C(G )
Constant Z = y ∈Y C ∈C(G ) ψC (yC )
Example
1
p(y ) = ψi (yi )ψj (yj )ψl (yl )ψi,j (yi , yj )
Z
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
12. Graphical Models Factor Graphs Test-time Inference Training
Graphical Models
Example 1
Yi Yj Yk
Cliques C(G ): set of vertex sets V with V ⊆ V ,
E ∩ (V × V ) = V × V
Here C(G ) = {{i}, {i, j}, {j}, {j, k}, {k}}
1
p(y ) = ψi (yi )ψj (yj )ψl (yl )ψi,j (yi , yj )
Z
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
13. Graphical Models Factor Graphs Test-time Inference Training
Graphical Models
Example 2
Yi Yj
Yk Yl
Here C(G ) = 2V : all subsets of V are cliques
1
p(y ) = ψA (yA ).
Z
A∈2{i,j,k,l}
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
14. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Factor Graphs
Graph: G = (V , F, E), E ⊆ V × F Yi Yj
variable nodes V ,
factor nodes F ,
edges E between variable and factor nodes.
scope of a factor,
N(F ) = {i ∈ V : (i, F ) ∈ E}
Yk Yl
Variable domains Yi
Factorization over potentials ψ at factors, Factor graph
1
p(y ) = ψF (yN(F ) )
Z
F ∈F
Constant Z = y ∈Y F ∈F ψF (yN(F ) )
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
15. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Factor Graphs
Graph: G = (V , F, E), E ⊆ V × F Yi Yj
variable nodes V ,
factor nodes F ,
edges E between variable and factor nodes.
scope of a factor,
N(F ) = {i ∈ V : (i, F ) ∈ E}
Yk Yl
Variable domains Yi
Factorization over potentials ψ at factors, Factor graph
1
p(y ) = ψF (yN(F ) )
Z
F ∈F
Constant Z = y ∈Y F ∈F ψF (yN(F ) )
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
16. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Why factor graphs?
Yi Yj Yi Yj Yi Yj
Yk Yl Yk Yl Yk Yl
Factor graphs are explicit about the factorization
Hence, easier to work with
Universal (just like MRFs and Bayesian networks)
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
17. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Capacity
Yi Yj Yi Yj
Yk Yl Yk Yl
Factor graph defines family of distributions
Some families are larger than others
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
18. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Four remaining pieces
1. Conditional distributions (CRFs)
2. Parameterization
3. Test-time inference
4. Learning the model from training data
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
19. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Four remaining pieces
1. Conditional distributions (CRFs)
2. Parameterization
3. Test-time inference
4. Learning the model from training data
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
20. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Conditional Distributions
We have discussed p(y ), Xi Xj
How do we define p(y |x)?
Potentials become a function of xN(F )
Partition function depends on x
Yi Yj
Conditional random fields (CRFs)
x is not part of the probability model, i.e. not conditional
treated as random variable distribution
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
21. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Conditional Distributions
We have discussed p(y ), Xi Xj
How do we define p(y |x)?
Potentials become a function of xN(F )
Partition function depends on x
Yi Yj
Conditional random fields (CRFs)
x is not part of the probability model, i.e. not conditional
treated as random variable distribution
1
p(y ) = ψF (yN(F ) )
Z
F ∈F
1
p(y |x) = ψF (yN(F ) ; xN(F ) )
Z (x)
F ∈F
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
22. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Conditional Distributions
We have discussed p(y ), Xi Xj
How do we define p(y |x)?
Potentials become a function of xN(F )
Partition function depends on x
Yi Yj
Conditional random fields (CRFs)
x is not part of the probability model, i.e. not conditional
treated as random variable distribution
1
p(y ) = ψF (yN(F ) )
Z
F ∈F
1
p(y |x) = ψF (yN(F ) ; xN(F ) )
Z (x)
F ∈F
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
23. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Potentials and Energy Functions
For each factor F ∈ F, YF = ×
i∈N(F )
Yi ,
EF : YN(F ) → R,
Potentials and energies (assume ψF (yF ) > 0)
ψF (yF ) = exp(−EF (yF )), and EF (yF ) = − log(ψF (yF )).
Then p(y ) can be written as
1
p(Y = y ) = ψF (yF )
Z
F ∈F
1
= exp(− EF (yF )),
Z
F ∈F
Hence, p(y ) is completely determined by E (y ) = F ∈F EF (yF )
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
24. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Potentials and Energy Functions
For each factor F ∈ F, YF = ×
i∈N(F )
Yi ,
EF : YN(F ) → R,
Potentials and energies (assume ψF (yF ) > 0)
ψF (yF ) = exp(−EF (yF )), and EF (yF ) = − log(ψF (yF )).
Then p(y ) can be written as
1
p(Y = y ) = ψF (yF )
Z
F ∈F
1
= exp(− EF (yF )),
Z
F ∈F
Hence, p(y ) is completely determined by E (y ) = F ∈F EF (yF )
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
25. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Potentials and Energy Functions
For each factor F ∈ F, YF = ×
i∈N(F )
Yi ,
EF : YN(F ) → R,
Potentials and energies (assume ψF (yF ) > 0)
ψF (yF ) = exp(−EF (yF )), and EF (yF ) = − log(ψF (yF )).
Then p(y ) can be written as
1
p(Y = y ) = ψF (yF )
Z
F ∈F
1
= exp(− EF (yF )),
Z
F ∈F
Hence, p(y ) is completely determined by E (y ) = F ∈F EF (yF )
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
26. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Energy Minimization
1
argmax p(Y = y ) = argmax exp(− EF (yF ))
y ∈Y y ∈Y Z
F ∈F
= argmax exp(− EF (yF ))
y ∈Y
F ∈F
= argmax − EF (yF )
y ∈Y
F ∈F
= argmin EF (yF )
y ∈Y
F ∈F
= argmin E (y ).
y ∈Y
Energy minimization can be interpreted as solving for the most likely
state of some factor graph model
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
27. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Parameterization
Factor graphs define a family of distributions
Parameterization: identifying individual members by parameters w
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
28. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Parameterization
Factor graphs define a family of distributions
Parameterization: identifying individual members by parameters w
distributions
indexed
by w pw1
pw2
distributions
in family
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
29. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Example: Parameterization
Image segmentation model
Pairwise “Potts” energy function
EF (yi , yj ; w1 ),
EF : {0, 1} × {0, 1} × R → R,
EF (0, 0; w1 ) = EF (1, 1; w1 ) = 0 image segmentation model
EF (0, 1; w1 ) = EF (1, 0; w1 ) = w1
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
30. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Example: Parameterization (cont)
Image segmentation model
Unary energy function EF (yi ; x, w ),
EF : {0, 1} × X × R{0,1}×D → R,
EF (0; x, w ) = w (0), ψF (x)
EF (1; x, w ) = w (1), ψF (x) image segmentation model
Features ψF : X → RD , e.g. image
filters
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
31. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Example: Parameterization (cont)
w(0), ψF (x)
... ... ...
w(1), ψF (x)
...
0 w1
w1 0
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
32. Graphical Models Factor Graphs Test-time Inference Training
Factor Graphs
Example: Parameterization (cont)
w(0), ψF (x)
... ... ...
w(1), ψF (x)
...
0 w1
w1 0
Total number of parameters: D + D + 1
Parameters are shared, but energies differ because of different ψF (x)
General form, linear in w ,
EF (yF ; xF , w ) = w (yF ), ψF (xF )
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
33. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Making Predictions
Making predictions: given x ∈ X , predict y ∈ Y
How to measure quality of prediction? (or function f : X → Y)
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
34. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Loss function
Define a loss function
∆ : Y × Y → R+ ,
so that ∆(y , y ∗ ) measures the loss incurred by predicting y when y ∗
is true.
The loss function is application dependent
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
35. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Test-time Inference
Loss function ∆(y , f (x)): correct label y , predict f (x)
∆:Y ×Y →R
True joint distribution d(X , Y ) and true conditional d(y |x)
Model distribution p(y |x)
Expected loss: quality of prediction
R∆ (x)
f = Ey ∼d(y |x) ∆(y , f (x))
= d(y |x) ∆(y , f (x)).
y ∈Y
≈ Ey ∼p(y |x;w ) ∆(y , f (x))
Assuming that p(y |x; w ) ≈ d(y |x)
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
36. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Test-time Inference
Loss function ∆(y , f (x)): correct label y , predict f (x)
∆:Y ×Y →R
True joint distribution d(X , Y ) and true conditional d(y |x)
Model distribution p(y |x)
Expected loss: quality of prediction
R∆ (x)
f = Ey ∼d(y |x) ∆(y , f (x))
= d(y |x) ∆(y , f (x)).
y ∈Y
≈ Ey ∼p(y |x;w ) ∆(y , f (x))
Assuming that p(y |x; w ) ≈ d(y |x)
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
37. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Test-time Inference
Loss function ∆(y , f (x)): correct label y , predict f (x)
∆:Y ×Y →R
True joint distribution d(X , Y ) and true conditional d(y |x)
Model distribution p(y |x)
Expected loss: quality of prediction
R∆ (x)
f = Ey ∼d(y |x) ∆(y , f (x))
= d(y |x) ∆(y , f (x)).
y ∈Y
≈ Ey ∼p(y |x;w ) ∆(y , f (x))
Assuming that p(y |x; w ) ≈ d(y |x)
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
38. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Example 1: 0/1 loss
Loss 0 iff perfectly predicted, 1 otherwise:
0 if y = y ∗
∆0/1 (y , y ∗ ) = I (y = y ∗ ) =
1 otherwise
Plugging it in,
y∗ := argmin Ey ∼p(y |x) ∆0/1 (y , y )
y ∈Y
= argmax p(y |x)
y ∈Y
= argmin E (y , x).
y ∈Y
Minimizing the expected 0/1-loss → MAP prediction (energy
minimization)
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
39. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Example 1: 0/1 loss
Loss 0 iff perfectly predicted, 1 otherwise:
0 if y = y ∗
∆0/1 (y , y ∗ ) = I (y = y ∗ ) =
1 otherwise
Plugging it in,
y∗ := argmin Ey ∼p(y |x) ∆0/1 (y , y )
y ∈Y
= argmax p(y |x)
y ∈Y
= argmin E (y , x).
y ∈Y
Minimizing the expected 0/1-loss → MAP prediction (energy
minimization)
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
40. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Example 2: Hamming loss
Count the number of mislabeled variables:
1
∆H (y , y ∗ ) = I (yi = yi∗ )
|V |
i∈V
Plugging it in,
y∗ := argmin Ey ∼p(y |x) [∆H (y , y )]
y ∈Y
= argmax p(yi |x)
yi ∈Yi
i∈V
Minimizing the expected Hamming loss → maximum posterior
marginal (MPM, Max-Marg) prediction
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
41. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Example 2: Hamming loss
Count the number of mislabeled variables:
1
∆H (y , y ∗ ) = I (yi = yi∗ )
|V |
i∈V
Plugging it in,
y∗ := argmin Ey ∼p(y |x) [∆H (y , y )]
y ∈Y
= argmax p(yi |x)
yi ∈Yi
i∈V
Minimizing the expected Hamming loss → maximum posterior
marginal (MPM, Max-Marg) prediction
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
42. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Example 3: Squared error
Assume a vector space on Yi (pixel intensities,
optical flow vectors, etc.).
Sum of squared errors
1
∆Q (y , y ∗ ) = yi − yi∗ 2 .
|V |
i∈V
Plugging it in,
y∗ := argmin Ey ∼p(y |x) [∆Q (y , y )]
y ∈Y
= p(yi |x)yi
yi ∈Yi
i∈V
Minimizing the expected squared error → minimum mean squared
error (MMSE) prediction
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
43. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Example 3: Squared error
Assume a vector space on Yi (pixel intensities,
optical flow vectors, etc.).
Sum of squared errors
1
∆Q (y , y ∗ ) = yi − yi∗ 2 .
|V |
i∈V
Plugging it in,
y∗ := argmin Ey ∼p(y |x) [∆Q (y , y )]
y ∈Y
= p(yi |x)yi
yi ∈Yi
i∈V
Minimizing the expected squared error → minimum mean squared
error (MMSE) prediction
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
44. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Inference Task: Maximum A Posteriori (MAP) Inference
Definition (Maximum A Posteriori (MAP) Inference)
Given a factor graph, parameterization, and weight vector w , and given
the observation x, find
y ∗ = argmax p(Y = y |x, w ) = argmin E (y ; x, w ).
y ∈Y y ∈Y
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
45. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Inference Task: Probabilistic Inference
Definition (Probabilistic Inference)
Given a factor graph, parameterization, and weight vector w , and given
the observation x, find
log Z (x, w ) = log exp(−E (y ; x, w )),
y ∈Y
µF (yF ) = p(YF = yf |x, w ), ∀F ∈ F, ∀yF ∈ YF .
This typically includes variable marginals
µi (yi ) = p(yi |x, w )
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
46. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Example: Man-made structure detection
Xi
ψi2
Yi 3
ψi,k Yk
ψi1
Left: input image x,
Middle: ground truth labeling on 16-by-16 pixel blocks,
Right: factor graph model
Features: gradient and color histograms
Estimate model parameters from ≈ 60 training images
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
47. Graphical Models Factor Graphs Test-time Inference Training
Test-time Inference
Example: Man-made structure detection
Left: input image x,
Middle (probabilistic inference): visualization of the variable
marginals p(yi = “manmade |x, w ),
Right (MAP inference): joint MAP labeling
y ∗ = argmaxy ∈Y p(y |x, w ).
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
48. Graphical Models Factor Graphs Test-time Inference Training
Training
Training the Model
What can be learned?
Model structure: factors
Model variables: observed variables fixed, but we can add
unobserved variables
Factor energies: parameters
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
49. Graphical Models Factor Graphs Test-time Inference Training
Training
Training the Model
What can be learned?
Model structure: factors
Model variables: observed variables fixed, but we can add
unobserved variables
Factor energies: parameters
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
50. Graphical Models Factor Graphs Test-time Inference Training
Training
Training: Overview
Assume a fully observed, independent and identically distributed
(iid) sample set
{(x n , y n )}n=1,...,N , (x n , y n ) ∼ d(X , Y )
Goal: predict well,
Alternative goal: first model d(y |x) well by p(y |x, w ), then predict
by minimizing the expected loss
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
51. Graphical Models Factor Graphs Test-time Inference Training
Training
Probabilistic Learning
Problem (Probabilistic Parameter Learning)
Let d(y |x) be the (unknown) conditional distribution of labels for a
problem to be solved. For a parameterized conditional distribution
p(y |x, w ) with parameters w ∈ RD , probabilistic parameter learning is
the task of finding a point estimate of the parameter w ∗ that makes
p(y |x, w ∗ ) closest to d(y |x).
We will discuss probabilistic parameter learning in detail.
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
52. Graphical Models Factor Graphs Test-time Inference Training
Training
Probabilistic Learning
Problem (Probabilistic Parameter Learning)
Let d(y |x) be the (unknown) conditional distribution of labels for a
problem to be solved. For a parameterized conditional distribution
p(y |x, w ) with parameters w ∈ RD , probabilistic parameter learning is
the task of finding a point estimate of the parameter w ∗ that makes
p(y |x, w ∗ ) closest to d(y |x).
We will discuss probabilistic parameter learning in detail.
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
53. Graphical Models Factor Graphs Test-time Inference Training
Training
Loss-Minimizing Parameter Learning
Problem (Loss-Minimizing Parameter Learning)
Let d(x, y ) be the unknown distribution of data in labels, and let
∆ : Y × Y → R be a loss function. Loss minimizing parameter learning is
the task of finding a parameter value w ∗ such that the expected
prediction risk
E(x,y )∼d(x,y ) [∆(y , fp (x))]
is as small as possible, where fp (x) = argmaxy ∈Y p(y |x, w ∗ ).
Requires loss function at training time
Directly learns a prediction function fp (x)
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models
54. Graphical Models Factor Graphs Test-time Inference Training
Training
Loss-Minimizing Parameter Learning
Problem (Loss-Minimizing Parameter Learning)
Let d(x, y ) be the unknown distribution of data in labels, and let
∆ : Y × Y → R be a loss function. Loss minimizing parameter learning is
the task of finding a parameter value w ∗ such that the expected
prediction risk
E(x,y )∼d(x,y ) [∆(y , fp (x))]
is as small as possible, where fp (x) = argmaxy ∈Y p(y |x, w ∗ ).
Requires loss function at training time
Directly learns a prediction function fp (x)
Sebastian Nowozin and Christoph H. Lampert
Part 2: Introduction to Graphical Models