Markov Logic Networks A logical KB is a set of hard constraints on the set of possible worlds
Let’s make them soft constraints: When a world violates a formula, It becomes less probable, not impossible
Give each formula a weight (Higher weight Stronger constraint)

Definition:

Definition A Markov Logic Network (MLN) is a set of pairs (F, w) where
F is a formula in first-order logic
w is a real number
Together with a finite set of constants, it defines a Markov network with
One node for each grounding of each predicate in the MLN
One feature for each grounding of each formula F in the MLN, with the corresponding weight w

Example of an MLN:

Example of an MLN Cancer(A) Smokes(A) Smokes(B) Cancer(B) Suppose we have two constants: Anna (A) and Bob (B)

Example of an MLN:

Example of an MLN Cancer(A) Smokes(A) Friends(A,A) Friends(B,A) Smokes(B) Friends(A,B) Cancer(B) Friends(B,B) Suppose we have two constants: Anna (A) and Bob (B)

Example of an MLN:

Example of an MLN Cancer(A) Smokes(A) Friends(A,A) Friends(B,A) Smokes(B) Friends(A,B) Cancer(B) Friends(B,B) Suppose we have two constants: Anna (A) and Bob (B)

Example of an MLN:

Example of an MLN Cancer(A) Smokes(A) Friends(A,A) Friends(B,A) Smokes(B) Friends(A,B) Cancer(B) Friends(B,B) Suppose we have two constants: Anna (A) and Bob (B)

More on MLNs:

More on MLNs Graph structure: Arc between two nodes iff predicates appear together in some formula
MLN is template for ground Markov nets
Typed variables and constants greatly reduce size of ground Markov net
Functions, existential quantifiers, etc.
MLN without variables = Markov network (subsumes graphical models)

MLNs and First-Order Logic:

MLNs and First-Order Logic Infinite weights First-order logic
Satisfiable KB, positive weights Satisfying assignments = Modes of distribution
MLNs allow contradictions between formulas
How to break KB into formulas?
Adding probability increases degrees of freedom
Knowledge engineering decision
Default: Convert to clausal form

Conditional Inference P(Formula|MLN,C) = ?
MCMC: Sample worlds, check formula holds
P(Formula1|Formula2,MLN,C) = ?
If Formula2 = Conjunction of ground atoms
First construct min subset of network necessary to answer query (generalization of KBMC)
Then apply MCMC

Grounding the Template:

Grounding the Template Initialize Markov net to contain all query preds
For each node in network
Add node’s Markov blanket to network
Remove any evidence nodes
Repeat until done

Markov Chain Monte Carlo Gibbs Sampler
1. Start with an initial assignment to nodes
2. One node at a time, sample node given others
3. Repeat
4. Use samples to compute P(X)
Apply to ground network
Many modes Multiple chains
Initialization: MaxWalkSat [Kautz et al., 1997]

MPE Inference:

MPE Inference Find most likely truth values of non-evidence ground atoms given evidence
Apply weighted satisfiability solver (maxes sum of weights of satisfied clauses)
MaxWalkSat algorithm [Kautz et al., 1997]
Start with random truth assignment
With prob p, flip atom that maxes weight sum; else flip random atom in unsatisfied clause
Repeat n times
Restart m times

Learning Data is a relational database
Closed world assumption
Learning structure
Corresponds to feature induction in Markov nets
Learn / modify clauses
ILP (e.g., CLAUDIEN [De Raedt & Dehaspe, 1997])
Better approach: Stanley will describe
Learning parameters (weights)

Learning Weights:

Learning Weights Like Markov nets, except with parameter tying over groundings of same formula
1st term: # true groundings of formula in DB
2nd term: inference required, as before (slow!) Feature count according to data Feature count according to model

Pseudo-Likelihood [Besag, 1975]:

Pseudo-Likelihood [Besag, 1975]
Likelihood of each ground atom given its Markov blanket in the data
Does not require inference at each step
Optimized using L-BFGS [Liu & Nocedal, 1989]

Gradient ofPseudo-Log-Likelihood:

Most terms not affected by changes in weights
After initial setup, each iteration takes O(# ground predicates x # first-order clauses) Gradient of Pseudo-Log-Likelihood where nsati(x=v) is the number of satisfied groundings of clause i in the training data when x takes value v

Domain University of Washington CSE Dept.
12 first-order predicates: Professor, Student, TaughtBy, AuthorOf, AdvisedBy, etc.
2707 constants divided into 10 types: Person (442), Course (176), Pub. (342), Quarter (20), etc.
4.1 million ground predicates
3380 ground predicates (tuples in database)

Systems Compared:

Systems Compared Hand-built knowledge base (KB)
ILP: CLAUDIEN [De Raedt & Dehaspe, 1997]
Markov logic networks (MLNs)
Using KB
Using CLAUDIEN
Using KB + CLAUDIEN
Bayesian network learner [Heckerman et al., 1995]
Naïve Bayes [Domingos & Pazzani, 1997]

Sample Clauses in KB:

Sample Clauses in KB Students are not professors
Each student has only one advisor
If a student is an author of a paper, so is her advisor
Advanced students only TA courses taught by their advisors
At most one author of a given paper is a professor

Methodology:

Methodology Data split into five areas: AI, graphics, languages, systems, theory
Leave-one-area-out testing
Task: Predict AdvisedBy(x, y)
All Info: Given all other predicates
Partial Info: With Student(x) and Professor(x) missing
Evaluation measures:
Conditional log-likelihood (KB, CLAUDIEN: Run WalkSat 100x to get probabilities)
Area under precision-recall curve

Results:

Results

Results: All Info:

Results: All Info

Results: Partial Info:

Results: Partial Info

Efficiency:

Efficiency Learning time: 16 mins
Time to infer all AdvisedBy predicates:
With complete info: 8 mins
With partial info: 15 mins
(124K Gibbs passes)

Other Applications:

Other Applications UW-CSE task: Link prediction
Collective classification
Link-based clustering
Social network models
Object identification
Etc.

Other SRL Approaches areSpecial Cases of MLNs:

Other SRL Approaches are Special Cases of MLNs Probabilistic relational models (Friedman et al, IJCAI-99)
Stochastic logic programs (Muggleton, SRL-00)
Bayesian logic programs (Kersting & De Raedt, ILP-01)
Relational Markov networks (Taskar et al, UAI-02)
Etc.

Open Problems: Inference:

Open Problems: Inference Lifted inference
Better MCMC (e.g., Swendsen-Wang)
Belief propagation
Selective grounding
Abstraction, summarization, multi-scale
Special cases

Open Problems: Learning:

Open Problems: Learning Discriminative training
Learning and refining structure
Learning with missing info
Faster optimization
Beyond pseudo-likelihood
Learning by reformulation

Open Problems: Applications:

Open Problems: Applications Information extraction & integration
Semantic Web
Social networks
Activity recognition
Parsing with world knowledge
Scene analysis with world knowledge
Etc.

Summary:

Summary Markov logic networks combine first-order logic and Markov networks
Syntax: First-order logic + Weights
Semantics: Templates for Markov networks
Inference: KBMC + MaxWalkSat + MCMC
Learning: ILP + Pseudo-likelihood
SRL problems easily formulated as MLNs
Many open research issues

You do not have the permission to view this presentation. In order to view it, please
contact the author of the presentation.

Send to Blogs and Networks

Processing ....

Premium member

Use HTTPs

HTTPS (Hypertext Transfer Protocol Secure) is a protocol used by Web servers to transfer and display Web content securely. Most web browsers block content or generate a “mixed content” warning when users access web pages via HTTPS that contain embedded content loaded via HTTP. To prevent users from facing this, Use HTTPS option.