A Tutorial on Inference and Learning in Bayesian Networks

Name: A Tutorial on Inference and Learning in Bayesian Networks
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Description: A Tutorial on Inference and Learning in Bayesian Networks

A Tutorial on Inference and Learning in Bayesian Networks
Irina Rish Moninder Singh IBM T.J.Watson Research Center

“Road map” Introduction: Bayesian networks Probabilistic inference
What are BNs: representation, types, etc Why use BNs: Applications (classes) of BNs Information sources, software, etc Probabilistic inference Exact inference Approximate inference Learning Bayesian Networks Learning parameters Learning graph structure Summary

Bayesian Networks Structured, graphical representation of probabilistic relationships between several random variables Explicit representation of conditional independencies Missing arcs encode conditional independence Efficient representation of joint pdf Allows arbitrary queries to be answered P (lung cancer=yes | smoking=no, dyspnoea=yes ) = ?

Example: Printer Troubleshooting (Microsoft Windows 95)
Print Output OK Correct Driver Uncorrupted Printer Path Net Cable Connected Net/Local Printing Printer On and Online Local Port Printer Selected Local Cable Application Output OK Print Spooling On Settings Printer Memory Adequate Network Up Spooled Data OK GDI Data Input OK PC to Printer Transport OK Spool Process OK Net Path OK Local Paper Loaded Local Disk Space Adequate [Heckerman, 95]

Example: Microsoft Pregnancy and Child Care)
[Heckerman, 95]

Independence Assumptions
Lung Cancer Smoking Bronchitis tail-to-tail Tuberculosis Visit to Asia Chest X-ray Head-to-tail Dyspnoea Lung Cancer Bronchitis Head-to-head

Nodes X and Y are d-connected by nodes in Z along a trail from X to Y if every head-to-head node along the trail is in Z or has a descendant in Z every other node along the trail is not in Z Nodes X and Y are d-separated by nodes in Z if they are not d-connected by Z along any trail from X to Y Nodes X and Y are d-separated by Z implies X and Y are conditionally independent given Z

A variable (node) is conditionally independent of its non-descendants given its parents Visit to Asia Smoking Lung Cancer Bronchitis Tuberculosis Chest X-ray Dyspnoea

Cancer Smoking Lung Tumor Diet Serum Calcium Age Gender Exposure to Toxins Cancer is independent of Diet given Exposure to Toxins and Smoking [Breese & Koller, 97]

Thus, the General Product rule for Bayesian Networks is P(X1,X2,…,Xn) = P P(Xi | Pa(Xi)) where Pa(Xi) is the set of parents of Xi n i=1

The Knowledge Acquisition Task
Variables: collectively exhaustive, mutually exclusive values clarity test: value should be knowable in principle Structure if data available, can be learned constructed by hand (using “expert” knowledge) variable ordering matters: causal knowledge usually simplifies Probabilities can be learned from data second decimal usually does not matter; relative probs sensitivity analysis

Fuel Gauge Start Battery TurnOver Variable Order is Important Causal Knowledge Simplifies Construction

Naive Baysian Classifiers [Duda&Hart; Langley 92] Selective Naive Bayesian Classifiers [Langley & Sage 94] Conditional Trees [Geiger 92; Friedman et al 97]

Selective Bayesian Networks [Singh & Provan, 95;96]

What are BNs useful for? Diagnosis: P(cause|symptom)=?
Prediction: P(symptom|cause)=? Classification: P(class|data) Decision-making (given a cost function) Data mining: induce best model from data Medicine Bio-informatics Computer troubleshooting Stock market Text Classification Speech recognition

What are BNs useful for? Cause Effect Cause Effect Decision Making -
Max. Expected Utility Unknown but important Imperfect Observations Value Decision Known Predisposing Factors Predictive Inference Effect Cause Effect Diagnostic Reasoning

(Value of Information)
What are BNs useful for? Value of Information Salient Observations Probability of fault “i” Expected Utility Do nothing Action 2 Action 1 Fault 1 Fault 2 Fault 3 . Assignment of Belief New Obs. Act Now! Yes Halt? No Next Best Observation (Value of Information)

Why use BNs? Explicit management of uncertainty
Modularity implies maintainability Better, flexible and robust decision making - MEU, VOI Can be used to answer arbitrary queries - multiple fault problems Easy to incorporate prior knowledge Easy to understand

Application Examples Intellipath
commercial version of Pathfinder lymph-node diseases (60), 100 findings APRI system developed at AT&T Bell Labs learns & uses Bayesian networks from data to identify customers liable to default on bill payments NASA Vista system predict failures in propulsion systems considers time criticality & suggests highest utility action dynamically decide what information to show

Application Examples Answer Wizard in MS Office 95/ MS Project
Bayesian network based free-text help facility uses naive Bayesian classifiers Office Assistant in MS Office 97 Extension of Answer wizard uses naïve Bayesian networks help based on past experience (keyboard/mouse use) and task user is doing currently This is the “smiley face” you get in your MS Office applications

Application Examples Microsoft Pregnancy and Child-Care
Available on MSN in Health section Frequently occuring children’s symptoms are linked to expert modules that repeatedly ask parents relevant questions Asks next best question based on provided information Presents articles that are deemed relevant based on information provided

Application Examples Printer troubleshooting
HP bought 40% stake in HUGIN. Developing printer troubleshooters for HP printers Microsoft has 70+ online troubleshooters on their web site use Bayesian networks - multiple faults models, incorporate utilities Fax machine troubleshooting Ricoh uses Bayesian network based troubleshooters at call centers Enabled Ricoh to answer twice the number of calls in half the time

Application Examples

Online/print resources on BNs
Conferences & Journals UAI, ICML, AAAI, AISTAT, KDD MLJ, DM&KD, JAIR, IEEE KDD, IJAR, IEEE PAMI Books and Papers Bayesian Networks without Tears by Eugene Charniak. AI Magazine: Winter 1991. Probabilistic Reasoning in Intelligent Systems by Judea Pearl. Morgan Kaufmann: 1998. Probabilistic Reasoning in Expert Systems by Richard Neapolitan. Wiley: 1990. CACM special issue on Real-world applications of BNs, March 1995

Online/Print Resources on BNs
Wealth of online information at Links to Electronic proceedings for UAI conferences Other sites with information on BNs and reasoning under uncertainty Several tutorials and important articles Research groups & companies working in this area Other societies, mailing lists and conferences

Publicly available s/w for BNs
List of BN software maintained by Russell Almond at bayes.stat.washington.edu/almond/belief.html several free packages: generally research only commercial packages: most powerful (& expensive) is HUGIN; others include Netica and Dxpress we are working on developing a Java based BN toolkit here at Watson - will also work within ABLE

What are BNs: representation, types, etc Why use BNs: Applications (classes) of BNs Information sources, software, etc Probabilistic inference Exact inference Approximate inference Learning Bayesian Networks Learning parameters Learning graph structure Summary

Probabilistic Inference Tasks
Belief updating: Finding most probable explanation (MPE) Finding maximum a-posteriory hypothesis Finding maximum-expected-utility (MEU) decision

Belief Updating P (lung cancer=yes | smoking=no, dyspnoea=yes ) = ?
Bronchitis X-ray Dyspnoea P (lung cancer=yes | smoking=no, dyspnoea=yes ) = ?

Bucket elimination Algorithm elim-bel (Dechter 1996)
Elimination operator bucket B: P(a) P(c|a) P(b|a) P(d|b,a) P(e|b,c) bucket C: bucket D: bucket E: bucket A: e=0 B C D E A W*=4 ”induced width” (max clique size) P(a|e=0)

Finding Algorithm elim-mpe (Dechter 1996)
Elimination operator bucket B: P(a) P(c|a) P(b|a) P(d|b,a) P(e|b,c) bucket C: bucket D: bucket E: bucket A: e=0 B C D E A W*=4 ”induced width” (max clique size) MPE

Generating the MPE-tuple
C: E: P(b|a) P(d|b,a) P(e|b,c) B: D: A: P(a) P(c|a) e=0

Complexity of inference
The effect of the ordering: “Moral” graph A D E C B B C D E A E D C B A

Other tasks and algorithms
MAP and MEU tasks: Similar bucket-elimination algorithms - elim-map, elim-meu (Dechter 1996) Elimination operation: either summation or maximization Restriction on variable ordering: summation must precede maximization (i.e. hypothesis or decision variables are eliminated last) Other inference algorithms: Join-tree clustering Pearl’s poly-tree propagation Conditioning, etc.

Relationship with join-tree clustering
BCE ADB A cluster is a set of buckets (a “super-bucket”) ABC

Relationship with Pearl’s belief propagation in poly-trees
“Causal support” “Diagnostic support” Pearl’s belief propagation for single-root query elim-bel using topological ordering and super-buckets for families Elim-bel, elim-mpe, and elim-map are linear for poly-trees.

Exact inference Approximate inference Learning Bayesian Networks Learning parameters Learning graph structure Summary

Inference is NP-hard => approximations
B C Approximations: Local inference Stochastic simulations Variational approximations etc.

Local Inference Idea

Bucket-elimination approximation: “mini-buckets”
Local inference idea: bound the size of recorded dependencies Computation in a bucket is time and space exponential in the number of variables involved Therefore, partition functions in a bucket into “mini-buckets” on smaller number of variables

Mini-bucket approximation: MPE task
Split a bucket into mini-buckets =>bound complexity

Approx-mpe(i) Example: approx-mpe(3) versus elim-mpe
Input: i – max number of variables allowed in a mini-bucket Output: [lower bound (P of a sub-optimal solution), upper bound] Example: approx-mpe(3) versus elim-mpe

Properties of approx-mpe(i)
Complexity: O(exp(2i)) time and O(exp(i)) time. Accuracy: determined by upper/lower (U/L) bound. As i increases, both accuracy and complexity increase. Possible use of mini-bucket approximations: As anytime algorithms (Dechter and Rish, 1997) As heuristics in best-first search (Kask and Dechter, 1999) Other tasks: similar mini-bucket approximations for: belief updating, MAP and MEU (Dechter and Rish, 1997)

Anytime Approximation

Empirical Evaluation (Dechter and Rish, 1997; Rish, 1999)
Randomly generated networks Uniform random probabilities Random noisy-OR CPCS networks Probabilistic decoding Comparing approx-mpe and anytime-mpe versus elim-mpe

Random networks Uniform random: 60 nodes, 90 edges (200 instances)
In 80% of cases, times speed-up while U/L<2 Noisy-OR – even better results Exact elim-mpe was infeasible; appprox-mpe took 0.1 to 80 sec.

CPCS networks – medical diagnosis (noisy-OR model)
Test case: no evidence 505.2 70.3 anytime-mpe( ), 110.5 1697.6 115.8 elim-mpe cpcs422 cpcs360 Algorithm Time (sec)

Effect of evidence More likely evidence=>higher MPE => higher accuracy (why?) Likely evidence versus random (unlikely) evidence

Probabilistic decoding
Error-correcting linear block code State-of-the-art: approximate algorithm – iterative belief propagation (IBP) (Pearl’s poly-tree algorithm applied to loopy networks)

approx-mpe vs. IBP Bit error rate (BER) as a function of noise (sigma):

Mini-buckets: summary
Mini-buckets – local inference approximation Idea: bound size of recorded functions Approx-mpe(i) - mini-bucket algorithm for MPE Better results for noisy-OR than for random problems Accuracy increases with decreasing noise in Accuracy increases for likely evidence Sparser graphs -> higher accuracy Coding networks: approx-mpe outperfroms IBP on low-induced width codes

Exact inference Approximate inference Local inference Stochastic simulations Variational approximations Learning Bayesian Networks Summary

Approximation via Sampling

Forward Sampling (logic sampling (Henrion, 1988))

Forward sampling (example)
Drawback: high rejection rate!

Likelihood Weighing (Fung and Chang, 1990; Shachter and Peot, 1990)
“Clamping” evidence+forward sampling+ weighing samples by evidence likelihood Works well for likely evidence!

Gibbs Sampling (Geman and Geman, 1984)
Markov Chain Monte Carlo (MCMC): create a Markov chain of samples Advantage: guaranteed to converge to P(X) Disadvantage: convergence may be slow

Gibbs Sampling (cont’d) (Pearl, 1988)
Markov blanket:

Exact inference Approximate inference Local inference Stochastic simulations Variational approximations Learning Bayesian Networks Summary

Variational Approximations
Idea: variational transformation of CPDs simplifies inference Advantages: Compute upper and lower bounds on P(Y) Usually faster than sampling techniques Disadvantages: More complex and less general: must be derived for each particular form of CPD functions

Variational bounds: example
log(x) This approach can be generalized for any concave (convex) function in order to compute its upper (lower) bounds: convex duality (Jaakkola and Jordan, 1997)

Convex duality (Jaakkola and Jordan, 1997)

Example: QMR-DT network (Quick Medical Reference – Decision-Theoretic (Shwe et al., 1991))
600 diseases 4000 findings Noisy-OR model:

Inference in QMR-DT Inference complexity: O(exp(min{p,k})) factorized
Positive evidence “couples” the disease nodes factorized Inference complexity: O(exp(min{p,k})) p = # of positive findings, k = max family size (Heckerman, 1989 (“Quickscore”), Rish and Dechter, 1998)

Variational approach to QMR-DT (Jaakkola and Jordan, 1997)
The effect of positive evidence is now factorized (diseases are “decoupled”)

Variational approximations
Bounds on local CPDs yield a bound on posterior Two approaches: sequential and block Sequential: applies variational transformation to (a subset of) nodes sequentially during inference using a heuristic node ordering; then optimizes across variational parameters Block: selects in advance nodes to be transformed, then selects variational parameters minimizing the KL-distance between true and approximate posteriors

Block approach

Inference in BN: summary
Exact inference is often intractable => need approximations Approximation principles: Approximating elimination – local inference, bounding size of dependencies among variables (cliques in a problem’s graph). Mini-buckets, IBP Other approximations: stochastic simulations, variational techniques, etc. Further research: Combining “orthogonal” approximation approaches Better understanding of “what works well where”: which approximation suits which problem structure Other approximation paradigms (e.g., other ways of approximating probabilities, constraints, cost functions)

Exact inference Approximate inference Learning Bayesian Networks Learning parameters Learning graph structure Summary

Why learn Bayesian networks?
< > < ?? ??> < ?? > <?? ??> ………………. Combining domain expert knowledge with data Efficient representation and inference Incremental learning: P(H) or Handling missing data: < ?? > S C Learning causal relationships:

Learning Bayesian Networks
Known graph C S B D X P(S) P(B|S) P(X|C,S) P(C|S) P(D|C,B) – learn parameters Complete data: parameter estimation (ML, MAP) Incomplete data: non-linear parametric optimization (gradient descent, EM) C S B D X Unknown graph – learn graph and parameters Complete data: optimization (search in space of graphs) Incomplete data: EM plus Multiple Imputation, structural EM, mixture models C S B D X

Learning Parameters: complete data
ML-estimate: - decomposable! X C B Multinomial counts MAP-estimate (Bayesian statistics) Conjugate priors - Dirichlet Equivalent sample size (prior knowledge)

Learning graph structure
Find NP-hard optimization Heuristic search: Complete data – local computations Incomplete data (score non-decomposable):stochastic methods C S B Add S->B C S B C S B Delete S->B C S B Reverse S->B Constrained-based methods Data impose independence relations (constraints)

Learning BNs: incomplete data
Learning parameters EM algorithm [Lauritzen, 95] Gibbs Sampling [Heckerman, 96] Gradient Descent [Russell et al., 96] Learning both structure and parameters Sum over missing values [Cooper & Herskovits, 92; Cooper, 95] Monte-Carlo approaches [Heckerman, 96] Gaussian approximation [Heckerman, 96] Structural EM [Friedman, 98] EM and Multiple Imputation [Singh 97,98,00]

Learning Parameters: incomplete data
Non-decomposable marginal likelihood (hidden nodes) S X D C B <? > <1 1 ? 0 1> < ? ?> <? ? 0 ? 1> ……… Data S X D C B ……….. Expected counts Expectation Inference: P(S|X=0,D=1,C=0,B=1) Initial parameters Current model Update parameters (ML, MAP) Maximization EM-algorithm: iterate until convergence

Learning Parameters: incomplete data
(Lauritzen, 95) Complete-data log-likelihood is E step Compute E( Nijk | Yobs,  M step Compute E( Nijk | Yobs, E( Nij | Yobs,  Nijk log ijk

Learning structure: incomplete data
Depends on the type of missing data - missing independent of anything else (MCAR) OR missing based on values of other variables (MAR) While MCAR can be resolved by decomposable scores, MAR cannot For likelihood-based methods, no need to explicitly model missing data mechanism Very few attempts at MAR: stochastic methods

Approximate EM by using Multiple Imputation to yield efficient Monte-Carlo method [Singh 97, 98, 00] trade-off between performance & quality learned network almost optimal approximate complete-data log-likelihood function using Multiple Imputation yields decomposable score, dependent only on each node & its parents converges to local maxima of observed-data likelihood

Scoring functions: Minimum Description Length (MDL)
Learning  data compression Other: MDL = -BIC (Bayesian Information Criterion) Bayesian score (BDe) - asymptotically equivalent to MDL < > < ?? ??> < ?? > <?? ??> ………………. DL(Data|model) DL(Model)

Learning Structure plus Parameters
No. of models is super exponential Alternatives: Model Selection or Model Averaging

Model Selection Generally, choose a single model M*.
Equivalent to saying P(M*|D) = 1 Task is now to: 1) define a metric to decide which model is best 2) search for that model through the space of all models

One Reasonable Score: Posterior Probability of a Structure
prior likelihood parameter prior 81

Global and Local Predictive Scores
[Spiegelhalter et al 93] Bayes’ factor m å log p ( D | S h ) = log p ( x | x , ... , x , S h ) l 1 l - 1 l = 1 = log p ( x | S h ) + log p ( x | x , S h ) + log p ( x | x , x , S h ) + L 1 2 1 3 1 2 Local is useful for diagnostic problems

Local Predictive Score Spiegelhalter et al. (1993)
Y disease X1 X2 Xn symptoms ... 83

Exact computation of p(D|Sh)
No missing data Cases are independent, given the model. Uniform priors on parameters discrete variables [Cooper & Herskovits, 92]

Bayesian Dirichlet Score Cooper and Herskovits (1991)
88

Learning BNs without specifying an ordering
n! ordering; ordering greatly affects the quality of network learned. use conditional independence tests, and d-separation to get an ordering [Singh & Valtorta’ 95]

Learning BNs via the MDL principle
Idea: best model is that which gives the most compact representation of the data So, encode the data using the model plus encode the model. Minimize this. [Lam & Bacchus, 93]

Learning BNs: summary Bayesian Networks – graphical probabilistic models Efficient representation and inference Expert knowledge + learning from data Learning: parameters (parameter estimation, EM) structure (optimization w/ score functions – e.g., MDL) Applications/systems: collaborative filtering (MSBN), fraud detection (AT&T), classification (AutoClass (NASA), TAN-BLT(SRI)) Future directions: causality, time, model evaluation criteria, approximate inference/learning, on-line learning, etc.

A Tutorial on Inference and Learning in Bayesian Networks

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