1. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. Learning I Excerpts from Tutorial at: http://robotics.stanford.edu/people/nir/tutorials/index.html

2. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-2 Bayesian Networks Qualitative part: statistical independence statements (causality!) u Directed acyclic graph (DAG) l Nodes - random variables of interest (exhaustive and mutually exclusive states) l Edges - direct (causal) influence Quantitative part: Local probability models. Set of conditional probability distributions. 0.90.1 e b e 0.20.8 0.01 0.99 0.90.1 be b b e BE P(A | E,B) Earthquake Radio Burglary Alarm Call

3. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-3 Learning Bayesian networks (reminder) Inducer Data + Prior information E R B A C.9.1 e b e.7.3.99.01.8.2 be b b e BEP(A | E,B)

4. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-4 The Learning Problem

5. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-5 Learning Problem ?? e b e ?? ? ? ?? be b b e BEP(A | E,B) E, B, A. Inducer E B A.9.1 e b e.7.3.99.01.8.2 be b b e BEP(A | E,B) E B A

6. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-6 Learning Problem ?? e b e ?? ? ? ?? be b b e BEP(A | E,B) E, B, A. Inducer E B A.9.1 e b e.7.3.99.01.8.2 be b b e BEP(A | E,B) E B A

7. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-7 Learning Problem ?? e b e ?? ? ? ?? be b b e BEP(A | E,B) E, B, A. Inducer E B A.9.1 e b e.7.3.99.01.8.2 be b b e BEP(A | E,B) E B A

8. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-8 Learning Problem ?? e b e ?? ? ? ?? be b b e BEP(A | E,B) E, B, A. Inducer E B A.9.1 e b e.7.3.99.01.8.2 be b b e BEP(A | E,B) E B A

9. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-9 Learning Parameters for the Burglary Story E B A C i.i.d. samples Network factorization We have 4 independent estimation problems

10. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-10 Incomplete Data Data is often incomplete u Some variables of interest are not assigned value This phenomena happen when we have u Missing values u Hidden variables

11. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-11 Missing Values u Examples: u Survey data u Medical records l Not all patients undergo all possible tests

12. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-12 Missing Values (cont.) Complicating issue: u The fact that a value is missing might be indicative of its value l The patient did not undergo X-Ray since she complained about fever and not about broken bones…. To learn from incomplete data we need the following assumption: Missing at Random (MAR):  The probability that the value of X i is missing is independent of its actual value given other observed values

13. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-13 Hidden (Latent) Variables u Attempt to learn a model with variables we never observe l In this case, MAR always holds u Why should we care about unobserved variables? X1X1 X2X2 X3X3 H Y1Y1 Y2Y2 Y3Y3 X1X1 X2X2 X3X3 Y1Y1 Y2Y2 Y3Y3 17 parameters 59 parameters

14. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-14 Learning Parameters from Incomplete Data (cont.). u In the presence of incomplete data, the likelihood can have multiple global maxima u Example: l We can rename the values of hidden variable H l If H has two values, likelihood has two global maxima u Similarly, local maxima are also replicated u Many hidden variables  a serious problem HY

15. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-15 Gradient Ascent u Main result  Requires computation: P(x i,Pa i |o[m],  ) for all i, m u Pros: l Flexible l Closely related to methods in neural network training u Cons: l Need to project gradient onto space of legal parameters l To get reasonable convergence we need to combine with “smart” optimization techniques

16. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-16 Expectation Maximization (EM) u A general purpose method for learning from incomplete data Intuition: u If we had access to counts, then we can estimate parameters u However, missing values do not allow to perform counts u “Complete” counts using current parameter assignment 1.3 0.4 1.7 1.6 X Z N (X,Y ) XY # HTHHTHTHHT Y ??HTT??HTT T??THT??TH HTHTHTHT HHTTHHTT P(Y=H|X=T,  ) = 0.4 Expected Counts P(Y=H|X=H,Z=T,  ) = 0.3 Data Current model

17. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-17 EM (cont.) Training Data X1X1 X2X2 X3X3 H Y1Y1 Y2Y2 Y3Y3 Initial network (G,  0 )  Expected Counts N(X 1 ) N(X 2 ) N(X 3 ) N(H, X 1, X 1, X 3 ) N(Y 1, H) N(Y 2, H) N(Y 3, H) Computation (E-Step) Reparameterize X1X1 X2X2 X3X3 H Y1Y1 Y2Y2 Y3Y3 Updated network (G,  1 ) (M-Step) Reiterate

18. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-18 EM (cont.) Formal Guarantees:  L(  1 :D)  L(  0 :D) l Each iteration improves the likelihood  If  1 =  0, then  0 is a stationary point of L(  :D) l Usually, this means a local maximum Main cost: u Computations of expected counts in E-Step u Requires a computation pass for each instance in training set l These are exactly the same as for gradient ascent!

19. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-19 Example: EM in clustering u Consider clustering example E-Step: Compute P(C[m]|X 1 [m],…,X n [m],  ) l This corresponds to “soft” assignment to clusters l Compute expected statistics: M-Step Re-estimate P(X i |C), P(C) Cluster X1X1... X2X2 XnXn

20. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-20 EM in Practice Initial parameters: u Random parameters setting u “Best” guess from other source Stopping criteria: u Small change in likelihood of data u Small change in parameter values Avoiding bad local maxima: u Multiple restarts u Early “pruning” of unpromising ones Speed up: u various methods to speed convergence

21. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-21 Why Struggle for Accurate Structure u Increases the number of parameters to be fitted u Wrong assumptions about causality and domain structure u Cannot be compensated by accurate fitting of parameters u Also misses causality and domain structure EarthquakeAlarm Set Sound Burglary EarthquakeAlarm Set Sound Burglary Earthquake Alarm Set Sound Burglary Adding an arcMissing an arc

22. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-22 Minimum Description Length (cont.) u Computing the description length of the data, we get u Minimizing this term is equivalent to maximizing # bits to encode G # bits to encode  G # bits to encode D using (G,  G )

23. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-23 Heuristic Search u We address the problem by using heuristic search u Define a search space: l nodes are possible structures l edges denote adjacency of structures u Traverse this space looking for high-scoring structures Search techniques: l Greedy hill-climbing l Best first search l Simulated Annealing l...

24. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-24 Heuristic Search (cont.) u Typical operations: S C E D S C E D S C E D S C E D Add C  D Reverse C  E Remove C  E

25. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-25 Exploiting Decomposability in Local Search u Caching: To update the score of after a local change, we only need to re-score the families that were changed in the last move S C E D S C E D S C E D S C E D

26. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-26 Greedy Hill-Climbing Simplest heuristic local search l Start with a given network empty network best tree a random network l At each iteration Evaluate all possible changes Apply change that leads to best improvement in score Reiterate l Stop when no modification improves score  Each step requires evaluating approximately n new changes

27. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-27 Greedy Hill-Climbing (cont.) u Greedy Hill-Climbing can get struck in: l Local Maxima: All one-edge changes reduce the score l Plateaus: Some one-edge changes leave the score unchanged u Both are occur in the search space

28. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-28 Greedy Hill-Climbing (cont.) To avoid these problems, we can use: u TABU-search Keep list of K most recently visited structures l Apply best move that does not lead to a structure in the list This escapes plateaus and local maxima and with “basin” smaller than K structures u Random Restarts l Once stuck, apply some fixed number of random edge changes and restart search l This can escape from the basin of one maxima to another

29. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-29 Other Local Search Heuristics u Stochastic First-Ascent Hill-Climbing l Evaluate possible changes at random l Apply the first one that leads “uphill” l Stop when a fix amount of “unsuccessful” attempts to change the current candidate u Simulated Annealing l Similar idea, but also apply “downhill” changes with a probability that is proportional to the change in score l Use a temperature to control amount of random downhill steps l Slowly “cool” temperature to reach a regime where performing strict uphill moves

30. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-30 Examples u Predicting heart disease l Features: cholesterol, chest pain, angina, age, etc. l Class: {present, absent} u Finding lemons in cars l Features: make, brand, miles per gallon, acceleration,etc. l Class: {normal, lemon} u Digit recognition l Features: matrix of pixel descriptors l Class: {1, 2, 3, 4, 5, 6, 7, 8, 9, 0} u Speech recognition l Features: Signal characteristics, language model l Class: {pause/hesitation, retraction}

31. © 1998, Nir Friedman, U.C. Berkeley, and Moises Goldszmidt, SRI International. All rights reserved. MP1-31 Some Applications u Biostatistics -- Medical Research Council (Bugs) u Data Analysis -- NASA (AutoClass) u Collaborative filtering -- Microsoft (MSBN) u Fraud detection -- ATT u Classification -- SRI (TAN-BLT) u Speech recognition -- UC Berkeley