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An introduction to machine learning and probabilistic graphical models Kevin Murphy MIT AI Lab Presented at Intel’s workshop on “Machine learning for the life sciences”, Berkeley, CA, 3 November 2003 . Overview Supervised learning Unsupervised learning Graphical models Learning relational models Thanks to Nir Friedman, Stuart Russell, Leslie Kaelbling and various web sources for letting me use many of their slides 2 Supervised learning yes no Color Shape Size Output Blue Torus Big Y Blue Square Small Y Blue Star Small Y Red Arrow Small N Learn to approximate function F(x1, x2, x3) -> t from a training set of (x,t) pairs 3 Supervised learning Training data X1 X2 X3 T B T B Y Learner B S S Y B S S Y Prediction R A S N T Testing data Y X1 X2 X3 T B A S ? Hypothesi N s Y C S ? 4 Key issue: generalization yes no ? ? Can’t just memorize the training set (overfitting) 5 Hypothesis spaces Decision trees Neural networks K-nearest neighbors Naïve Bayes classifier Support vector machines (SVMs) Boosted decision stumps … 6 Perceptron (neural net with no hidden layers) Linearly separable data 7 Which separating hyperplane? 8 The linear separator with the largest margin is the best one to pick margin 9 What if the data is not linearly separable? 10 Kernel trick x2 x 2 xy y 2 y z3 x2 kernel x1 z2 z1 Kernel implicitly maps from 2D to 3D, making problem linearly separable 11 Support Vector Machines (SVMs) Two key ideas: Large margins Kernel trick 12 Boosting Simple classifiers (weak learners) can have their performance boosted by taking weighted combinations Boosting maximizes the margin 13 Supervised learning success stories Face detection Steering an autonomous car across the US Detecting credit card fraud Medical diagnosis … 14 Unsupervised learning What if there are no output labels? 15 K-means clustering 1. Guess number of clusters, K 2. Guess initial cluster centers, 1, 2 Reiterate 3. Assign data points xi to nearest cluster center 4. Re-compute cluster centers based on assignments 16 AutoClass (Cheeseman et al, 1986) EM algorithm for mixtures of Gaussians “Soft” version of K-means Uses Bayesian criterion to select K Discovered new types of stars from spectral data Discovered new classes of proteins and introns from DNA/protein sequence databases 17 Hierarchical clustering 18 Principal Component Analysis (PCA) PCA seeks a projection that best represents the data in a least-squares sense. PCA reduces the dimensionality of feature space by restricting attention to those directions along which the scatter of the cloud is greatest. . Discovering nonlinear manifolds 20 Combining supervised and unsupervised learning 21 Discovering rules (data mining) Occup. Income Educ. Sex Married Age Student $10k MA M S 22 Student $20k PhD F S 24 Doctor $80k MD M M 30 Retired $30k HS F M 60 Find the most frequent patterns (association rules) Num in household = 1 ^ num children = 0 => language = English Language = English ^ Income < $40k ^ Married = false ^ num children = 0 => education {college, grad school} 22 Unsupervised learning: summary Clustering Hierarchicalclustering Linear dimensionality reduction (PCA) Non-linear dim. Reduction Learning rules 23 Discovering networks ? From data visualization to causal discovery 24 Networks in biology Most processes in the cell are controlled by networks of interacting molecules: Metabolic Network Signal Transduction Networks Regulatory Networks Networks can be modeled at multiple levels of detail/ realism Molecular level Concentration level Decreasing detail Qualitative level 25 Molecular level: Lysis-Lysogeny circuit in Lambda phage Arkin et al. (1998), Genetics 149(4):1633-48 5 genes, 67 parameters based on 50 years of research Stochastic simulation required supercomputer 26 Concentration level: metabolic pathways Usually modeled with differential equations g1 w12 w55 g2 g5 w23 g4 g3 27 Qualitative level: Boolean Networks 28 Probabilistic graphical models Supports graph-based modeling at various levels of detail Models can be learned from noisy, partial data Can model “inherently” stochastic phenomena, e.g., molecular-level fluctuations… But can also model deterministic, causal processes. "The actual science of logic is conversant at present only with things either certain, impossible, or entirely doubtful. Therefore the true logic for this world is the calculus of probabilities." -- James Clerk Maxwell "Probability theory is nothing but common sense reduced to calculation." -- Pierre Simon Laplace 29 Graphical models: outline What are graphical models? Inference Structure learning 30 Simple probabilistic model: linear regression Y = + X + noise Deterministic (functional) relationship Y X 31 Simple probabilistic model: linear regression Y = + X + noise Deterministic (functional) relationship Y “Learning” = estimating parameters , , from (x,y) pairs. Is the empirical mean Can be estimate by least squares X Is the residual variance 32 Piecewise linear regression Latent “switch” variable – hidden process at work 33 Probabilistic graphical model for piecewise linear regression input X •Hidden variable Q chooses which set of parameters to use for predicting Y. Q •Value of Q depends on value of input X. •This is an example of “mixtures of experts” Y output Learning is harder because Q is hidden, so we don’t know which data points to assign to each line; can be solved with EM (c.f., K-means) 34 Classes of graphical models Probabilistic models Graphical models Directed Undirected Bayes nets MRFs DBNs 35 Bayesian Networks Compact representation of probability distributions via conditional independence Family of Alarm Qualitative part: Earthquake Burglary E B P(A | E,B) Directed acyclic graph (DAG) e b 0.9 0.1 Nodes - random variables e b 0.2 0.8 Radio Alarm e b 0.9 0.1 Edges - direct influence e b 0.01 0.99 Call Together: Quantitative part: Define a unique distribution Set of conditional in a factored form probability distributions P (B , E , A, C , R ) P (B )P (E )P (A | B , E )P (R | E )P (C | A) 36 Example: “ICU Alarm” network Domain: Monitoring Intensive-Care Patients 37 variables MINVOLSET 509 parameters PULMEMBOLUS INTUBATION KINKEDTUBE VENTMACH DISCONNECT …instead of 254 PAP SHUNT VENTLUNG VENITUBE PRESS MINOVL FIO2 VENTALV ANAPHYLAXIS PVSAT ARTCO2 TPR SAO2 INSUFFANESTH EXPCO2 HYPOVOLEMIA LVFAILURE CATECHOL LVEDVOLUME STROEVOLUME HISTORY ERRBLOWOUTPUT HR ERRCAUTER CVP PCWP CO HREKG HRSAT HRBP BP 37 Success stories for graphical models Multiple sequence alignment Forensic analysis Medical and fault diagnosis Speech recognition Visual tracking Channel coding at Shannon limit Genetic pedigree analysis … 38 Graphical models: outline What are graphical models? p Inference Structure learning 39 Probabilistic Inference Posterior probabilities Probability of any event given any evidence P(X|E) Earthquake Burglary Radio Alarm Call 40 Viterbi decoding Compute most probable explanation (MPE) of observed data Hidden Markov Model (HMM) X1 X2 X3 hidden Y1 Y3 observed Y2 “Tomato” 41 Inference: computational issues Easy Hard Dense, loopy graphs Chains Trees MINVOLSET INTUBATION KINKEDTUBE Grids PULMEMBOLUS DISCONNECT VENTMACH PAP SHUNT VENTLUNG VENITUBE PRESS MINOVL VENTALV PVSAT ARTCO2 TPR SAO2 EXPCO2 INSUFFANESTH LVFAILURE CATECHOL HYPOVOLEMIA STROEVOLUME ERRBLOWOUTPUT LVEDVOLUME HISTORY HRERRCAUTER CVP PCWP CO HREKGHRSAT HRBP BP 42 Inference: computational issues Easy Hard Dense, loopy graphs Chains Trees MINVOLSET INTUBATION KINKEDTUBE Grids PULMEMBOLUS DISCONNECT VENTMACH PAP SHUNT VENTLUNG VENITUBE PRESS MINOVL VENTALV PVSAT ARTCO2 TPR SAO2 EXPCO2 INSUFFANESTH LVFAILURE CATECHOL HYPOVOLEMIA STROEVOLUME ERRBLOWOUTPUT LVEDVOLUME HISTORY HRERRCAUTER CVP PCWP CO HREKGHRSAT HRBP BP Many difference inference algorithms, both exact and approximate 43 Bayesian inference Bayesian probability treats parameters as random variables Learning/ parameter estimation is replaced by probabilistic inference P(|D) Example: Bayesian linear regression; parameters are = (, , ) Parameters are tied (shared) across repetitions of the data X1 Xn Y1 Yn 44 Bayesian inference + Elegant – no distinction between parameters and other hidden variables + Can use priors to learn from small data sets (c.f., one-shot learning by humans) - Math can get hairy - Often computationally intractable 45 Graphical models: outline What are graphical models? p Inference p Structure learning 46 Why Struggle for Accurate Structure? Earthquake Alarm Set Burglary Sound Missing an arc Adding an arc Earthquake Alarm Set Burglary Earthquake Alarm Set Burglary Sound Sound Cannot be compensated Increases the number of for by fitting parameters parameters to be estimated Wrong assumptions about Wrong assumptions about domain structure domain structure 47 Score-based Learning Define scoring function that evaluates how well a structure matches the data E, B, A <Y,N,N> <Y,Y,Y> <N,N,Y> <N,Y,Y> . . E B E E <N,Y,Y> A A A B B Search for a structure that maximizes the score 48 Learning Trees Can find optimal tree structure in O(n2 log n) time: just find the max-weight spanning tree If some of the variables are hidden, problem becomes hard again, but can use EM to fit mixtures of trees 49 Heuristic Search Learning arbitrary graph structure is NP-hard. So it is common to resort to heuristic search Define a search space: search states are possible structures operators make small changes to structure Traverse space looking for high-scoring structures Search techniques: Greedy hill-climbing Best first search Simulated Annealing ... 50 Local Search Operations Typical operations: S C E S C D E score = S({C,E} D) D - S({E} D) S C S C E E D D 51 Problems with local search Easy to get stuck in local optima “truth” you S(G|D) 52 P(G|D) Problems with local search II Picking a single best model can be misleading E B R A C 53 P(G|D) Problems with local search II Picking a single best model can be misleading E B E B E B E B E B R A R A R A R A R A C C C C C Small sample size many high scoring models Answer based on one model often useless Want features common to many models 54 Bayesian Approach to Structure Learning Posteriordistribution over structures Estimate probability of features Edge XY Path X… Y Bayesian score … for G P (f | D ) f (G )P (G | D ) G Feature of G, Indicator function e.g., XY for feature f 55 Bayesian approach: computational issues Posterior distribution over structures P (f | D ) f (G )P (G | D ) G How compute sum over super-exponential number of graphs? •MCMC over networks •MCMC over node-orderings (Rao-Blackwellisation) 56 Structure learning: other issues Discovering latent variables Learning causal models Learning from interventional data Active learning 57 Discovering latent variables a) 17 parameters b) 59 parameters There are some techniques for automatically detecting the possible presence of latent variables 58 Learning causal models So far, we have only assumed that X -> Y -> Z means that Z is independent of X given Y. However, we often want to interpret directed arrows causally. This is uncontroversial for the arrow of time. But can we infer causality from static observational data? 59 Learning causal models We can infer causality from static observational data if we have at least four measured variables and certain “tetrad” conditions hold. See books by Pearl and Spirtes et al. However, we can only learn up to Markov equivalence, not matter how much data we have. X Y Z X Y Z X Y Z X Y Z 60 Learning from interventional data The only way to distinguish between Markov equivalent networks is to perform interventions, e.g., gene knockouts. We need to (slightly) modify our learning algorithms. smoking smoking Cut arcs coming into nodes which were set by intervention Yellow Yellow fingers fingers P(smoker|observe(yellow)) >> prior P(smoker | do(paint yellow)) = prior 61 Active learning Which experiments (interventions) should we perform to learn structure as efficiently as possible? This problem can be modeled using decision theory. Exact solutions are wildly computationally intractable. Can we come up with good approximate decision making techniques? Can we implement hardware to automatically perform the experiments? “AB: Automated Biologist” 62 Learning from relational data Can we learn concepts from a set of relations between objects, instead of/ in addition to just their attributes? 63 Learning from relational data: approaches Probabilistic relational models (PRMs) Reify a relationship (arcs) between nodes (objects) by making into a node (hypergraph) Inductive Logic Programming (ILP) Top-down, e.g., FOIL (generalization of C4.5) Bottom up, e.g., PROGOL (inverse deduction) 64 ILP for learning protein folding: input yes no TotalLength(D2mhr, 118) ^ NumberHelices(D2mhr, 6) ^ … 100 conjuncts describing structure of each pos/neg example 65 ILP for learning protein folding: results PROGOL learned the following rule to predict if a protein will form a “four-helical up-and-down bundle”: English: “The protein P folds if it contains a long In helix h1 at a secondary structure position between 1 and 3 and h1 is next to a second helix” 66 ILP: Pros and Cons + Can discover new predicates (concepts) automatically + Can learn relational models from relational (or flat) data - Computationally intractable - Poor handling of noise 67 The future of machine learning for bioinformatics? Oracle 68 The future of machine learning for bioinformatics Prior knowledge Hypotheses Replicated experiments Learner Biological literature Real world Expt. design •“Computer assisted pathway refinement” 69 The end 70 Decision trees blue? yes oval? no big? no yes 71 Decision trees blue? yes oval? + Handles mixed variables + Handles missing data no + Efficient for large data sets big? + Handles irrelevant attributes + Easy to understand - Predictive power no yes 72 Feedforward neural network input Hidden layer Output Weights on each arc Sigmoid function at each node f ( J i si ), f ( x) 1/(1 e cx ) i 73 Feedforward neural network input Hidden layer Output - Handles mixed variables - Handles missing data - Efficient for large data sets - Handles irrelevant attributes - Easy to understand + Predicts poorly 74 Nearest Neighbor Remember all your data When someone asks a question, find the nearest old data point return the answer associated with it 75 Nearest Neighbor ? - Handles mixed variables - Handles missing data - Efficient for large data sets - Handles irrelevant attributes - Easy to understand + Predictive power 76 Support Vector Machines (SVMs) Two key ideas: Large margins are good Kernel trick 77 SVM: mathematical details Training data : l-dimensional vector with flag of true or false {xi, yi }, xi Rl , yi {1,1} Separating hyperplane : wx b 0 Margin : d 2/ w Inequalities : yi (xi w b) 1 0, i Support vector expansion: w i x i i Support vectors : Decision: margin 78 Replace all inner products with kernels Kernel function 79 SVMs: summary - Handles mixed variables - Handles missing data - Efficient for large data sets - Handles irrelevant attributes - Easy to understand + Predictive power General lessons from SVM success: •Kernel trick can be used to make many linear methods non-linear e.g., kernel PCA, kernelized mutual information •Large margin classifiers are good 80 Boosting: summary Can boost any weak learner Most commonly: boosted decision “stumps” + Handles mixed variables + Handles missing data + Efficient for large data sets + Handles irrelevant attributes - Easy to understand + Predictive power 81 Supervised learning: summary Learn mapping F from inputs to outputs using a training set of (x,t) pairs F can be drawn from different hypothesis spaces, e.g., decision trees, linear separators, linear in high dimensions, mixtures of linear Algorithms offer a variety of tradeoffs Many good books, e.g., “The elements of statistical learning”, Hastie, Tibshirani, Friedman, 2001 “Pattern classification”, Duda, Hart, Stork, 2001 82 Inference Posterior probabilities Probability of any event given any evidence Most likely explanation Scenario that explains evidence Rational decision making Earthquake Burglary Maximize expected utility Value of Information Radio Alarm Effect of intervention Call 83 Assumption needed to make learning work We need to assume “Future futures will resemble past futures” (B. Russell) Unlearnable hypothesis: “All emeralds are grue”, where “grue” means: green if observed before time t, blue afterwards. 84 Structure learning success stories: gene regulation network (Friedman et al.) Yeast data [Hughes et al 2000] 600 genes 300 experiments 85 Structure learning success stories II: Phylogenetic Tree Reconstruction (Friedman et al.) Input: Biological sequences Human CGTTGC… Uses structural EM, with max-spanning-tree Chimp CCTAGG… in the inner loop Orang CGAACG… …. Output: a phylogeny leaf 86 Instances of graphical models Probabilistic models Graphical models Naïve Bayes classifier Directed Undirected Bayes nets MRFs Mixtures DBNs of experts Kalman filter model Ising model Hidden Markov Model (HMM) 87 ML enabling technologies Faster computers More data The web Parallel corpora (machine translation) Multiple sequenced genomes Gene expression arrays New ideas Kernel trick Large margins Boosting Graphical models … 88