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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 .
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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
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Learn to approximate function from a training set of (x,t) pairs
Supervised learning yes no Color Shape Size Output Blue Torus Big Y Square Small Star Red Arrow N F(x1, x2, x3) -> t Learn to approximate function from a training set of (x,t) pairs
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Supervised learning X1 X2 X3 T B Y S R A N Learner T Y N X1 X2 X3 T B
Training data X1 X2 X3 T B Y S R A N Learner Prediction T Y N Testing data X1 X2 X3 T B A S ? Y C Hypothesis
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Key issue: generalization
yes no ? ? Can’t just memorize the training set (overfitting)
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Hypothesis spaces Decision trees Neural networks K-nearest neighbors
Naïve Bayes classifier Support vector machines (SVMs) Boosted decision stumps …
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Perceptron (neural net with no hidden layers)
Linearly separable data
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Which separating hyperplane?
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The linear separator with the largest margin is the best one to pick
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What if the data is not linearly separable?
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kernel Kernel trick z3 x2 x1 z2 z1
Kernel implicitly maps from 2D to 3D, making problem linearly separable
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Support Vector Machines (SVMs)
Two key ideas: Large margins Kernel trick
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Boosting maximizes the margin
Simple classifiers (weak learners) can have their performance boosted by taking weighted combinations Boosting maximizes the margin
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Supervised learning success stories
Face detection Steering an autonomous car across the US Detecting credit card fraud Medical diagnosis …
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Unsupervised learning
What if there are no output labels?
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K-means clustering Guess number of clusters, K
Guess initial cluster centers, 1, 2 Assign data points xi to nearest cluster center Re-compute cluster centers based on assignments Reiterate
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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
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Hierarchical clustering
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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. .
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Discovering nonlinear manifolds
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Combining supervised and unsupervised learning
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Discovering rules (data mining)
Occup. Income Educ. Sex Married Age Student $10k MA M S 22 $20k PhD F 24 Doctor $80k MD 30 Retired $30k HS 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}
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Unsupervised learning: summary
Clustering Hierarchical clustering Linear dimensionality reduction (PCA) Non-linear dim. Reduction Learning rules
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From data visualization to causal discovery
Discovering networks ? From data visualization to causal discovery
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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 Qualitative level Decreasing detail
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Molecular level: Lysis-Lysogeny circuit in Lambda phage
Arkin et al. (1998), Genetics 149(4): 5 genes, 67 parameters based on 50 years of research Stochastic simulation required supercomputer
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Concentration level: metabolic pathways
Usually modeled with differential equations w23 g1 g2 g3 g4 g5 w12 w55
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Qualitative level: Boolean Networks
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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
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Graphical models: outline
What are graphical models? Inference Structure learning
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Simple probabilistic model: linear regression
Y = + X + noise Deterministic (functional) relationship Y X
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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
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Piecewise linear regression
Latent “switch” variable – hidden process at work
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Probabilistic graphical model for piecewise linear regression
input X Y Q Hidden variable Q chooses which set of parameters to use for predicting Y. Value of Q depends on value of input X. This is an example of “mixtures of experts” 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)
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Classes of graphical models
Probabilistic models Graphical models Undirected Directed Bayes nets MRFs DBNs
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Bayesian Networks Compact representation of probability distributions via conditional independence Family of Alarm 0.9 0.1 e b 0.2 0.8 0.01 0.99 B E P(A | E,B) Qualitative part: Directed acyclic graph (DAG) Nodes - random variables Edges - direct influence Earthquake Burglary Radio Alarm Call Together: Define a unique distribution in a factored form Quantitative part: Set of conditional probability distributions
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Example: “ICU Alarm” network
Domain: Monitoring Intensive-Care Patients 37 variables 509 parameters …instead of 254 PCWP CO HRBP HREKG HRSAT ERRCAUTER HR HISTORY CATECHOL SAO2 EXPCO2 ARTCO2 VENTALV VENTLUNG VENITUBE DISCONNECT MINVOLSET VENTMACH KINKEDTUBE INTUBATION PULMEMBOLUS PAP SHUNT ANAPHYLAXIS MINOVL PVSAT FIO2 PRESS INSUFFANESTH TPR LVFAILURE ERRBLOWOUTPUT STROEVOLUME LVEDVOLUME HYPOVOLEMIA CVP BP
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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 …
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Graphical models: outline
What are graphical models? p Inference Structure learning
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Probabilistic Inference
Posterior probabilities Probability of any event given any evidence P(X|E) Earthquake Radio Burglary Alarm Call Radio Call
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Viterbi decoding X1 X2 X3 Y1 Y3 Y2
Compute most probable explanation (MPE) of observed data Hidden Markov Model (HMM) X1 X2 X3 hidden Y1 Y3 observed Y2 “Tomato”
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Inference: computational issues
Easy Hard Dense, loopy graphs Chains Trees PCWP CO HRBP HREKG HRSAT ERRCAUTER HR HISTORY CATECHOL SAO2 EXPCO2 ARTCO2 VENTALV VENTLUNG VENITUBE DISCONNECT MINVOLSET VENTMACH KINKEDTUBE INTUBATION PULMEMBOLUS PAP SHUNT MINOVL PVSAT PRESS INSUFFANESTH TPR LVFAILURE ERRBLOWOUTPUT STROEVOLUME LVEDVOLUME HYPOVOLEMIA CVP BP Grids
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Inference: computational issues
Easy Hard Dense, loopy graphs Chains Trees PCWP CO HRBP HREKG HRSAT ERRCAUTER HR HISTORY CATECHOL SAO2 EXPCO2 ARTCO2 VENTALV VENTLUNG VENITUBE DISCONNECT MINVOLSET VENTMACH KINKEDTUBE INTUBATION PULMEMBOLUS PAP SHUNT MINOVL PVSAT PRESS INSUFFANESTH TPR LVFAILURE ERRBLOWOUTPUT STROEVOLUME LVEDVOLUME HYPOVOLEMIA CVP BP Grids Many difference inference algorithms, both exact and approximate
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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
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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
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Graphical models: outline
What are graphical models? Inference Structure learning p p
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Why Struggle for Accurate Structure?
Earthquake Alarm Set Sound Burglary Truth Missing an arc Adding an arc Earthquake Alarm Set Sound Burglary Earthquake Alarm Set Sound Burglary Cannot be compensated for by fitting parameters Wrong assumptions about domain structure Increases the number of parameters to be estimated Wrong assumptions about domain structure
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Scorebased 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 E B E A A B A B Search for a structure that maximizes the score
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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
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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 ...
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Local Search Operations
Typical operations: S C E D Add C D S C E D score = S({C,E} D) - S({E} D) Reverse C E Delete C E S C E D S C E D
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Problems with local search
Easy to get stuck in local optima “truth” you S(G|D)
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Problems with local search II
P(G|D) Picking a single best model can be misleading E R B A C
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Problems with local search II
P(G|D) Picking a single best model can be misleading E R B A C E R B A C E R B A C E R B A C E R B A C Small sample size many high scoring models Answer based on one model often useless Want features common to many models
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Bayesian Approach to Structure Learning
Posterior distribution over structures Estimate probability of features Edge XY Path X… Y … Bayesian score for G Feature of G, e.g., XY Indicator function for feature f
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Bayesian approach: computational issues
Posterior distribution over structures How compute sum over super-exponential number of graphs? MCMC over networks MCMC over node-orderings (Rao-Blackwellisation)
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Structure learning: other issues
Discovering latent variables Learning causal models Learning from interventional data Active learning
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Discovering latent variables
a) 17 parameters b) 59 parameters There are some techniques for automatically detecting the possible presence of latent variables
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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?
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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
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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 fingers Yellow fingers P(smoker|observe(yellow)) >> prior P(smoker | do(paint yellow)) = prior
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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”
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Learning from relational data
Can we learn concepts from a set of relations between objects, instead of/ in addition to just their attributes?
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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)
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ILP for learning protein folding: input
yes no TotalLength(D2mhr, 118) ^ NumberHelices(D2mhr, 6) ^ … 100 conjuncts describing structure of each pos/neg example
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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”: In English: “The protein P folds if it contains a long helix h1 at a secondary structure position between 1 and 3 and h1 is next to a second helix”
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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
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The future of machine learning for bioinformatics?
Oracle
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The future of machine learning for bioinformatics
Prior knowledge Hypotheses Replicated experiments Learner Biological literature Real world Expt. design “Computer assisted pathway refinement”
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The end
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Decision trees blue? yes oval? big? no no yes
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Decision trees blue? yes oval? big? no no yes
+ Handles mixed variables + Handles missing data + Efficient for large data sets + Handles irrelevant attributes + Easy to understand - Predictive power big? no no yes
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Feedforward neural network
input Hidden layer Output Weights on each arc Sigmoid function at each node
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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
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Nearest Neighbor Remember all your data When someone asks a question,
find the nearest old data point return the answer associated with it
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Nearest Neighbor ? - Handles mixed variables - Handles missing data
- Efficient for large data sets - Handles irrelevant attributes - Easy to understand + Predictive power
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Support Vector Machines (SVMs)
Two key ideas: Large margins are good Kernel trick
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SVM: mathematical details
Training data : l-dimensional vector with flag of true or false Separating hyperplane : Margin : margin Inequalities : Support vector expansion: Support vectors : Decision:
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Replace all inner products with kernels
Kernel function
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General lessons from SVM success: Large margin classifiers are good
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
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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
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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
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Inference Posterior probabilities Most likely explanation
Probability of any event given any evidence Most likely explanation Scenario that explains evidence Rational decision making Maximize expected utility Value of Information Effect of intervention Earthquake Radio Burglary Alarm Call Radio Call
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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.
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Structure learning success stories: gene regulation network (Friedman et al.)
Yeast data [Hughes et al 2000] 600 genes 300 experiments
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Input: Biological sequences
Structure learning success stories II: Phylogenetic Tree Reconstruction (Friedman et al.) Input: Biological sequences Human CGTTGC… Chimp CCTAGG… Orang CGAACG… …. Output: a phylogeny Uses structural EM, with max-spanning-tree in the inner loop leaf 10 billion years
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Instances of graphical models
Probabilistic models Graphical models Naïve Bayes classifier Undirected Directed Bayes nets MRFs Mixtures of experts DBNs Kalman filter model Ising model Hidden Markov Model (HMM)
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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 …
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