6.047 / Computational Biology: Genomes, Networks, Evolution Fall 2008

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1 MIT OpenCourseWare / Computational Biology: Genomes, Networks, Evolution Fall 2008 For information about citing these materials or our Terms of Use, visit:

2 6.047/6.878 Computational Biology: Genomes, Networks, Evolution Introduction to Bayesian Networks

3 Overview We have looked at a number of graphical representations of probability distributions DAG example: HMM Undirected graph example: CRF Today we will look at a very general graphical model representation Bayesian Networks One application modeling gene expression Aviv Regev guest lecture an extension of this basic idea

4 Probabilistic Reconstruction Expression data gives us information about what genes tend to be expressed with others In probability terms, information about the joint distribution over gene states X: P(X)=P(X 1,X 2,X 3,X 4,,X m ) Can we model this joint distribution?

5 Bayesian Networks Directed graph encoding joint distribution variables X P(X) = P(X1,X2,X3,,XN) Learning approaches Inference algorithms Captures information about dependency structure of P(X)

6 Example 1 Icy Roads I Icy Roads W Watson Crashes Causal impact H Holmes Crashes Assume we learn that Watson has crashed Given this causal network, one might fear Holmes has crashed too. Why?

7 Example 1 Icy Roads I Icy Roads W Watson Crashes H Holmes Crashes Now imagine we have learned that roads are not icy We would no longer have an increased fear that Holmes has crashed

8 Conditional Independence We Know I I Icy Roads No Info on I W Watson Crashes H Holmes Crashes If we know nothing about I, W and H are dependent If we know I, W and H are conditionally independent

9 Conditional Independency Independence of 2 random variables X Y PXY (, ) = PXPY ( ) ( ) Conditional independence given a third X Y Z P( X, Y Z) = P( X Z) P( Y Z) but PXY (, ) PXPY ( ) ( ) necessarily

10 Example 2 Rain/Sprinkler R Rain S Sprinkler W Watson Wet H Holmes Wet Holmes discovers his house is wet. Could be rain or his sprinkler.

11 Example 2 Rain/Sprinkler R Rain S Sprinkler W Watson Wet H Holmes Wet Now imagine Holmes sees Watsons grass is wet Now we conclude it was probably rain And probably not his sprinkler

12 Explaining Away R Rain S Sprinkler W Watson Wet H Holmes Wet Initially we had two explanations for Holmes wet grass. But once we had more evidence for R, this explained away H and thus no reason for increase in S

13 Conditional Dependence Don t know H R Rain S Sprinkler W Watson Wet H Holmes Wet Know H If we don t know H, R and S are independent But if we know H, R and S are conditionally dependent

14 Graph Semantics Three basic building blocks X Z X Z X Z Y Y Y Serial Diverging Converging Each implies a particular independence relationship

15 Chain/Linear Conditional Independence X Y Z X Z X Y Z X Z Y

16 Diverging Conditional Independence X Y Z X Z X Y Z X Z Y

17 Converging Conditional Dependence - Explaining Away X Y Z X Z X Y Z X Z Y

18 Graph Semantics Three basic building blocks X Z Y A B. Converging

19 D-Separation Three semantics combine in concept of d-separation Definition : Two nodes A and B are d- separated (or blocked) if for every path p between A and B there is a node V such that either 1. The connection is serial or diverging and V is known 2. The connection is converging and V and all of its descendants are unknown If A and B are not d-separated, they are d- connected

20 Equivalence of Networks Two structures are equivalent if they represent same independence relationship - they encode the same space of probability distributions Example X Z X Z X Z Y Y Y Will return to this when we consider causal vs probabilistic networks

21 Bayesian Networks A Bayesian network (BN) for X = {X 1, X 2, X 3,, X n } consists of: A network structure S Directed acyclic graph (DAG) Nodes => random variables X Encodes graph independence sematics Set of probability distributions P Conditional Probability Distribution (CPD) Local distributions for X

22 Example Bayesian Network R Rain S Sprinkler W Watson Wet H Holmes Wet P( X) = PRPS ( ) ( RPW ) ( RSPH, ) ( RSW,, ) = PRPSPW ( ) ( ) ( RPH ) ( SR, )

23 BN Probability Distribution Only need distributions over nodes and their parents n PX ( ) = PX ( X,..., X ) = i= 1 n i= 1 i 1 i 1 PX ( pax ( )) i i

24 BNs are Compact Descriptions of P(X) Independencies allow us to factorize distribution X2 X1 X3 Example - Assume 2 states per node PX ( ) = P(X )P(X X )P(X X,X ) P(X X,X,X )P(X5 X,X,X,X ) = 62 entries X4 X5 n PX ( ) = PX ( pax ( )) = i= 1 i P(X1)P(X2 X1)P(X3 X1) i P(X4 X2)P(X5 X3) = 18 entries

25 Recall from HMM/CRF Lecture Y 1 Y 2 Y 3 Y 4 Y 1 Y 2 Y 3 Y 4 X 1 X 2 X 3 X 4 X 2 Directed Graph Semantics Potential Functions over Cliques (conditioned on X) Markov Random Field P(X,Y) = Factorization all nodes v P(v parents(v)) ( ) ( ) = P Y Y P X Y i i-1 i i P(Y X) = Factorization all nodes v P(v clique(v),x) ( ) = P Y Y,X i i-1

26 CPDs Discrete R S R S H P(H R,S) W Continuous H

27 Bayesian Networks for Inference Observational inference Observe values (evidence on) of a set of nodes, want to predict state of other nodes Exact Inference Junction Tree Algorithm Approximate Inference Variational approaches, Monte Carlo sampling

28 Walking Through an Example R S P 0 (R) W H P 0 (S) S=y S=n P 0 (W R) W=y W=n P 0 (H R,S) S=y 1,0 0.9,0.1 S=n 0,0 0,1

29 Walking Through an Example R S We define two clusters: -WR, RHS W H The key idea: the clusters only communicate through R If they agree on R, all is good WR R RHS

30 Walking Through an Example We will find it easier to work on this representation: WR R RHS We then need P(WR) and P(RHS): P(WR) =P(R)P(W R) P(RHS) =P(R)P(S)P(H R,S) P 0 (R) P 0 (S) S=y S=n P 0 (W R) W=y W=n P 0 (H R,S) S=y 1,0 0.9,0.1 S=n 0,0 0,1

31 Walking Through an Example We will find it easier to work on this representation: P 0 (R) WR R RHS We then need P(WR) and P(RHS): P(WR) =P(R)P(W R) P(RHS) =P(R)P(S)P(H R,S) P 0 (W,R) W=y W=n P 0 (R,H,S) S=y 0.02, ,0.008 S=n 0.18,0 0,0.72

32 Walking Through an Example P 0 (R) R=y R=n WR R RHS Note that by marginalizing out W from P 0 (W,R) we get P 0 (W)=(0.36,0.64) This is our initial belief in Watsons grass being (wet,not wet) P 0 (W,R) W=y W=n P 0 (R,H,S) S=y 0.02, ,0.008 S=n 0.18,0 0,0.72

33 Walking Through an Example H=y P 0 (R) WR R RHS Now we observe H=y We need to do three things: 1. Update RHS with this info 2. Calculate a new P 1 (R) 3. Transmit P 1 (R) to update WR P 0 (W,R) W=y W=n P 0 (R,H,S) S=y 0.02, ,0.008 S=n 0.18,0 0,0.72

34 Walking Through an Example H=y P 0 (R) WR R RHS Updating RHS with H=y We can simply - Zero out all entries in RHS where H=n P 0 (W,R) W=y W=n P 0 (R,H,S) S=y 0.02, ,0 S=n 0.18,0 0,0

35 Walking Through an Example H=y P 0 (R) WR R RHS Updating RHS with H=y We can simply - Zero out all entries in RHS where H=n - Re-normalize But you can see that this changes P(R) from the perspective of RHS P 0 (W,R) W=y W=n P 1 (R,H,S) S=y 0.074, ,0 S=n 0.662,0 0,0

36 Walking Through an Example H=y P 0 (R) WR R RHS Calculate new P 1 (R) Marginalize out H,S from RHS for: P 1 (R) = (0.736,0.264) Note also P 1 (S) =(0.339,0.661) P 0 (S) =(0.1,0.9) P 0 (W,R) W=y W=n P 1 (R,H,S) S=y 0.074, ,0 S=n 0.662,0 0,0

37 Walking Through an Example P 1 (R) H=y P 1 (R) WR R RHS Transmit P 1 (R) to update WR P (W,R)=P(W R)P (R) =P 0(W,R) P 0 P(R) (R) P 0 (W,R) W=y W=n P 1 (R,H,S) S=y 0.074, ,0 S=n 0.662,0 0,0

38 Walking Through an Example P 1 (R) H=y P 1 (R) WR R RHS Transmit P 1 (R) to update WR P (W,R)=P(W R)P (R) =P 0(W,R) P 0 P(R) (R) P 1 (W,R) W=y W=n P 1 (R,H,S) S=y 0.074, ,0 S=n 0.662,0 0,0

39 Walking Through an Example P 1 (R)/P 0 (R) P 1 (R) H=y P 1 (R) WR R RHS Transmit P 1 (R) to update WR P (W,R)=P(W R)P (R) =P 0(W,R) P 0 P(R) (R) P 1 (W,R) W=y W=n P 1 (R,H,S) S=y 0.074, ,0 S=n 0.662,0 0,0

40 Walking Through an Example P 1 (R)/P 0 (R) P 1 (R) H=y P 1 (R) WR R RHS Transmit P 1 (R) to update WR P (W,R)=P(W R)P (R) 1 1 P 1 0 P(R) 1 =P 0(W,R) P(R) (W=y)=0.788 P(W=y)= P 1 (W,R) W=y W=n P 1 (R,H,S) S=y 0.074, ,0 S=n 0.662,0 0,0

41 Walking Through an Example W=y P 2 (R) P 2 (R)/P 1 (R) H=y P 1 (R) WR R RHS Now we observe W=y 1. Update WR with this info 2. Calculate a new P 2 (R) 3. Transmit P 2 (R) to update WR P 1 (W,R) W=y W=n P 1 (R,H,S) S=y 0.074, ,0 S=n 0.662,0 0,0

42 Walking Through an Example W=y P 2 (R) P 2 (R)/P 1 (R) H=y P 2 (R) WR R RHS P 2 (S=y)=0.161 P 1 (S=y)=0.339 P 0 (S=y)=0.1 - R is almost certain - We have explained away H=y -S goes low again P 2 (W,R) W=y W=n 0 0 P 2 (R,H,S) S=y 0.094, ,0 S=n 0.839,0 0,0

43 Message Passing in Junction Trees P*(S)/P(S) e X S Y Separator State of separator S is information shared between X and Y When Y is updated, it sends a message to X Message has information to update X to agree with Y on state of S

44 Message Passing in Junction Trees 2 G 3 4 H 1 e e F E D 1 1 A B C A node can send one message to a neighbor, only after receiving all messages from each other neighbor When messages have been passed both ways along a link, it is consistent Passing continues until all links are consistent

45 HUGIN Algorithm A simple algorithm for coordinated message passing in junction trees Select one node, V, as root Call CollectEvidence(V): Ask all neighbors of V to send message to V. If not possible, recursively pass message to all neighbors but V Call DistributeEvidence(V): Send message to all neighbors of V Recursively send message to all neighbors but V

46 BN to Junction Tree R S A topic by itself. In summary: W H Moral Graph undirected graph with links between all parents of all nodes Triangulate add links so all cycles>3 have cord Cliques become nodes of Junction Tree WR R RHS

47 Recall from HMM/CRF Lecture Y 1 Y 2 Y 3 Y 4 Y 1 Y 2 Y 3 Y 4 X 1 X 2 X 3 X 4 X 2 Directed Graph Semantics Potential Functions over Cliques (conditioned on X) Markov Random Field P(X,Y) = Factorization all nodes v P(v parents(v)) ( ) ( ) = P Y Y P X Y i i-1 i i P(Y X) = Factorization all nodes v P(v clique(v),x) ( ) = P Y Y,X i i-1

48 Applications G9 G8 G10 G7 G6 G1 G2 G4 G3 G5 Measure some gene expression predict rest

49 Latent Variables G9 G8 Drugs D1 G10 G7 G6 D2 G1 G2 G4 G3 G5 D3 D4

50 Latent Variables Env Drugs ph G9 G8 D1 O 2 G10 G7 G6 D2 G1 G2 G4 G3 G5 D3 D4

51 Latent Variables Env Drugs ph G9 G8 D1 O 2 G10 G7 G6 D2 G1 G2 G4 G3 G5 D3 D4 Metabolites M1 M2 M3 M4

52 Observation vs Intervention Arrows not necessarily causal BN models probability and correlation between variables For applications so far, we observe evidence and want to know states of other nodes most likely to go with observation What about interventions?

53 Example Sprinkler W Season Rain H Sprinkler If we observe the Sprinkler on Holmes grass more likely wet And more likely summer But what if we force sprinkler on Intervention cut arrows from parents

54 Causal vs Probabilistic X Y Z This depends on getting the arrows correct! X Y Z Flipping all arrows does not change independence relationships But changes causality for interventions

55 Learning Bayesian Networks Given a set of observations D (i.e. expression data set) on X, we want to find: 1. A network structure S 2. Parameters, Θ, for probability distributions on each node, given S Relatively Easy

56 Learning Θ Given S, we can choose maximum likelihood parameter Θ n θ arg max PD ( θ, S) PX ( pax ( ), θ) = = θ i= 1 i i We can also choose to include prior information P(Θ) in a bayesian approach P( θ S,D) = P( S, D θ) P( θ) θbayes = θp( θ S, D) dθ

57 Learning Bayesian Networks Given a set of observations D (i.e. expression data set) on X, we want to find: 1. A network structure S NP-Hard 2. Parameters, Θ, for probability distributions on each node, given S

58 Learning S Find optimal structure S given D PS ( D) PD ( SPS ) ( ) P( D S) = P( D, S) P( S) d θ θ θ In special circumstances, integral analytically tractable (e.g. no missing data, multinomial, dirichlet priors)

59 Learning S Heuristic Search Compute P(S D) for all networks S Select S* that maximizes P(S D) Problem 1: number of S grows super-exponentially with number of nodes no exhaustive search, use hill climbing, etc.. Problem 2: Sparse data and overfitting P(S D) P(S D) S* S S* S Lots of data Sparse data

60 Model Averaging Rather than select only one model as result of search, we can draw many samples from P(M D) and model average PE ( D) = PExy ( DSPS, ) ( D) xy = samples samples 1 ( SPS ) ( D) xy How do we sample...?

61 Sampling Models - MCMC Sample from Markov Chain Monte Carlo Method Direct approach intractable due to partition function MCMC PD ( Ss) PS ( k) PS ( D) = P( D Ss ) P( Sk ) Propose Given S old, propose new S new with probability Q(S new S old ) Accept/Reject Accept S new as sample with PD ( Snew) PS ( new) QS ( old S ) new p = min 1, PD ( Sold ) PS ( old ) QS ( new Sold ) k