WELCOME BAYESIAN NETWORK

Transcription

1 WELCOME BAYESIAN NETWORK 1

2 OUTLINE HOW IS IT DIFFERENT? RIVEW OF PROBABILITY THEORY INDEPENDENCE OF EVENTS JOINT PROBABILITY DISTRIBUTION MARKOV CONDITION BAYESIAN NETWORK NAÏVE BAYES CLASSIFIER 2

3 DIFFERENCES DEPENDENCE AMONG FEATURES ABLE TO PREDICT VALUE OF ANY FEATURE MORE INSIGHT OF THE DATA ASSESSMENT ON ACCURACY OF PREDICTION 3

4 PROBABILITY THEORY P(A) = A / S A <= S 0 <= P(A) <= 1 4

5 INDEPENDENCE OF EVENTS TWO EVENTS ARE INDEPENDENT WHEN KNOWING ONE DOES NOT HELP US TO KNOW THE OTHER ONE BETTER P ( E F ) = P ( E ) I P ( E, F ) P ( E, F ) = P ( E ) P ( F ) P ( E F, G ) = P ( E G ) I P ( E, F G) P ( E, F G ) = P ( E G ) P ( F G ) 5

6 EXAMPLES FROM CARD PROBLEMS P (Queen) = 4 / 52 = 1 / 13 P (Spade) = 13 / 52 = 1 / 4 P (Queen Spade) = 1 / 13 = P (Queen) P (Queen, Spade) = 1 / 52 = P(Queen)P(Spade) P(Royal) = 12 / 52 = 3 / 13 P(Queen Royal) = 4 / 12 = 1 / 3 > P(Queen) 6

7 MORE EXAMPLES P(One) = 5 / 13 P(One Square) = 3 / 8 P(One Black) = 3 / 9 = 1 / 3 P(One Square, Black) = 2 / 6 = 1 / 3 P(One White) = 2 / 4 = 1 / 2 P(One Square, White) = 1 / 2 7

8 RANDOM VARIABLE IT IS EASY TO EXPRESS EVENTS IN TERMS OF RANDOM VARIABLES 8

9 JOINT PROBABILITY DISTRIBUTION (JPD) ALL POSSIBLE COMBINATIONS OF THE VALUES OF RANDOM VARIABLES P (x, y) = P (X = x, Y = y) 9

10 PROBLEM WITH JPD IF WE HAVE 10 RANDOM VARIABLES EACH WITH 10 DIFFERENT VALUES EXPONENTIAL COMPLEXITY CONDITIONAL INDEPENDENCE CAN HELP 10

11 CAUSAL GRAPH - DAG IT IS A DIRECTED ACYCLIC GRAPH EDGE REPRESENTS CAUSE EFFECT RELATIONSHIP DESCENDANT & NONDESCENDANT 11

12 MARKOV CONDITION WE HAVE A DAG, G = ( V, E ) PA X = SET OF PARENTS OF X ND X = SET OF NONDESCENDANTS OF X FOR ALL X є V, I P ( X, ND X PA X ) WHEN A DAG SATISFIES THE MARKOV CONDITION, IT IS CALLED BAYESIAN NETWORK 12

13 EXAMPLE P ( v, s, c ) = P ( v c ) P (s c) P (c) P (One, Square, Black) = P (One Black) P (Square Black) P(Black) = ( 1 / 3 ) ( 2 / 3) ( 9 / 13 ) = 2 / 13 13

14 MORE EXAMPLES NOTICE THAT WE NEED ONLY 11 PROBABILITIES INSTEAD OF 32 14

15 THE PROBLEM FINDING THE STRUCTURE CALCULATING THE CONDITIONAL PROBABILITY TABLES 15

16 Biological basis DNA (Storage of Genetic Information) RNA Polymerase (Copy DNA in RNA) Ribosome (Translate Genetic Information in Proteins) mrna (Storage & Transport of Genetic Information) Proteins (Expression of Genetic Information) *-PDB file 1L3A, Transcriptional Regulator Pbf-2 16

17 Biological basis How do proteins get regulated? E. coli operon lactose example : In normal time, E. coli uses glucose to get energy, but how does it react if there is no more glucose but only lactose? 17

18 Biological basis E. coli environment Glucose X Lactose Lactose decomposor (β-galactosidase) Lactose getter (permease) RNA Polymerase Gene Lac I associated protein Polymerase action is blocked because of a DNA lock 18

19 Biological basis E. coli environment Glucose X Lactose X Lactose decomposor (β-galactosidase) Lactose getter (permease) RNA Polymerase Lactose Lactose Lactose recruits gene laci associated protein unlocking the DNA that is then accessible to the polymerase 19

20 Biological basis Lactose decomposor (β-galactosidase) Lactose getter (permease) = inhibit 20

21 Biological basis E.coli environment Glucose Lactose X RNA Polymerase Lactose decomposor (β-galactosidase) c-amp CAP Lactose getter (permease) Lactose In absence of glucose, a polymerase magnet binds to the DNA to accelerate the products of information that help lactose decomposition 21

22 Biological basis Lactose decomposor (β-galactosidase) Lactose getter (permease) = activate = inhibit Research goal: Infer these links 22

23 Why? Get insights about cellular processes Help understand diseases Find drug targets 23

24 [mrna] How? Using gene expression data and tools for learning Bayesian networks Experiments * + Tools for Learning Bayesian networks Lactose decomposor (β-galactosidase) Lactose getter (permease) *-Spellman et al.(1998) Mol Biol Cell 9:

25 [mrna] What is gene expression data? Data showing the concentration of a specific mrna at a given time of the cell life. * Experiments A real value is coming from one spot and tells if the concentration Every of columns a specific are mrna the result is higher(+) of one image or lower(-) than the normal value *-Spellman et al.(1998) Mol Biol Cell 9:

26 What is Bayesian networks? Graphic representation of a joint distribution over a set of random variables. A C E B D P(A,B,C,D,E) = P(A)*P(B) *P(C A)*P(D A,B) *P(E D) Nodes represent gene expression while edges encode the interactions (cf. inhibition, activation) 26

27 Bayesian networks little problem A Bayesian network should be a DAG (Direct Acyclic Graph), but there are a lot of example of regulatory networks having directed cycles. Histeric oscillator * Transcription factor dimer Switch *-Husmeier D.,Bioinformatics,Vol. 19 no , pages

28 How can we deal with that? Using DBN (Dynamic Bayesian Networks*) and sequential gene expression data t t+1 A A1 A2 We unfold the network in time B B1 B2 DBN = BN with constraints on parents and children nodes *-Friedman, Murphy, Russell,Learning the Structure of Dynamic Probabilitic Networks 28

29 What are we searching for? A Bayesian network that is most probable given the data D (gene expression) We found this BN like that : BN* = argmax BN {P(BN D)} Where: Marginal likelihood Prior on network structure P( BN D) P( D BN ) P( BN ) P( D) Data probability Naïve approach to the problem : try all possible dags and keep the best one! 29

30 It is impossible to try all possible DAGs because The number of dags increases super-exponentially with the number of nodes n = 3 25 dags n = dags n = dags n = dags n = dags n = dags We are interested in problem having around 60 nodes. 30

31 Learning Bayesian Networks from data? Choosing search space method and a conditional distribution representation Networks space search methods Greedy hill-climbing Beam-search Stochastic hill-climbing Simulated annealing MCMC simulation Basically add, remove and reverse edges Conditional distribution representation Linear Gaussian Multinomial, binomial A B C P(a) =? P(b) =? P(c a,b) =? 31

32 Learning Bayesian Networks from data? Choosing search space method and a conditional distribution representation Networks space search methods Greedy hill-climbing Beam-search Stochastic hill-climbing Simulated annealing MCMC simulation Basically add, remove and reverse edges Conditional distribution representation Linear Gaussian Multinomial, binomial A B C P(a) =? P(b) =? P(c a,b) =? 32

33 We use three types of gene expression level? Sort Split data in 3 equal buckets Discretized data 33

34 Return on: Marginal likelihood Prior on network structure P( BN D) P( D BN ) P( BN ) P( D) Data probability 34

35 Insight on each terms P(BN) prior on network In our research, we always use a prior equals to 1 We could incorporate knowledge using it Eg. : we know the presence of an edge. If the edge is in the BN, P(BN) = 1 else P(BN) = 0 Efforts are made to reduce the search space by using knowledge eg. limit the number of parents or children 35

36 Insight on each terms P(D BN) marginal likelihood Easy to calculate using Multinomial distribution with Dirichlet prior * P( d bn) n qi ( N ij ) ( a s i 1 j 1 ( Nij Mij) k 1 ( aijk) ri ijk ijk ) *-Heckerman,A Tutorial on Learning With Bayesian Networks and Neapolitan,Learning Bayesian Networks 36

37 MCMC (Markov Chain Monte Carlo) simulation Markov Chain part: Zoom on a node of the chain C A B 1/5 P(BN new ) 1/5 C A B C A B 0 A 1/5 C A B 1/5 1/5 A C A B C B C B 37

38 MCMC (Markov Chain Monte Carlo) simulation Monte Carlo part: Choose next BN with probability P(BNnew) Accept the new BN with the following Metropolis Hastings acceptance criterion : PMH P( BNnew D) min 1, * P( BNnew) P( BNold D) P( D BNnew) P( BNnew) P( D) min 1, * P( BNnew) P( D BNold) P( BNold) P( D) P( D BNnew) P( BNnew) min 1, * P( BNnew) P( D BNold) P( BNold) P(D) is gone! 38

39 Monte Carlo part example : 1. Choose a random path. Each path having a P(BN new ) of 1/5 2. Choose another random number. If it is smaller than the Metropolis-Hasting criterion, accept BN new else return to BN old C A B 1/5 P(BN new ) 1/5 C A B C A B 0 A 1/5 C A B 1/5 1/5 A C A B C B C B 39

40 log(p(d BN)P(BN)) MCMC (Markov Chain Monte Carlo) simulation recap: Choose a starting BN at random Burning phase (generate 5*N BN from MCMC without storing them) Storing phase (get 100*N BN structure from MCMC) = Burning phase = Storing phase Iteration 40

41 Why 100*N BN and not only 1: Cause we don t have enough data and there are a lot of high scoring networks Instead, we associate confidence to edge. Eg. : how many time in the sample can we find edge going from A to B? We could fix a threshold on confidence and retrieve a global network construct with edges having confidence over the threshold 41

42 What we are working on: Mixing both sequential and non-sequential data to retrieve interesting relation between genes How? Using DBN and MCMC for sequential data + BN and MCMC for non-sequential 100*N networks from DBN 100*N networks from BN Information tuner Learn network 42

43 How to test the approach: Problem : There is no way to test it on real data cause there is no completely known network Solution : Work on realistic simulation where we know the network structure Example : * Simulate *-Hartemink A. Using Bayesian Network Inference Algorithms to Recover Molecular Genetic Regulatory Networks 43

44 How to test the approach: Info tuner BN MCMC DBN MCMC Sequential data Non-Sequential data Compare using ROC curves * Simulate *-Hartemink A. Using Bayesian Network Inference Algorithms to Recover Molecular Genetic Regulatory Networks 44

45 Test description: Generate 60 sequential data Generate 120 non-sequential data (~reality proportion) Run DBN MCMC on sequential data keep 100*N sample net Run BN MCMC on non-sequential data keep 100*N sample net Test performance using weight on sample 0 BN 1 DBN.05 BN 0.95 DBN 0.95 BN.05 DBN 1 BN 0 DBN The metric used is the area under ROC curve. Perfect learner gets 1.0, random gets 0.5 and the worst one gets 0. 45

46 Area under ROC curve Results: BN 1 DBN 46

47 Perspective: Working on more sophisticated ways to mix sequential and nonsequential data Working on real cases: Yeast cell-cycle Arabidopsis Thaliana circadian rhythm Real data also means missing values Evaluate missing values solution (EM, KNNImpute) 47

48 Acknowledgements: François Major 48

49 Why are there missing datas? Experimental problems Low correlation 49

50 ROC Curve Receiver Operating Characteristic curve * *- 50

51 MCMC simulation and number of sampled networks ROC area ROC curve area in function of the number of sample networks from M CM C simulation for N= # of samples from MCMC

52 GRN Models Directed and undirected graphs Bayesian networks Boolean networks Generalized logical networks Non-linear ordinary differential equations Piecewise linear differential equations Qualitative differential equations Partial differential equations Stochastic master equations Rule based formalisms

53 GRN Models Hidde de Jong, Modeling and simulation of genetic regulatory systems: a literature review; J Comput Biol. 2002;9(1): Review. Dynamics Data Node States Computation Complexity 53 23/2/2008

54 What class of models should be chosen? The selection should be made in view of data requirements goals of modeling and analysis /2/2008

55 Classical Tradeoff A fine model with many parameters may be able to capture detailed low-level phenomena (protein concentrations, reaction kinetics); requires very large amounts of data for inference, lest the model be overfit. A coarse model with lower complexity may succeed in capturing high-level phenomena (which genes are ON/OFF); requires smaller amounts of data /2/2008

56 Occam s Razor 56 23/2/2008

57 Model Reliability and Adequacy P is the set of all possible observations S set of all observations made on the study system M is the set of all model outputs Q=S пm Q P 57 23/2/2008

58 Model Reliability and Adequacy P P Q S Useless Model Dream Model 58 23/2/2008

59 Model Reliability and Adequacy Q P Q P M S 59 Incomplete model Model reliability: Q / M Model adequacy: Q / S Complete, but erring model 23/2/2008

60 GRN Models Directed and undirected graphs Bayesian networks Boolean networks Generalized logical networks Non-linear ordinary differential equations Piecewise linear differential equations Qualitative differential equations Partial differential equations Stochastic master equations Rule based formalisms 60 23/2/2008

61 Directed and undirected Graphs Probably most straightforward way to model a GRN G=<V,E> V set of vertices Set of edges E=<i,j> where i,j є V, head and tail of edge Additional labels denote positive/negative influence 61 23/2/2008

62 Directed and undirected Graphs Advantages: Intuitive way of visualization Common and well explored graph algorithms can make biologically relevant predictions about GRSes: paths between genes may reveal missing regulatory interactions or provide clues about redundancy cycles in the network point at feedback relations connectivity characteristics give indication of the complexity loosely connected subgraphs point at functional modules Disadvantages: Time does not play a role Too much abstraction: very simplified model far from reality 62 23/2/2008

63 GRN Models Directed and undirected graphs Bayesian networks Boolean networks More popular and efficient Generalized logical networks Non-linear ordinary differential equations Piecewise linear differential equations Qualitative differential equations Partial differential equations Stochastic master equations Rule based formalisms 63 23/2/2008

64 Boolean Network Model A Boolean network is defined by a set of nodes, V = {x 1, x 2,..., x n }, and a list of Boolean functions, F= {f 1, f 2,..., f n } Each x k represents the state (expression) of a gene, g k, where x k = 1 the gene is expressed or x k = 0, the gene is not expressed 64 23/2/2008

65 Boolean Network t x x x At any given time, combining the gene states gives a gene activity pattern (GAP). GAP 65 23/2/2008

66 Boolean Network Given a GAP at time t, a deterministic function (a set of logical rules) provides the GAP at time t +1. t x x x GAP t GAP t /2/2008

67 Boolean Network t x x x /2/2008

68 Boolean Network Example t x x x /2/2008

69 Boolean Network t x x x /2/2008

70 Boolean Network Example t t x 1 x 2 x 3 x x t+1 x 1 x /2/2008

71 Boolean Network Example t t t+1 x 1 x 1 or x 2 x 3 x x x /2/2008

72 Boolean Network Example t t x 1 x 2 x 3 x x t+1 x 1 x /2/2008

73 Boolean Network Example t t x 1 x 2 x 3 x x t+1 x 1 x /2/2008

74 Boolean Network Example t t t+1 x 1 x 1 or x 2 x 3 x x x For each node there will be 2^2^k possible functions 74 23/2/2008

75 Boolean Network Example t t+1 x 1 x 2 x 3 x 1 x 2 x 3 or nor nand t x x x /2/2008

76 Boolean Network Example AND NAND NOT 76 I. Shmulevich et al., Bioinformatics (2002), 18 (2): /2/2008

77 Boolean Networks Summary 77 Advantages Efficient analysis of large RN Positive/negative feedback-cycles can be modeled with BN s Disadvantages Strong simplifying assumptions gene is either on or off, no in between states The computation time is very high or often impractical to construct large-scale gene networks Very susceptible to noise There are situations where boolean idealisation is not appropriate more general methods required 23/2/2008

78 Bayesian Networks A gene regulatory network is represented by directed acyclic graph: Vertices correspond to genes. Edges correspond to direct influence or interaction. For each gene x i, a conditional distribution p(x i ancestors(x i ) ) is defined. The graph and the conditional distributions, uniquely specify the joint probability distribution /2/2008

79 Bayesian Network Example x 1 x 2 x 4 x 3 Conditional distributions: p(x 1 ), p(x 2 ), p(x 3 x 2 ), p(x 4 x 1, x 2 ), p(x 5 x 4 ) x 5 p(x) =p(x) = p(x 1 ) p(x 2 ) p(x 3 x 2 ) p(x 4 x 1, x 2 ) p(x 5 x 4 ) 79 23/2/2008

80 Learning Bayesian Models Using gene expression data, the goal is to find the bayesian network that best matches the data. Recovering optimal conditional probability distributions when the graph is known is easy. Recovering the structure of the graph is NP hard (non-deterministic polynomial ). But, good statistics are available: What is the likelihood of a specific assignment? What is the distribution of x i given x j? 80 23/2/2008

81 Issues with Bayesian Models Computationally intensive. Requires lots of data. Does not allow for feedback loops which are known to play an important role in biological networks. Does not make use of the temporal aspect of the data. Dynamical Bayesian Networks aim at solving some of these issues but they require even more data /2/2008

82 Differential Equations Typically uses linear differential equations to model the gene trajectories: dx i (t) / dt = a 0 + a i,1 x 1 (t)+ a i,2 x 2 (t)+ + a i,n x n (t) Several reasons for that choice: lower number of parameters implies that we are less likely to over fit the data sufficient to model complex interactions between the genes 82 23/2/2008

83 Small Network Example + x 1 x 4 + x 2 + _ x 3 83 dx 1 (t) / dt = x 1 (t) dx 2 (t) / dt = x 3 (t) x 4 (t) dx 3 (t) / dt = x 1 (t) x 3 (t) dx 4 (t) / dt = x 1 (t) x 3 (t) x 4 (t) 23/2/2008

84 Small Network Example _ 84 _ + x 1 x 4 + x 2 + _ x 3 dx 1 (t) / dt = x 1 (t) dx 2 (t) / dt = x 3 (t) x 4 (t) dx 3 (t) / dt = x 1 (t) x 3 (t) dx 4 (t) / dt = x 1 (t) x 3 (t) x 4 (t) one interaction coefficient 23/2/2008

85 Small Network Example _ 85 _ + x 1 x 4 + x 2 + _ x 3 dx 1 (t) / dt = x 1 (t) dx 2 (t) / dt = x 3 (t) x 4 (t) dx 3 (t) / dt = x 1 (t) x 3 (t) dx 4 (t) / dt = x 1 (t) x 3 (t) x 4 (t) constant coefficients 23/2/2008

86 Problem Revisited a 0,i a 1,i a 2,i a 3,i a 4,i x x x x Given the time-series data, can we find the interactions coefficients? 23/2/2008

87 Issues with Differential Equations Even under the simplest linear model, there are m(m+1) unknown parameters to estimate: m(m-1) directional effects m self effects m constant effects Number of data points is mn and we typically have that n << m (few time-points). To avoid over fitting, extra constraints must be incorporated into the model such as: Smoothness of the equations Sparseness of the network (few non-null interaction coefficients) 87 23/2/2008

88 OUTLINES What is the Gene Regulatory Network? GRN Application of GRN GRN Construction Methodology GRN modeling steps GRN Models GRS Software Next work Reference 88 23/2/2008

89 GRN Software GNA: Genetic Network Analyzer Helix Bioinformatics /2/2008

90 GRN Software Probabilistic Boolean Networks (PBN) Matlab Tool Box Ilya Shmulevich Institute for Systems Biology 90 23/2/2008

91 OUTLINES What is the Gene Regulatory Network? GRN Application of GRN GRN Construction Methodology GRN modeling steps GRN Models GRS Software Next work Reference 91 23/2/2008

92 Future Work: Literature Review Study the noisy natural of Microarray Data. Study in depth the existing modeling methodology. Focus on specialized problem like cancer.

93 Future Work : GSP Statistics Books Genomics signal processing and statistics, Edward,2006 Introduction to genomics signal processing with control, Ily,2006 Computational and Statistical Approaches to Genomics (Springer, 2006), Ily

94 Future Work : Statistics Books Handbook of Computational Statistics An Introduction to Statistical Signal Processing, Robert M. Gray,2007 fundamentals of statistical signal processing :estimation theory, steven kay nonlinear signal processing a statistical approach, Gonzalo R,2005 Inference_in_HMM, Olivier Cappe, /2/2008

95 Future Work : Modeling Books Modeling and Control of Complex Systems (Control Engineering) by Petros A. Ioannou, Andreas Pitsillides,2008 MODELING BIOLOGICAL SYSTEMS: Principles and Applications2005 gene regulation and metabolism postgenomic computational approaches, Julio, /2/2008

96 Future Work: Resources IEEE Transactions on Computational Biology and Bioinformatics IEEE International Workshop on Genomic Signal Processing and Statistics IEEE Journal of Selected Topics in Signal Processing: Special Issue on Genomic and Proteomic Signal Processing EURASIP Journal of Bioinformatics and Systems Biology Special issue of the on Genetic Regulatory Networks IEEE Signal Processing Magazine on Signal Processing Special issue of the Methods in Genomics and Proteomics IEEE Transactions on Signal Processing Special Genomic Signal Processing issue of the Workshop on Discrete Models for Genetic Regulatory Networks 96 23/2/2008

97 OUTLINES What is the Gene Regulatory Network? GRN Application of GRN GRN Construction Methodology GRN modeling steps GRN Models GRS Software Next work Reference 97 23/2/2008

98 Reference Hidde de Jong, Modeling and simulation of genetic regulatory systems: a literature review; J Comput Biol. 2002;9(1): Review. BAYESIAN ROBUSTNESS IN THE CONTROL OF GENE REGULATORY NETWORKS Ranadip Pal1, Aniruddha Datta2, Edward R. Dougherty Anastassiou, D. (2001). Genomic Signal Processing. IEEE Signal Processing Dougherty, E. R. and A. Datta (2005). "Genomic signal processing: diagnosis and therapy." Signal Processing Magazine, IEEE 22(1): Vaidyanathan, P. P. (2004). Genomics and Proteomics: A Signal Processorapos's Tour. Circuits and Systems Magazine, IEEE. 4: /2/2008

99 Reference Vaidyanathan, P. P. and B.-J. Yoon (2004). "The role of signal-processing concepts in genomics and proteomics." Journal of the Franklin Institute.(Special Issue on Genomics) /2/2008

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101 NAÏVE BAYES FORMULATION WE NEED P ( C F1, F2,, Fn) IT CAN BE SHOWN THAT - 101

102 THANK YOU ANY QUESTION? 102