Introduction to Probabilistic Models for Bioinformatics
- 1. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Introduction to Probabilistic Models for Bioinformatics Igor Bogicevic (igor.bogicevic@sbgenomics.com) July 3, 2011 EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 2. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Short introduction to Bioinformatics Bioinformatics is the application of statistics and computer science to the field of molecular biology. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 3. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Short introduction to Bioinformatics Bioinformatics is the application of statistics and computer science to the field of molecular biology. Major research efforts in the field include sequence alignment, gene finding, genome assembly, drug design, drug discovery, protein structure alignment, protein structure prediction, prediction of gene expression and protein-protein interactions, genome-wide association studies and the modeling of evolution. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 4. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Short introduction to Bioinformatics Bioinformatics is the application of statistics and computer science to the field of molecular biology. Major research efforts in the field include sequence alignment, gene finding, genome assembly, drug design, drug discovery, protein structure alignment, protein structure prediction, prediction of gene expression and protein-protein interactions, genome-wide association studies and the modeling of evolution. At the current moment, given the enormous volumes of sequenced data, one of the biggest challenges is not producing, but actually understanding the data. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 5. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us What are the Probabilistic Models? There are 2 basic definitions: EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 6. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us What are the Probabilistic Models? There are 2 basic definitions: Statistical analysis tool that estimates, on the basis of past (historical) data, the probability of an event occurring again. Probabilistic model is a system that simulates the object under the consideration and produces different outcomes with different probabilities. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 7. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us What are the Probabilistic Models? There are 2 basic definitions: Statistical analysis tool that estimates, on the basis of past (historical) data, the probability of an event occurring again. Probabilistic model is a system that simulates the object under the consideration and produces different outcomes with different probabilities. Simple example - rolling a die. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 8. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us What are the Probabilistic Models? There are 2 basic definitions: Statistical analysis tool that estimates, on the basis of past (historical) data, the probability of an event occurring again. Probabilistic model is a system that simulates the object under the consideration and produces different outcomes with different probabilities. Simple example - rolling a die. A bit more relevant example - random sequence model in DNA . Biological sequences are strings from a finite alphabet of residues, most commonly either four nucleotides, or twenty amino acids. Imagine that a residue a occurs with probability qa , if protein or DNA sequence is denoted x1 ...xn , then probability of the whole sequence is: n Y qx1 qx2 ...qxn = qxi i=1 EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 9. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Sequence Alignment Sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 10. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Sequence Alignment Sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences. A variety of computational algorithms have been applied to the sequence alignment problem, i.e. dynamic programming, heuristic algorithms, probabilistic methods. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 11. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Sequence Alignment Sequence alignment is a way of arranging the sequences of DNA, RNA, or protein to identify regions of similarity that may be a consequence of functional, structural, or evolutionary relationships between the sequences. A variety of computational algorithms have been applied to the sequence alignment problem, i.e. dynamic programming, heuristic algorithms, probabilistic methods. Common formats for representing alignments are FASTA and GenBank format EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 12. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 13. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Pairwise Alignment Pairwise sequence alignment methods are used to find the best-matching piecewise (local) or global alignments of two query sequences. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 14. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Pairwise Alignment Pairwise sequence alignment methods are used to find the best-matching piecewise (local) or global alignments of two query sequences. The three primary methods of producing pairwise alignments are dot-matrix methods, dynamic programming, and word methods. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 15. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Pairwise Alignment Pairwise sequence alignment methods are used to find the best-matching piecewise (local) or global alignments of two query sequences. The three primary methods of producing pairwise alignments are dot-matrix methods, dynamic programming, and word methods. Needleman-Wunsch algorithm (Global Alignment) EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 16. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Pairwise Alignment Pairwise sequence alignment methods are used to find the best-matching piecewise (local) or global alignments of two query sequences. The three primary methods of producing pairwise alignments are dot-matrix methods, dynamic programming, and word methods. Needleman-Wunsch algorithm (Global Alignment) Smith-Waterman algorithm (Local Alignment) EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 17. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Pairwise Alignment Pairwise sequence alignment methods are used to find the best-matching piecewise (local) or global alignments of two query sequences. The three primary methods of producing pairwise alignments are dot-matrix methods, dynamic programming, and word methods. Needleman-Wunsch algorithm (Global Alignment) Smith-Waterman algorithm (Local Alignment) FASTA/BLAST Algorithms (k-tuple heuristic methods, often combined with dynamic models) EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 18. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Pairwise Alignment Pairwise sequence alignment methods are used to find the best-matching piecewise (local) or global alignments of two query sequences. The three primary methods of producing pairwise alignments are dot-matrix methods, dynamic programming, and word methods. Needleman-Wunsch algorithm (Global Alignment) Smith-Waterman algorithm (Local Alignment) FASTA/BLAST Algorithms (k-tuple heuristic methods, often combined with dynamic models) Gap Penalities - modeling a cost of a gap in matched sequences (linear, affine, etc.) EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 19. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Example - Smith-Waterman: A matrix H is built as follows: H(i, 0) = 0, 0 ≤ i ≤ m H(0, j) = 0, 0 ≤ j ≤ n if ai = bj then w (ai , bj ) = w (match) or if ai ! = bj then w (ai , bj ) = w (mismatch) 8 9 > > 0 > > H(i − 1, j − 1) + w (ai , bj ) Match/Mismatch < = H(i, j) = max , 1 ≤ i ≤ m, 1 ≤ j ≤ n > H(i − 1, j) + w (ai , −) > Deletion > > H(i, j − 1) + w (−, bj ) Insertion : ; EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 20. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Sequence 1 = ACACACTA, Sequence 2 = AGCACACA EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 21. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Sequence 1 = ACACACTA, Sequence 2 = AGCACACA w(match) = +2 w(a,-) = w(-,b) = w(mismatch) = -1 − A C A C A C T A 0 1 B− 0 0 0 0 0 0 0 0 0C BA 0 2 1 2 1 2 1 0 2C B C BG 0 1 1 1 1 1 1 0 1C B C BC 0 0 3 2 3 2 3 2 1C B C H=B BA 0 2 2 5 4 5 4 3 4C C BC 0 1 4 4 7 6 7 6 5C B C BA 0 2 3 6 6 9 8 7 8C B C @C 0 1 4 5 8 8 11 10 9A A 0 2 3 6 7 10 10 10 12 EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 22. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Sequence 1 = ACACACTA, Sequence 2 = AGCACACA w(match) = +2 w(a,-) = w(-,b) = w(mismatch) = -1 − A C A C A C T A 0 1 B− 0 0 0 0 0 0 0 0 0C BA 0 2 1 2 1 2 1 0 2C B C BG 0 1 1 1 1 1 1 0 1C B C BC 0 0 3 2 3 2 3 2 1C B C H=B BA 0 2 2 5 4 5 4 3 4C C BC 0 1 4 4 7 6 7 6 5C B C BA 0 2 3 6 6 9 8 7 8C B C @C 0 1 4 5 8 8 11 10 9A A 0 2 3 6 7 10 10 10 12 In the example, the highest value corresponds to the cell in position (8,8). The walk back corresponds to (8,8), (7,7), (7,6), (6,5), (5,4), (4,3), (3,2), (2,1), (1,1), and (0,0) Sequence 1 = A-CACACTA, Sequence 2 = AGCACAC-A EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 23. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Multiple Sequence Alignment Models A multiple sequence alignment (MSA) is a sequence alignment of three or more biological sequences, commonly protein, DNA, or RNA. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 24. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Multiple Sequence Alignment Models A multiple sequence alignment (MSA) is a sequence alignment of three or more biological sequences, commonly protein, DNA, or RNA. We usually want to do multiple alignments to find a homologous sequences that point to a shared evolutionary origins that can be used for further phylogenetic analysis. Progressive Alignment Methods - constructing succession of a pairwise alignment. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 25. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Multiple Sequence Alignment Models A multiple sequence alignment (MSA) is a sequence alignment of three or more biological sequences, commonly protein, DNA, or RNA. We usually want to do multiple alignments to find a homologous sequences that point to a shared evolutionary origins that can be used for further phylogenetic analysis. Progressive Alignment Methods - constructing succession of a pairwise alignment. Hidden Markov Models - representation of MSA as DAG, observed states are individual alignment columns and the hidden states represent the presumed ancestral sequence. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 26. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 27. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us What is Phylogenetics? Phylogenetics is the study of evolutionary relatedness among groups of organisms (e.g. species, populations), which is discovered through molecular sequencing data and morphological data matrices. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 28. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us What is Phylogenetics? Phylogenetics is the study of evolutionary relatedness among groups of organisms (e.g. species, populations), which is discovered through molecular sequencing data and morphological data matrices. Evolution is regarded as a branching process, whereby populations are altered over time and may speciate into separate branches, hybridize together, or terminate by extinction. This may be visualized in a phylogenetic tree. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 29. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us What is Phylogenetics? Phylogenetics is the study of evolutionary relatedness among groups of organisms (e.g. species, populations), which is discovered through molecular sequencing data and morphological data matrices. Evolution is regarded as a branching process, whereby populations are altered over time and may speciate into separate branches, hybridize together, or terminate by extinction. This may be visualized in a phylogenetic tree. Ernst Haeckel’s recapitulation theory (”ontogeny recapitulates phylogeny”) is a hypothesis that in developing from embryo to adult, animals go through stages resembling or representing successive stages in the evolution of their remote ancestors. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 30. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Building Phylogenetic Trees Phylogenetic trees among a nontrivial number of input sequences are constructed using computational phylogenetics methods. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 31. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Building Phylogenetic Trees Phylogenetic trees among a nontrivial number of input sequences are constructed using computational phylogenetics methods. Common method is to search for maximum likelihood, often within a Bayesian Framework, and apply an explicit model of evolution to phylogenetic tree estimation. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 32. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Building Phylogenetic Trees Phylogenetic trees among a nontrivial number of input sequences are constructed using computational phylogenetics methods. Common method is to search for maximum likelihood, often within a Bayesian Framework, and apply an explicit model of evolution to phylogenetic tree estimation. Identifying the optimal tree using many of these techniques is NP-hard, so heuristic search and optimization methods are used in combination with tree-scoring functions to identify a reasonably good tree that fits the data. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 33. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Building Phylogenetic Trees Phylogenetic trees among a nontrivial number of input sequences are constructed using computational phylogenetics methods. Common method is to search for maximum likelihood, often within a Bayesian Framework, and apply an explicit model of evolution to phylogenetic tree estimation. Identifying the optimal tree using many of these techniques is NP-hard, so heuristic search and optimization methods are used in combination with tree-scoring functions to identify a reasonably good tree that fits the data. They do not necessarily accurately represent the species evolutionary history as the data on which they are based is noisy; the analysis can be confounded by horizontal gene transfer, hybridisation between species that were not nearest neighbors on the tree before hybridisation takes place, convergent evolution, and conserved sequences. EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 34. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 35. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Other Models Transformational Grammars (Chomsky Hierarchy) RNA Structure Analysis Models (RNA contains the interactions - rather than preserving the sequence) EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics
- 36. Short introduction to Bioinformatics What are the Probabilistic Models? Sequence Alignment Pairwise Alignment Multiple Sequence Alignment Models What is Phylogenetics? Building Phylogenetic Trees Other Models Conctact Us Contact Us We are Hiring! EVEN BRIDGES G E N O M I C S, LLC Igor Bogicevic (igor.bogicevic@sbgenomics.com) Introduction to Probabilistic Models for Bioinformatics