1. Algebraic Statistics for Computational Biology Lior Pachter and Bernd Sturmfels Ch.5: Parametric Inference R. Mihaescu Παρουσίαση: Aγγελίνα Βιδάλη Αλγεβρικοί & Γεωμετρικοί Αλγόριθμοι στη Μοριακή Βιολογία Διδάσκων: Ι. Εμίρης

2. Convenient algebraic structure for stating dynamic programming algorithms: the tropical semiring Tropical arithmetic (Convex hull) (Minkowski sum) The polytope agebra ( P d natural higher-dimensional generalization:

3. Inference From Observed random variables Y 1 = σ 1,…,Y n = σ n we want to infer values for the Hidden random variables Χ 1,…,Χ m : Unknown biological data, i.e.: How do two sequences allign? MAP estimation: given an observation σ 1,…,σ n which is the most probable explanation X 1 =h 1,…, Χ m =h m ? Model parameters give transition probabilities p hσ : hidden state hσ observed state Observation: σ 1,…,σ n : Known biological data

4. Observation: σ 1,…,σ n We want to compute an explanation for the observation: the sequence h 1,…,h m which yields the maximum a prosteriori probability (MAP): We can efficiently compute the marginal probabilities: Hidden Markov Model (HMM)

5. Computation of the marginal probabilities: p σ has the decomposition which gives the “Forward algorithm”. Markov chain: Independent probabilities

6. Viterbi algorithm problem of computing p σ Tropicalization:u ij =-log(p’ ij )v ij =-log(p ij ) We can now efficiently find an explanation h 1,…,h m for the observation σ 1,…,σ n using the recursion: It is again the Forward algorithm.

7. Pair Hidden Markov Model (pHMM) The algebraic statistical model for sequence alignment, known as the pair hidden Markov model, is the image of the map where A n,m is the set of all alignments of the sequences σ 1, σ 2.

8. The Needleman-Wunsch algorithm for finding the shortest path in the alignment graph is the tropicalization of the pair hidden Markov model for sequence allignment. gttta- gt--gc g t g c gttta Example: n=5, m=4 **

9. The polytope propagation algorithm Tropical sum-product algorithm in general fashion. f is the density function for a statistical model. From the d monomials find the one that maximizes Solution: Tropicalization: w i =-logp i & Computation in the ploytope algebra

10. Density function for a statistical model: f(p 1,p 2 )=p 1 3 +p 1 2 p 2 2 +p 1 p 2 2 +p 1 +p 2 4 Find the index j of the monomial that minimizes the function e j. w. Find an explanation Find the index j of the monomial with maximal value Tropicalization: w i =-logp i

11. Explanations are vertices of the Newton Polytope of f p13p13 p11p11 f(p 1,p 2 )=p 1 3 +p 1 2 p 2 2 +p 1 p 2 2 +p 1 +p 2 4 we find a point for each exponent vector of a monomial

12. Normal fan The normal fan partitions the parameter space into regions such that: the explanation(s) for all sets of parameters in a given region is given by the polytope vertex(face) associated to that region.

13. Parametric MAP estimation problem Local: given a choice of parameters determine the set of all parameters with the same MAP estimate. Solution: Computation of the normal cone of the Newton Polytope. Global: asks for a partition of the space of parameters such that any two parameters lie in the same part iff they yield the same MAP estimate. Solution: Computation of the normal fan of the Newton Polytope.