1. Comp. Genomics Recitation 12 Bayesian networks Taken from Artificial Intelligence course, MIT, 6.034 http://courses.csail.mit.edu/6.034s/handouts/6034-review-sol.pdf

2. Question 1.1 Draw a Bayesian network among the following binary variables that model the outcome of an election: I: candidate is Incumbent M: has lots of Money for advertising A: uses advertisements that focus on Attacking the candidate’s opponent Q: uses advertisements that focus on the candidate’s Qualifications L: candidate is Liked D: opponent is Distrusted E: candidate is Elected

3. Question 1.1 – cont’d Your network should encode the following beliefs: Incumbents tend to raise lots of money. Money can be used to buy advertising that either focuses on the candidate’s qualifications or that attacks the candidate’s opponent. But if one does the first, there is less money to do the latter. Attack advertisements tend to make voters distrust the opponent but they also make the voters tend not to like the candidate. Advertisement focusing on qualifications tends to make the voters like the candidate. Candidates that people like tend to get elected. Candidates whose opponent people distrust tent to get elected.

4. Question 1.1 - solution

5. Question 1.2 For each of the following, say whether it is or is not asserted by the network structure you drew (without assuming anything about the numerical entries in the CPTs). 1.P(L | A,Q,D) = P(L | A,Q) 2.P(A | M,Q) = P(A | M) 3.P(L,D | A,Q) = P(L | A,Q) P(D | A,Q)

6. Question 1.2 - solution 1.P(L | A,Q,D) = P(L | A,Q) Asserted 2.P(A | M,Q) = P(A | M) Not asserted 3.P(L,D | A,Q) = P(L | A,Q) P(D | A,Q) Asserted

7. Question 2 Show a Bayesian network structure that encodes the following relationships: A is independent of B A is dependent on B given C A is dependent on D A is independent of D given C

8. Question 2 - solution Nodes A and B have no parents Node C has two parents: A and B Node D has one parent: C

9. Question 3 Which of the following conditional independence assumptions are true? 1.A and E are independent 2.A and E are independent given D 3.B and C are independent 4.B and C are independent given A 5.B and C are independent given D 6.A and E are independent given B 7.A and E are independent given F 8.B and C are independent given E

10. Question 3 - solution A and E are independent False A and E are independent given D True B and C are independent False B and C are independent given A True B and C are independent given D False A and E are independent given B False A and E are independent given F False B and C are independent given E False

11. Question 4 For each statement, name all of the graph structures, G1-G4, or “none” that imply it.

12. Question 4 – cont’d 1.A is conditionally independent of B given C 2.A is conditionally independent of B given D 3.B is conditionally independent of D given A 4.B is conditionally independent of D given C 5.B is independent of C 6.B is conditionally independent of C given A

13. Question 4 - solution A is conditionally independent of B given C G2 A is conditionally independent of B given D none B is conditionally independent of D given A G3,G4 B is conditionally independent of D given C none B is independent of C G2,G3 B is conditionally independent of C given A G1,G2,G4

14. HW solution – ass. 2, q. 5 Let G = (G 1, …, G n ) be n contiguous DNA regions representing genes. For each G i we define the mRNA concentration of the gene as P i, s.t. their sum is equal to 1. P = (P 1, …, P n ) can be interpreted as the normalized expression levels for the regions in G.

15. HW solution – q. 5 – cont’d Our model assumes that reads are generated by randomly picking a region R from G according to the distribution P, and then copying this region. The copying process is error-prone. This process is repeated until we have a set of m reads R = r 1, …, r m generated according to the model described above.

16. HW solution – q. 5 – cont’d For each region G j and read r i, we have a probability p ij = P(r j | G i ), the probability of observing r j given that the locus of the read was gene G i. In practice, for each read r j, this probability will be close to zero for all but a few regions.

17. Likelihood function Write the likelihood of observing the m reads.

18. Q function Write the Q(P | P (t) ) term.

19. M-step Write the M-step term using argmax function.

20. Update rule Infer from c the update step for P.