cs262 pset 2 - CS262 Problem Set 2 Sequence Alignment and...

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CS262: Problem Set 2 Sequence Alignment and Dynamic Programming Jan, 20 2011 Kate Swanson K PS 2 Kate Swanson Worked with David Zhang
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CS262: Problem Set 2 Sequence Alignment and Dynamic Programming Jan, 20 2011 Kate Swanson Problem 1 A. False. The states are hidden, the emissions are observed. B. False. We can have states which are very unlikely to be reached from any state, or impossible to reach, so all incoming edges would be very low probability (eg. state representing the world getting taken over by bunnies). Or we can have a state that is guaranteed to be reached from multiple states. C. False, the edge from a to b gives the probability that we transition to b given we are in state a, not the other way around. D. True. In viterbi each cell contains the max over all ways to generate the path, whereas forward algorithm sums over all the possible ways. E. True. Given a path with N states and a model with K states the runtime of both Viterbi and Forward algorithms is O(K^2*N). For each output we observe (N outputs) each pair of K states is compared (K^2). F. False. Duration modeling of vanilla HMM's is pretty good for intronic regions. Exons behave differently than introns in that they can translate into different proteins which are made up of 3 amino acids so they need to look more than one state back, thereby violating the markov property. Furthermore, duration of introns follow a geometric distribution which is modeled by the duration model (mean 1/(1-p). G. False. The Markov property implies that at a time t the current state depends only on the previous state, however, we could extend to allow dependencies on more than just the previoius state, however, our current programming solution would no longer be possible. H. True. Baum-Welch is a generalied EM algorithm and as such is guaranteed to converge to a *local* optimum, since we choose a starting point randomly and iterate until change < some threshold, it is not necesssary that we are converging at the global optimum (if we repeated the algorithm randomly choosing starting positions, eventually we could find the true optimum).
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CS262: Problem Set 2 Sequence Alignment and Dynamic Programming Jan, 20 2011 Kate Swanson I. True. They are better than cake. Also less calories. Problem 2 A. Proof by induction: Show that using Baum-Welch on a sequence with an initial transition probability a kl = 0, a kl = 0 after training. Consider equation 3.18 from Durbin p. 63. For a kl to be non-zero A kl clearly must be nonzero. (1) a kl = A kl /(sum over l’ A kl’) using equation 3.20, we can calculate A kl using (2) A kl = sum over j (1/P(x j ) sum over I (f i k (i) a kl e l (x j i+1 )b j i (i+1) Base case: a kl =0 Inductive Step: If a kl = 0 at iteration i, show that after one iteration it still = 0.
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