Stats254_Hw2

Stats254_Hw2 - = (0 . 24 , . 26 , . 26 , . 24) for { A, C,...

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Stats M254 Homework 2 NOTE: Please submit the names of your presentation group members with this homework assignment. 1. The transition probability matrix of a Markov chain { X n } on state space { 1 , 2 } is T = ± 0 . 7 0 . 3 0 . 1 0 . 9 ² . (a) Determine T ( ) = lim n →∞ T n . (b) Find the following probabilites as n → ∞ : (i) P ( X n = 2 ,X n +1 = 1 | X 1 = 1); (ii) P ( X n - 1 = 2 | X n = 1). 2. We observed ten binding sites of the transcription factor Nanog and summarized them into a count matrix X (Table 1). A position-specific weight matrix Θ is used as the model for the binding sites. Table 1: The observed count matrix X position 1 2 3 4 5 6 7 8 9 A 1 2 1 9 0 0 8 5 1 C 4 7 3 0 0 0 0 1 4 G 4 1 6 1 0 0 1 1 5 T 1 0 0 0 10 10 1 3 0 (a) If we use a flat prior on Θ (i.e. all pseudo-counts =1), what is the posterior distribution of Θ given the observed count matrix X ? (b) Assume that the background model is an i.i.d multinomial distribution with
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Unformatted text preview: = (0 . 24 , . 26 , . 26 , . 24) for { A, C, G, T } . Suppose we know that the sequence S = TCACTATTATCCCTGTTA contains at most one Nanog binding site. Let a be the start position of the site if it exists, and dene a = 0 if S does not contain a site. Only consider the forward strand, i.e., a { , 1 , , 10 } . Assuming that the prior probability ( a = i ) is uniform, calculate P ( a = i | S,X, ) for i = 0 , 1 , , 10. Hint: P ( a = i | S,X, ) P ( S,a = i | X, ) = Z P ( S,a = i | , ) p ( | X ) d . 3. Show that the Gibbs sampler is a special case of the Metropolis-Hastings algorithm. 1...
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