04-ex-solution.pdf - Problem Set 4 MS&E 221 Due Monday:59PM...

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Problem Set 4 MS&E 221 Due: Monday, February 26 11:59PM Question 4.1 (Cayley Tree) : The Cayley tree of degree k and depth d is a tree with undirected edges, on 1+ k + k ( k - 1)+ · · · + k ( k - 1) d - 1 nodes, where every non-leaf node has k adjacent edges. (Assume k > 2 and d > 0 .) Assume that a particle does a random walk on a Cayley tree of degree k and depth d : from any node, the particle moves to one of the neighbors of that node chosen uniformly at random. The following notation may be helpful in this question. For a node i , let N ( i ) denote the neighbors of i ; note that any non-leaf node has exactly k neighbors. For a leaf node v , let p ( v ) denote the parent of v in the graph; i.e., p ( v ) is the unique node adjacent to v . Figure 1: A Cayley tree with k = 4 and d = 3 ; note that it has 1 + k + k ( k - 1) + k ( k - 1) 2 = 53 nodes, and that each non-leaf node has k = 4 adjacent edges. (a) Describe the movement of the particle as a Markov chain. (b) Is this Markov chain irreducible? (c) Is this Markov chain aperiodic? 1
(d) Compute the stationary distribution of the random walk. Answer:
Question 4.2 (Expected failure via MCMC) : Consider a video streaming service with a faulty server. Let X denote the availability of the service ( 1 if available, 0 if server is down), which we model as a Bernoulli distribution with probability θ . We incorporate our prior knowledge about server failures by viewing θ as a random variable and assigning a prior distribution on θ : P ( θ = θ j ) = p j , for j = 1 , . . . , m. (we assume p j > 0 for all j = 1 , . . . , m ) Suppose that we have observed server availabilities X 1 = x 1 , . . . , X n = x n . (a) Compute the posterior pmf P ( θ = θ j | X n 1 = x n 1 ) in terms of θ j , p j and X 1 , . . . , X n ’s. We now wish to see how Markov chain Monte Carlo (MCMC) can approximate samples from the posterior distribution computed in part ( a ) . 2
(b) Consider the Metropolis-Hastings algorithm with some proposal distribution Q = ( Q ( z 1 , z 2 ) : z 1 , z 2 S ) where S = { θ 1 , . . . , θ m } . Construct a Markov chain Z = ( Z n : n 0) with state space S and stationary distribution π ( θ j ) = P ( θ = θ j | X n 1 = x n 1 ) . Argue that lim n →∞ P ( Z n = θ j | Z 0 = z ) = π ( θ j ) for any z S (1) if Q ( x, y ) > 0 for all x, y S .

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