Inference_for_Graphs_and_Networks.pdf

# Em starts with some initial selection for the model

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EM starts with some initial selection for the model parameters, which we denote θ old . Initial Parameters. To obtain θ old , we proceed by assuming that the high-state emission density, Pr( X t = x t | Z t = 1 , φ ) only emits positive counts. This, in effect, makes Z t an observed random variable. Let b t, 0 = I ( X t = 0), b t, 1 = I ( X t > 0). We use initial transition probabilities defined by the maximum likelihood estimators (MLEs) of the observed Markov chain: ˜ p 01 = n 01 n 01 + n 00 ˜ p 10 = n 10 n 10 + n 11 , where n ij is the number of times that the consecutive pair ( b t 1 ,i , b t,j ) was observed in x . An initial estimate for π is the steady-state probability given by ˜ π = ˜ p 01 ˜ p 01 p 10 . To obtain initial estimates for the high-state emission parameters φ , we collect the samples of X such that X t > 0, and call that collection Y . We then calculate ˜ µ and ˜ σ 2 , the sample mean and variance of Y . Finally, we reparameterize from (˜ µ, ˜ σ 2 ) to (˜ µ, ˜ s ) where ˜ s is the initial size param- eter, via ˜ s = ˜ µ 2 ˜ σ 2 ˜ µ . This approach ignores the fact that the high-state distribution can emit zeros, but for our application these initial values were suﬃcient for the EM algorithm to converge in a reasonable number of iterations. The E step. In the E step, we take these initial parameter values and find the posterior distribution of the latent variables Pr( Z = z | X = x , θ old ). This posterior distribution is then used to evaluate the expectation of the logarithm of the complete-data likelihood function, as a function of the parameters θ , to give the function Q ( θ , θ old ) defined by Q ( θ , θ old ) = Z Pr( Z = z | X = x , θ old ) log Pr( X = x , Z = z | θ ) . (3.5) It has been shown (Baum and Sell, 1968; Baker, 1975) that maximization of Q ( θ , θ old ) results in increased likelihood. To evaluate Q ( θ , θ old ), we intro- duce some notation. Let γ ( z t ) be the marginal posterior of z t and ξ ( z t 1 , z t ) be the joint posterior of two successive latent variables, so γ ( z t ) = Pr( Z t = z t | X = x , θ old ) ξ ( z t 1 , z t ) = Pr( Z t 1 = z t 1 , Z t = z t | X = x , θ old ) . Now for k = 0 , 1, the two states of the Markov chain, we denote z tk = I ( z t = k ), which is 1 if z t is in state k and 0 otherwise. Let γ ( z tk ) Copyright © 2014. Imperial College Press. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 2/16/2016 3:37 AM via CGC-GROUP OF COLLEGES (GHARUAN) AN: 779681 ; Heard, Nicholas, Adams, Niall M..; Data Analysis for Network Cyber-security Account: ns224671

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Statistical Detection of Intruders Within Computer Networks 85 be the conditional probability that z tk = 1, with a similar notation for ξ ( z t 1 ,j , z tk ). Since expectation of a binary random variable is just the probability that it takes value 1, γ ( z tk ) = Ez tk = Z γ ( Z ) z tk ξ ( z t 1 ,j , z tk ) = Ez t 1 ,j , z tk = Z γ ( Z ) z t 1 ,j z tk .
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• Spring '12
• Kushal Kanwar
• Graph Theory, Statistical hypothesis testing, Imperial College Press, applicable copyright law

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