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Monte Carlo Markov Chain Algorithm Convergence
Published in Richard M. Golden, Statistical Machine Learning, 2020
Let p(0) be a probability vector specifying the initial probability distribution of x~(0). Also, if x~(t) takes on the jth state, ej, at time t with probability one, then p(t) is the jth m-dimensional column vector of the m-dimensional identity matrix. Thus, using (11.2), the kth element of the row vector p(1)T = p(0)TP is the probability that the Markov chain will transition from the initial state ej to the state ek. Thus, p(1) specifies the probability of reaching each possible state in the state space from the initial state p(0) after the first step of the Markov chain.
Stochastic Learning Automata (SLA)
Published in Phil Mars, J.R. Chen, Raghu Nambiar, Learning Algorithms, 1996
Phil Mars, J.R. Chen, Raghu Nambiar
The existing learning algorithms for learning automata can be broadly classified into two groups: Ergodic and Absolutely expedient. Ergodic learning algorithms result in the optimal action probability vector converging in distribution independent of the initial action probability distribution. In non-stationary environments, if the optimal action changes with time, an ergodic SLA can track the change. Absolutely expedient learning schemes, on the other hand, possess absorbing barriers. If an automaton enters an absorbing barrier, then it is locked into that state for all time. Thus, convergence to one of these absorbing states can be proven. Since all extremities of the simplex S (Equation [2.5]) are absorbing states, there exists, a finite probability of convergence to the wrong action and thus the algorithm is ϵ-optimal. Estimator algorithms, however, use the enhanced definition of the state (Equation [2.28]) and use this extra information for the updating algorithms. This ensures, with a large probability, that the unit vector corresponding to the optimal action forms the only absorbing barrier. Thus, convergence to the optimal action in probability is established [TS85].
Bayesian Classification of Genomic Big Data
Published in Ervin Sejdić, Tiago H. Falk, Signal Processing and Machine Learning for Biomedical Big Data, 2018
Ulisses M. Braga-Neto, Emre Arslan, Upamanyu Banerjee, Arghavan Bahadorinejad
where the probability vector p = (p1,…,pM) follows a Dirichlet distribution with parameter αD = (αD1,…,αDM), and N=∑j=1Mxj is the total number of sequencing reads, which is assumed to have a Poisson distribution with parameter λ, which in turn follows a Gamma distribution with parameters α, β. See Figure 20.5 for a diagram of the model.
Reliability modeling for multi-component system subject to dependent competing failure processes with phase-type distribution considering multiple shock sources
Published in Quality Engineering, 2023
Hao Lyu, Shuai Wang, Zaiyou Yang, Hongchen Qu, Li Ma
In an absorbable Markov chain with m transients and an absorbing state, the distribution of transition times N before the Markov chain enters the absorbing state is a discrete phase-type distribution and denoted by The probability mass function for a discrete phase-type distribution random variable N is (Ozkut and Eryilmaz 2019): Where, Q = (qij)m×m is the transition probability matrix, which is composed of transition probabilities between m transients; and u′ = (I – Q)e′ is the transition probability vector, which consists of m transition probabilities from transient state to absorbing state; e = (1,1,…,1)1×m; I is the identity matrix.
Neural Visual Social Comment on Image-Text Content
Published in IETE Technical Review, 2021
Yue Yin, Hanzhou Wu, Xinpeng Zhang
The RCNN topic classification model is trained with cross-entropy loss function to learn a topic embedding space where the topic feature vectors of post-comment pairs with the same topic are close to each other. Since Euclidean distance between topic feature vectors is adopted to measure the similarity between topics of post-comment pairs, we expect topic feature vectors to reflect general features of comments of a specific topic but keep some degree of distinction. Although the probability vector indicates probability distribution over predefined topics, building our perceptual loss function based on the low dimensional probability vector probably ignores many differences between comments of similar topics. That will result in reducing the diversity of generated comments. By contrast, the output vector h of pooling layer is high dimensional feature vector containing specific topics’ general feature information but still keeping some post-comment pair’s own distinction. Therefore, we select the high dimensional output vector h of the pooling layer as topic feature vector instead of output probabilities of the softmax function.
Weak Gibbs measures and equilibrium states
Published in Dynamical Systems, 2020
Maria Carvalho, Sebastián A. Pérez
Consider a Markov unilateral chain associated to an irreducible 2-dimensional matrix A. The measure of maximal entropy μ of the shift map acting on , which is the equilibrium state of , is a 0-Gibbs measure (cf. [1, Theorem 1.16]) determined by a probability vector which is fixed under the action of a stochastic matrix (cf. [9, Theorem 8.10]), that is, pP = p. Accordingly, the μ-measure of a cylinder where and for every , is given by . Take now a probability vector such that . Then the corresponding ν is not σ-invariant, though it is a Markov measure and 0-Gibbs as well. Indeed, for every cylinder , one has and so where and .