The latent roots of certain Markov chains arising in genetics: A new approach, II. Further haploid models

1975 ◽  
Vol 7 (02) ◽  
pp. 264-282 ◽  
Author(s):  
C. Cannings
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Reproductive Structure ◽  
Variable Size ◽  
New Approach ◽  
Age Structured ◽  
Drift Models ◽  
Latent Roots

The method developed for the treatment of the classical drift models of Wright and Moran, and their generalizations, in Cannings (1974) are extended to more complex haploid models. The possibility of subdivision of the population, as for migration models and age-structured models, is incorporated. Models with variable size or reproductive structure determined by another Markov chain are analysed.

The latent roots of certain Markov chains arising in genetics: A new approach, II. Further haploid models

1975 ◽  
Vol 7 (2) ◽  
pp. 264-282 ◽  
Author(s):  
C. Cannings
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Reproductive Structure ◽  
Variable Size ◽  
New Approach ◽  
Age Structured ◽  
Drift Models ◽  
Latent Roots

The method developed for the treatment of the classical drift models of Wright and Moran, and their generalizations, in Cannings (1974) are extended to more complex haploid models. The possibility of subdivision of the population, as for migration models and age-structured models, is incorporated. Models with variable size or reproductive structure determined by another Markov chain are analysed.

The latent roots of certain Markov chains arising in genetics: A new approach, I. Haploid models

1974 ◽  
Vol 6 (2) ◽  
pp. 260-290 ◽  
Author(s):  
C. Cannings
Keyword(s):  
Markov Chains ◽  
Genetic Drift ◽  
Overlapping Generations ◽  
Branching Processes ◽  
Multiple Alleles ◽  
New Approach ◽  
Constant Size ◽  
Latent Roots

Haploid models of genetic drift in populations of constant size are considered. Generalizations of the models of Moran and Wright have been developed by Karlin and McGregor (for multiple alleles and non-overlapping generations), by Chia and Watterson (for two alleles and overlapping or non-overlapping generations) and by Chia (for multiple alleles and overlapping or non-overlapping generations), using conditioned branching processes. A new approach is developed which contains the models mentioned above and provides simpler expressions for the latent roots. A greater dependence between the birth events and death events can be permitted, and non-independent mutations treated.

Asymptotic Cumulants for Distributions of k-State Markov Chains

1962 ◽  
Vol 58 (2) ◽  
pp. 427-430 ◽  
Author(s):  
P. V. Krishna Iyer ◽  
N. S. Shakuntala
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Probability Distributions ◽  
Exact Probability ◽  
Determinantal Equation ◽  
Asymptotic Values ◽  
Asymptotic Cumulants ◽  
Latent Roots

The expectation, variance and covariance for different states of a k-state Markov chain have been given by Patankar (6), Whittle (7), Good (3) and Bhat (l). Patankar's results involve the k latent roots of the determinantal equation. As it is not easy to determine the latent roots when k > 2, the actual asymptotic values of variances and covariances cannot be readily evaluated. Whittle gives exact probability distributions for the transitions, but the moments have been obtained after some gross approximations. Good (3) and Bhat(l) have given the first two moments and product moment for the frequency of different states. By using certain methods developed by Iyer and Kapur(4), the first four cumulants and product cumulants for the transition numbers of a two-state Markov chain were calculated and presented in an earlier publication(5).

The latent roots of certain Markov chains arising in genetics: A new approach, I. Haploid models

1974 ◽  
Vol 6 (02) ◽  
pp. 260-290 ◽  
Author(s):  
C. Cannings
Keyword(s):  
Markov Chains ◽  
Genetic Drift ◽  
Overlapping Generations ◽  
Branching Processes ◽  
Multiple Alleles ◽  
New Approach ◽  
Constant Size ◽  
Latent Roots

Haploid models of genetic drift in populations of constant size are considered. Generalizations of the models of Moran and Wright have been developed by Karlin and McGregor (for multiple alleles and non-overlapping generations), by Chia and Watterson (for two alleles and overlapping or non-overlapping generations) and by Chia (for multiple alleles and overlapping or non-overlapping generations), using conditioned branching processes. A new approach is developed which contains the models mentioned above and provides simpler expressions for the latent roots. A greater dependence between the birth events and death events can be permitted, and non-independent mutations treated.

Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains

1990 ◽  
Vol 27 (03) ◽  
pp. 545-556 ◽  
Author(s):  
S. Kalpazidou
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
Probabilistic Interpretation ◽  
Stationary Markov Chain

The asymptotic behaviour of the sequence (𝒞 n (ω), wc,n (ω)/n), is studied where 𝒞 n (ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n ) until time n and wc,n (ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ω c ) which represents uniquely the chain (ξ n ) as a circuit chain, and ω c is given a probabilistic interpretation.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

scholarly journals Proportional lumpability and proportional bisimilarity

2021 ◽  
Author(s):  
Andrea Marin ◽  
Carla Piazza ◽  
Sabina Rossi
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Upper And Lower Bounds ◽  
General Definition ◽  
Performance Indices ◽  
State Aggregation ◽  
Structural Regularity ◽  
Original Definition ◽  
Aggregation Technique ◽  
Definition Of

AbstractIn this paper, we deal with the lumpability approach to cope with the state space explosion problem inherent to the computation of the stationary performance indices of large stochastic models. The lumpability method is based on a state aggregation technique and applies to Markov chains exhibiting some structural regularity. Moreover, it allows one to efficiently compute the exact values of the stationary performance indices when the model is actually lumpable. The notion of quasi-lumpability is based on the idea that a Markov chain can be altered by relatively small perturbations of the transition rates in such a way that the new resulting Markov chain is lumpable. In this case, only upper and lower bounds on the performance indices can be derived. Here, we introduce a novel notion of quasi-lumpability, named proportional lumpability, which extends the original definition of lumpability but, differently from the general definition of quasi-lumpability, it allows one to derive exact stationary performance indices for the original process. We then introduce the notion of proportional bisimilarity for the terms of the performance process algebra PEPA. Proportional bisimilarity induces a proportional lumpability on the underlying continuous-time Markov chains. Finally, we prove some compositionality results and show the applicability of our theory through examples.

scholarly journals A Bayesian model for binary Markov chains

2004 ◽  
Vol 2004 (8) ◽  
pp. 421-429 ◽  
Author(s):  
Souad Assoudou ◽  
Belkheir Essebbar
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Bayesian Estimation ◽  
Bayesian Model ◽  
Transition Probabilities ◽  
Simulated Data ◽  
Bayesian Estimator ◽  
Jeffreys Prior ◽  
Data Set

This note is concerned with Bayesian estimation of the transition probabilities of a binary Markov chain observed from heterogeneous individuals. The model is founded on the Jeffreys' prior which allows for transition probabilities to be correlated. The Bayesian estimator is approximated by means of Monte Carlo Markov chain (MCMC) techniques. The performance of the Bayesian estimates is illustrated by analyzing a small simulated data set.

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