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Статті в журналах з теми "Markov Equivalence"

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Автор: Grafiati

Опубліковано: 10 грудня 2022

Оновлено: 28 січня 2023

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1

Tuncel, Selim. "Markov measures determine the zeta function." Ergodic Theory and Dynamical Systems 7, no. 2 (June 1987): 303–11. http://dx.doi.org/10.1017/s014338570000403x.

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AbstractWith the purpose of understanding when two subshifts of finite type are equivalent from the point of view of their spaces of Markov measures we propose the notion of Markov equivalence. We show that a Markov equivalence must respect the cycles (periodic orbits) of the subshifts. In particular, Markov equivalent subshifts of finite type have the same zeta function.

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2

DOYEN, LAURENT, THOMAS A. HENZINGER, and JEAN-FRANÇOIS RASKIN. "EQUIVALENCE OF LABELED MARKOV CHAINS." International Journal of Foundations of Computer Science 19, no. 03 (June 2008): 549–63. http://dx.doi.org/10.1142/s0129054108005814.

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We consider the equivalence problem for labeled Markov chains (LMCs), where each state is labeled with an observation. Two LMCs are equivalent if every finite sequence of observations has the same probability of occurrence in the two LMCs. We show that equivalence can be decided in polynomial time, using a reduction to the equivalence problem for probabilistic automata, which is known to be solvable in polynomial time. We provide an alternative algorithm to solve the equivalence problem, which is based on a new definition of bisimulation for probabilistic automata. We also extend the technique to decide the equivalence of weighted probabilistic automata. Then, we consider the equivalence problem for labeled Markov decision processes (LMDPs), which asks given two LMDPs whether for every scheduler (i.e. way of resolving the nondeterministic decisions) for each of the processes, there exists a scheduler for the other process such that the resulting LMCs are equivalent. The decidability of this problem remains open. We show that the schedulers can be restricted to be observation-based, but may require infinite memory.

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3

MATSUMOTO, KENGO, and HIROKI MATUI. "Continuous orbit equivalence of topological Markov shifts and dynamical zeta functions." Ergodic Theory and Dynamical Systems 36, no. 5 (February 6, 2015): 1557–81. http://dx.doi.org/10.1017/etds.2014.128.

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For continuously orbit equivalent one-sided topological Markov shifts $(X_{A},{\it\sigma}_{A})$ and $(X_{B},{\it\sigma}_{B})$, their eventually periodic points and cocycle functions are studied. As a result, we directly construct an isomorphism between their ordered cohomology groups $(\bar{H}^{A},\bar{H}_{+}^{A})$ and $(\bar{H}^{B},\bar{H}_{+}^{B})$. We also show that the cocycle functions for the continuous orbit equivalences give rise to positive elements of their ordered cohomology groups, so that the zeta functions of continuously orbit equivalent topological Markov shifts are related. The set of Borel measures is shown to be invariant under continuous orbit equivalence of one-sided topological Markov shifts.

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4

Ali, R. Ayesha, Thomas S. Richardson, and Peter Spirtes. "Markov equivalence for ancestral graphs." Annals of Statistics 37, no. 5B (October 2009): 2808–37. http://dx.doi.org/10.1214/08-aos626.

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5

Kabayashi, Kingo, Shun-Ichi Amari, and Hisashi Ito. "Equivalence of hidden Markov models." Electronics and Communications in Japan (Part III: Fundamental Electronic Science) 74, no. 7 (1991): 85–94. http://dx.doi.org/10.1002/ecjc.4430740709.

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6

Buchholz, Peter, and Miklós Telek. "Rational Processes Related to Communicating Markov Processes." Journal of Applied Probability 49, no. 1 (March 2012): 40–59. http://dx.doi.org/10.1239/jap/1331216833.

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We define a class of stochastic processes, denoted as marked rational arrival processes (MRAPs), which is an extension of matrix exponential distributions and rational arrival processes. Continuous-time Markov processes with labeled transitions are a subclass of this more general model class. New equivalence relations between processes are defined, and it is shown that these equivalence relations are natural extensions of strong and weak lumpability and the corresponding bisimulation relations that have been defined for Markov processes. If a general rational process is equivalent to a Markov process, it can be used in numerical analysis techniques instead of the Markov process. This observation allows one to apply MRAPs like Markov processes and since the new equivalence relations are more general than lumpability and bisimulation, it is sometimes possible to find smaller representations of given processes. Finally, we show that the equivalence is preserved by the composition of MRAPs and can therefore be exploited in compositional modeling.

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7

Buchholz, Peter, and Miklós Telek. "Rational Processes Related to Communicating Markov Processes." Journal of Applied Probability 49, no. 01 (March 2012): 40–59. http://dx.doi.org/10.1017/s0021900200008858.

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Анотація:

We define a class of stochastic processes, denoted as marked rational arrival processes (MRAPs), which is an extension of matrix exponential distributions and rational arrival processes. Continuous-time Markov processes with labeled transitions are a subclass of this more general model class. New equivalence relations between processes are defined, and it is shown that these equivalence relations are natural extensions of strong and weak lumpability and the corresponding bisimulation relations that have been defined for Markov processes. If a general rational process is equivalent to a Markov process, it can be used in numerical analysis techniques instead of the Markov process. This observation allows one to apply MRAPs like Markov processes and since the new equivalence relations are more general than lumpability and bisimulation, it is sometimes possible to find smaller representations of given processes. Finally, we show that the equivalence is preserved by the composition of MRAPs and can therefore be exploited in compositional modeling.

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8

Verley, Gatien. "Dynamical equivalence classes for Markov jump processes." Journal of Statistical Mechanics: Theory and Experiment 2022, no. 2 (February 1, 2022): 023211. http://dx.doi.org/10.1088/1742-5468/ac4981.

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Abstract Two different Markov jump processes driven out of equilibrium by constant thermodynamic forces may have identical current fluctuations in the stationary state. The concept of dynamical equivalence classes emerges from this statement as proposed by Andrieux for discrete-time Markov chains on simple graphs. We define dynamical equivalence classes in the context of continuous-time Markov chains on multigraphs using the symmetric part of the rate matrices that define the dynamics. The freedom on the skew-symmetric part is at the core of the freedom inside a dynamical equivalence class. It arises from different splittings of the thermodynamic forces onto the system’s transitions.

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9

Larget, Bret. "A canonical representation for aggregated Markov processes." Journal of Applied Probability 35, no. 2 (June 1998): 313–24. http://dx.doi.org/10.1239/jap/1032192850.

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A deterministic function of a Markov process is called an aggregated Markov process. We give necessary and sufficient conditions for the equivalence of continuous-time aggregated Markov processes. For both discrete- and continuous-time, we show that any aggregated Markov process which satisfies mild regularity conditions can be directly converted to a canonical representation which is unique for each class of equivalent models, and furthermore, is a minimal parameterization of all that can be identified about the underlying Markov process. Hidden Markov models on finite state spaces may be framed as aggregated Markov processes by expanding the state space and thus also have canonical representations.

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10

Larget, Bret. "A canonical representation for aggregated Markov processes." Journal of Applied Probability 35, no. 02 (June 1998): 313–24. http://dx.doi.org/10.1017/s0021900200014972.

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Анотація:

A deterministic function of a Markov process is called an aggregated Markov process. We give necessary and sufficient conditions for the equivalence of continuous-time aggregated Markov processes. For both discrete- and continuous-time, we show that any aggregated Markov process which satisfies mild regularity conditions can be directly converted to a canonical representation which is unique for each class of equivalent models, and furthermore, is a minimal parameterization of all that can be identified about the underlying Markov process. Hidden Markov models on finite state spaces may be framed as aggregated Markov processes by expanding the state space and thus also have canonical representations.

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11

Kämpke, T. "Reversibility and equivalence in directed markov fields." Mathematical and Computer Modelling 23, no. 3 (February 1996): 87–101. http://dx.doi.org/10.1016/0895-7177(95)00235-9.

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12

Mogensen, Søren Wengel, and Niels Richard Hansen. "Markov equivalence of marginalized local independence graphs." Annals of Statistics 48, no. 1 (February 2020): 539–59. http://dx.doi.org/10.1214/19-aos1821.

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13

Hayashi, Masahito. "Local equivalence problem in hidden Markov model." Information Geometry 2, no. 1 (June 2019): 1–42. http://dx.doi.org/10.1007/s41884-019-00016-z.

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14

LAMBROPOULOU, SOFIA. "L-MOVES AND MARKOV THEOREMS." Journal of Knot Theory and Its Ramifications 16, no. 10 (December 2007): 1459–68. http://dx.doi.org/10.1142/s0218216507005919.

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Given a knot theory (virtual, singular, knots in a 3-manifold etc.), there are deep relations between the diagrammatic knot equivalence in this theory, the braid structures and a corresponding braid equivalence. The L-moves between braids, due to their fundamental nature, may be adapted to any diagrammatic situation in order to formulate a corresponding braid equivalence. In this short paper, we discuss and compare various diagrammatic set-ups and results therein, in order to draw the underlying logic relating diagrammatic isotopy, braid structures, Markov theorems and L-move analogues. Finally, we apply our conclusions to singular braids.

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15

Ghassami, AmirEmad, Saber Salehkaleybar, Negar Kiyavash, and Kun Zhang. "Counting and Sampling from Markov Equivalent DAGs Using Clique Trees." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 3664–71. http://dx.doi.org/10.1609/aaai.v33i01.33013664.

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A directed acyclic graph (DAG) is the most common graphical model for representing causal relationships among a set of variables. When restricted to using only observational data, the structure of the ground truth DAG is identifiable only up to Markov equivalence, based on conditional independence relations among the variables. Therefore, the number of DAGs equivalent to the ground truth DAG is an indicator of the causal complexity of the underlying structure–roughly speaking, it shows how many interventions or how much additional information is further needed to recover the underlying DAG. In this paper, we propose a new technique for counting the number of DAGs in a Markov equivalence class. Our approach is based on the clique tree representation of chordal graphs. We show that in the case of bounded degree graphs, the proposed algorithm is polynomial time. We further demonstrate that this technique can be utilized for uniform sampling from a Markov equivalence class, which provides a stochastic way to enumerate DAGs in the equivalence class and may be needed for finding the best DAG or for causal inference given the equivalence class as input. We also extend our counting and sampling method to the case where prior knowledge about the underlying DAG is available, and present applications of this extension in causal experiment design and estimating the causal effect of joint interventions.

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16

DUTKAY, DORIN ERVIN, and PALLE E. T. JORGENSEN. "Representations of Cuntz algebras associated to quasi-stationary Markov measures." Ergodic Theory and Dynamical Systems 35, no. 7 (June 30, 2014): 2080–93. http://dx.doi.org/10.1017/etds.2014.37.

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In this paper, we answer the question of equivalence, or singularity, of two given quasi-stationary Markov measures on one-sided infinite words, as well as the corresponding question of equivalence of associated Cuntz algebra${\mathcal{O}}_{N}$-representations. We do this by associating certain monic representations of${\mathcal{O}}_{N}$to quasi-stationary Markov measures and then proving that equivalence for a pair of measures is decided by unitary equivalence of the corresponding pair of representations.

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17

Mayta, Jorge, and Maritza Moreno. "Equivalence of the stability of discrete-time Markov jump linear systems." Selecciones Matemáticas 7, no. 2 (December 30, 2020): 234–41. http://dx.doi.org/10.17268/sel.mat.2020.02.05.

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18

Matúš, F. "On equivalence of Markov properties over undirected graphs." Journal of Applied Probability 29, no. 3 (September 1992): 745–49. http://dx.doi.org/10.2307/3214910.

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The dependence of coincidence of the global, local and pairwise Markov properties on the underlying undirected graph is examined. The pairs of these properties are found to be equivalent for graphs with some small excluded subgraphs. Probabilistic representations of the corresponding conditional independence structures are discussed.

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19

ZHAO, Hui. "On the Markov equivalence of maximal ancestral graphs." Science in China Series A 48, no. 4 (2005): 548. http://dx.doi.org/10.1360/04ys0023.

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20

Lambropoulou, S., and C. P. Rourke. "Algebraic Markov equivalence for links in three-manifolds." Compositio Mathematica 142, no. 04 (July 2006): 1039–62. http://dx.doi.org/10.1112/s0010437x06002144.

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21

Vanluyten, Bart, Jan C. Willems, and Bart De Moor. "Equivalence of state representations for hidden Markov models." Systems & Control Letters 57, no. 5 (May 2008): 410–19. http://dx.doi.org/10.1016/j.sysconle.2007.10.004.

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22

Matúš, F. "On equivalence of Markov properties over undirected graphs." Journal of Applied Probability 29, no. 03 (September 1992): 745–49. http://dx.doi.org/10.1017/s0021900200043552.

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Анотація:

The dependence of coincidence of the global, local and pairwise Markov properties on the underlying undirected graph is examined. The pairs of these properties are found to be equivalent for graphs with some small excluded subgraphs. Probabilistic representations of the corresponding conditional independence structures are discussed.

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23

Gorban, Alexander N. "Local equivalence of reversible and general Markov kinetics." Physica A: Statistical Mechanics and its Applications 392, no. 5 (March 2013): 1111–21. http://dx.doi.org/10.1016/j.physa.2012.11.028.

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24

Gillispie, Steven B. "Formulas for counting acyclic digraph Markov equivalence classes." Journal of Statistical Planning and Inference 136, no. 4 (April 2006): 1410–32. http://dx.doi.org/10.1016/j.jspi.2004.10.007.

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25

MacKay, David J. C. "Equivalence of Linear Boltzmann Chains and Hidden Markov Models." Neural Computation 8, no. 1 (January 1996): 178–81. http://dx.doi.org/10.1162/neco.1996.8.1.178.

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Several authors have studied the relationship between hidden Markov models and “Boltzmann chains” with a linear or “time-sliced” architecture. Boltzmann chains model sequences of states by defining state-state transition energies instead of probabilities. In this note I demonstrate that under the simple condition that the state sequence has a mandatory end state, the probability distribution assigned by a strictly linear Boltzmann chain is identical to that assigned by a hidden Markov model.

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26

Barthe, Gilles, Raphaëlle Crubillé, Ugo Dal Lago, and Francesco Gavazzo. "On Feller continuity and full abstraction." Proceedings of the ACM on Programming Languages 6, ICFP (August 29, 2022): 826–54. http://dx.doi.org/10.1145/3547651.

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We study the nature of applicative bisimilarity in λ-calculi endowed with operators for sampling from contin- uous distributions. On the one hand, we show that bisimilarity, logical equivalence, and testing equivalence all coincide with contextual equivalence when real numbers can be manipulated through continuous functions only. The key ingredient towards this result is a notion of Feller-continuity for labelled Markov processes, which we believe of independent interest, giving rise a broad class of LMPs for which coinductive and logically inspired equivalences coincide. On the other hand, we show that if no constraint is put on the way real numbers are manipulated, characterizing contextual equivalence turns out to be hard, and most of the aforementioned notions of equivalence are even unsound.

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27

Talvitie, Topi, and Mikko Koivisto. "Counting and Sampling Markov Equivalent Directed Acyclic Graphs." Proceedings of the AAAI Conference on Artificial Intelligence 33 (July 17, 2019): 7984–91. http://dx.doi.org/10.1609/aaai.v33i01.33017984.

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Exploring directed acyclic graphs (DAGs) in a Markov equivalence class is pivotal to infer causal effects or to discover the causal DAG via appropriate interventional data. We consider counting and uniform sampling of DAGs that are Markov equivalent to a given DAG. These problems efficiently reduce to counting the moral acyclic orientations of a given undirected connected chordal graph on n vertices, for which we give two algorithms. Our first algorithm requires O(2nn4) arithmetic operations, improving a previous superexponential upper bound. The second requires O(k!2kk2n) operations, where k is the size of the largest clique in the graph; for bounded-degree graphs this bound is linear in n. After a single run, both algorithms enable uniform sampling from the equivalence class at a computational cost linear in the graph size. Empirical results indicate that our algorithms are superior to previously presented algorithms over a range of inputs; graphs with hundreds of vertices and thousands of edges are processed in a second on a desktop computer.

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28

Matsumoto, Kengo. "Relative Morita equivalence of Cuntz–Krieger algebras and flow equivalence of topological Markov shifts." Transactions of the American Mathematical Society 370, no. 10 (May 9, 2018): 7011–50. http://dx.doi.org/10.1090/tran/7272.

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29

Chen, Xinxin, Adam Dor-On, Langwen Hui, Christopher Linden, and Yifan Zhang. "Doob equivalence and non-commutative peaking for Markov chains." Journal of Noncommutative Geometry 15, no. 4 (November 9, 2021): 1469–84. http://dx.doi.org/10.4171/jncg/444.

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30

Moura, J. M. F., and M. G. S. Bruno. "DCT/DST and Gauss-Markov fields: conditions for equivalence." IEEE Transactions on Signal Processing 46, no. 9 (1998): 2571–74. http://dx.doi.org/10.1109/78.709549.

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31

Andersson, Steen A., David Madigan, and Michael D. Perlman. "A characterization of Markov equivalence classes for acyclic digraphs." Annals of Statistics 25, no. 2 (April 1997): 505–41. http://dx.doi.org/10.1214/aos/1031833662.

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32

Givan, Robert, Thomas Dean, and Matthew Greig. "Equivalence notions and model minimization in Markov decision processes." Artificial Intelligence 147, no. 1-2 (July 2003): 163–223. http://dx.doi.org/10.1016/s0004-3702(02)00376-4.

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33

Richardson, Thomas. "A characterization of Markov equivalence for directed cyclic graphs." International Journal of Approximate Reasoning 17, no. 2-3 (August 1997): 107–62. http://dx.doi.org/10.1016/s0888-613x(97)00020-0.

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34

Andersson, Steen A., and Michael D. Perlman. "Characterizing Markov equivalence classes for AMP chain graph models." Annals of Statistics 34, no. 2 (April 2006): 939–72. http://dx.doi.org/10.1214/009053606000000173.

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35

Matsumoto, Kengo. "On flow equivalence of one-sided topological Markov shifts." Proceedings of the American Mathematical Society 144, no. 7 (March 17, 2016): 2923–37. http://dx.doi.org/10.1090/proc/13074.

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36

Radhakrishnan, Adityanarayanan, Liam Solus, and Caroline Uhler. "Counting Markov equivalence classes for DAG models on trees." Discrete Applied Mathematics 244 (July 2018): 170–85. http://dx.doi.org/10.1016/j.dam.2018.03.015.

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37

COSTA, ALFREDO, and BENJAMIN STEINBERG. "A categorical invariant of flow equivalence of shifts." Ergodic Theory and Dynamical Systems 36, no. 2 (September 30, 2014): 470–513. http://dx.doi.org/10.1017/etds.2014.74.

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We prove that the Karoubi envelope of a shift—defined as the Karoubi envelope of the syntactic semigroup of the language of blocks of the shift—is, up to natural equivalence of categories, an invariant of flow equivalence. More precisely, we show that the action of the Karoubi envelope on the Krieger cover of the shift is a flow invariant. An analogous result concerning the Fischer cover of a synchronizing shift is also obtained. From these main results, several flow equivalence invariants—some new and some old—are obtained. We also show that the Karoubi envelope is, in a natural sense, the best possible syntactic invariant of flow equivalence of sofic shifts. Another application concerns the classification of Markov–Dyck and Markov–Motzkin shifts: it is shown that, under mild conditions, two graphs define flow equivalent shifts if and only if they are isomorphic. Shifts with property ($\mathscr{A}$) and their associated semigroups, introduced by Wolfgang Krieger, are interpreted in terms of the Karoubi envelope, yielding a proof of the flow invariance of the associated semigroups in the cases usually considered (a result recently announced by Krieger), and also a proof that property ($\mathscr{A}$) is decidable for sofic shifts.

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38

Pearl, Judea, and Azaria Paz. "Confounding Equivalence in Causal Inference." Journal of Causal Inference 2, no. 1 (March 1, 2014): 75–93. http://dx.doi.org/10.1515/jci-2013-0020.

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AbstractThe paper provides a simple test for deciding, from a given causal diagram, whether two sets of variables have the same bias-reducing potential under adjustment. The test requires that one of the following two conditions holds: either (1) both sets are admissible (i.e. satisfy the back-door criterion) or (2) the Markov boundaries surrounding the treatment variable are identical in both sets. We further extend the test to include treatment-dependent covariates by broadening the back-door criterion and establishing equivalence of adjustment under selection bias conditions. Applications to covariate selection and model testing are discussed.

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39

Wang, Yu, Nima Roohi, Matthew West, Mahesh Viswanathan, and Geir E. Dullerud. "Verifying Stochastic Hybrid Systems with Temporal Logic Specifications via Model Reduction." ACM Transactions on Embedded Computing Systems 20, no. 6 (November 30, 2021): 1–27. http://dx.doi.org/10.1145/3483380.

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We present a scalable methodology to verify stochastic hybrid systems for inequality linear temporal logic (iLTL) or inequality metric interval temporal logic (iMITL). Using the Mori–Zwanzig reduction method, we construct a finite-state Markov chain reduction of a given stochastic hybrid system and prove that this reduced Markov chain is approximately equivalent to the original system in a distributional sense. Approximate equivalence of the stochastic hybrid system and its Markov chain reduction means that analyzing the Markov chain with respect to a suitably strengthened property allows us to conclude whether the original stochastic hybrid system meets its temporal logic specifications. Based on this, we propose the first statistical model checking algorithms to verify stochastic hybrid systems against correctness properties, expressed in iLTL or iMITL. The scalability of the proposed algorithms is demonstrated by a case study.

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40

Li, Yanying. "Characterizing the Minimal Essential Graphs of Maximal Ancestral Graphs." International Journal of Pattern Recognition and Artificial Intelligence 34, no. 04 (August 5, 2019): 2059009. http://dx.doi.org/10.1142/s0218001420590090.

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Learning ancestor graph is a typical NP-hard problem. We consider the problem to represent a Markov equivalence class of ancestral graphs with a compact representation. Firstly, the minimal essential graph is defined to represent the equivalent class of maximal ancestral graphs with the minimum number of invariant arrowheads. Then, an algorithm is proposed to learn the minimal essential graph of ancestral graphs based on the detection of minimal collider paths. It is the first algorithm to use necessary and sufficient conditions for Markov equivalence as a base to seek essential graphs. Finally, a set of orientation rules is presented to orient edge marks of a minimal essential graph. Theory analysis shows our algorithm is sound, and complete in the sense of recognizing all minimal collider paths in a given ancestral graph. And the experiment results show we can discover all invariant marks by these orientation rules.

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41

Matsumoto, Kengo. "Continuous orbit equivalence, flow equivalence of Markov shifts and circle actions on Cuntz–Krieger algebras." Mathematische Zeitschrift 285, no. 1-2 (May 31, 2016): 121–41. http://dx.doi.org/10.1007/s00209-016-1700-3.

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42

Madsen, Line Meldgaard, Gianluca Fiandaca, Anders Vest Christiansen, and Esben Auken. "Resolution of well-known resistivity equivalences by inclusion of time-domain induced polarization data." GEOPHYSICS 83, no. 1 (January 1, 2018): E47—E54. http://dx.doi.org/10.1190/geo2017-0009.1.

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The principle of equivalence is known to cause nonuniqueness in interpretations of direct current (DC) resistivity data. Low- or high-resistivity equivalences arise when a thin geologic layer with a low/high resistivity is embedded in a relative high-/low-resistivity background formation causing strong resistivity-thickness correlations. The equivalences often make it impossible to resolve embedded layers. We found that the equivalence problem could be significantly reduced by combining the DC data with full-decay time-domain induced polarization (IP) measurements. We applied a 1D Markov chain Monte Carlo algorithm to invert synthetic DC data of models with low- and high-resistivity equivalences. By applying this inversion method, it is possible to study the space of equivalent models that have an acceptable fit to the observed data, and to make a full sensitivity analysis of the model parameters. Then, we include a contrast in chargeability into the model, modeled in terms of spectral Cole-Cole IP parameters, and invert the DC and IP data in combination. The results show that the addition of IP data largely resolves the DC equivalences. Furthermore, we present a field example in which DC and IP data were measured on a sand formation with an embedded clay layer known from a borehole drilling. Inversion results show that the DC data alone do not resolve the clay layer due to equivalence problems, but by adding the IP data to the inversion, the layer is resolved.

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43

Ridder, Ad, and Jean Walrand. "Some Large Deviations Results in Markov Fluid Models." Probability in the Engineering and Informational Sciences 6, no. 4 (October 1992): 543–60. http://dx.doi.org/10.1017/s0269964800002722.

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Markov modulated fluid models are studied in this paper. When the input of the fluid model is represented by one Markov chain, two approaches are given that result in asymptotic expressions for the overflow probability. Both approaches are based on large deviations theories. The equivalence of the expressions is proved. When the input is represented by N similar Markov chains, a reduction property is derived.

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44

Matsumoto, Kengo. "Strongly continuous orbit equivalence of one-sided topological Markov shifts." Journal of Operator Theory 74, no. 2 (September 2015): 457–83. http://dx.doi.org/10.7900/jot.2014aug19.2063.

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45

Fang, Zhuangyan, Yue Liu, Zhi Geng, Shengyu Zhu, and Yangbo He. "A local method for identifying causal relations under Markov equivalence." Artificial Intelligence 305 (April 2022): 103669. http://dx.doi.org/10.1016/j.artint.2022.103669.

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46

Reuter, G. E. H. "On Kendall's Conjecture Concerning 0+ -Equivalence of Markov Transition Functions." Journal of the London Mathematical Society s2-35, no. 2 (April 1987): 377–84. http://dx.doi.org/10.1112/jlms/s2-35.2.377.

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47

Strubbe, Stefan, and Arjan van der Schaft. "STOCHASTIC EQUIVALENCE OF CPDP-AUTOMATA AND PIECEWISE DETERMINISTIC MARKOV PROCESSES." IFAC Proceedings Volumes 38, no. 1 (2005): 25–30. http://dx.doi.org/10.3182/20050703-6-cz-1902.00289.

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48

Marcus, Brian, and Selim Tuncel. "Matrices of polynomials, positivity, and finite equivalence of Markov chains." Journal of the American Mathematical Society 6, no. 1 (January 1, 1993): 131. http://dx.doi.org/10.1090/s0894-0347-1993-1168959-x.

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49

Ohtsubo, Yoshio, and Kenji Toyonaga. "Equivalence classes for optimizing risk models in Markov decision processes." Mathematical Methods of Operational Research 60, no. 2 (October 2004): 239–50. http://dx.doi.org/10.1007/s001860400361.

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50

Apostolov, S. S., Z. A. Mayzelis, O. V. Usatenko, and V. A. Yampol'skii. "Equivalence of the Markov chains and two-sided symbolic sequences." Europhysics Letters (EPL) 76, no. 6 (December 2006): 1015–21. http://dx.doi.org/10.1209/epl/i2006-10410-4.

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