5 An Introduction to Imprecise Markov Chains
177
the goal there is not to represent uncertainty and change these parameters to compute
robust bounds on quantities of interest. Rather, their aim is to optimise the process
evolution towards some operational target.
Finally, we again emphasise that our treatment uses epistemic irrelevance, which
we distinguish from using strong independence. There is, however, an extended
body of literature also on the latter. These alternative models are known as Markov
chains under strong independence, e.g. in [27], as interval Markov chains [20, 29,
42, 43] or as Markov set chains [23–25].
Acknowledgments The author wishes to express his sincere gratitude to Gert de Cooman and
Jasper De Bock, for their helpful comments and suggestions during the writing of this chapter. He
also wants to thank the reviewer, whose comments and suggestions further helped to improve this
work.
References
1. W.J. Anderson, Continuous-Time Markov Chains, An Applications-Oriented Approach.
Springer Series in Statistics (Springer, New York, 1991)
2. A. Antonucci, C.P. de Campos, M. Zaffalon, Probabilistic graphical models, in Introduction to
Imprecise Probabilities, ed. by T. Augustin, F.P.A. Coolen, G. De Cooman, M.C.M. Troffaes
(Wiley, New York, 2014)
3. T. Augustin, F.P.A. Coolen, G. De Cooman, M.C.M. Troffaes, Introduction to Imprecise
Probabilities (Wiley, New York, 2014)
4. A. Benavoli, M. Zaffalon, E. Miranda, Robust filtering through coherent lower previsions.
IEEE Trans. Autom. Control 56(7), 1567–1581 (2011)
5. I. Couso, S. Moral, P. Walley, A survey of concepts of independence for imprecise probabilities.
Risk Decis. Policy 5(2), 165–181 (2000)
6. F.G. Cozman, Credal networks. Artif. Intell. 120, 199–233 (2000)
7. F.G. Cozman, Graphical models for imprecise probabilities. Int. J. Approx. Reason. 39(2–3),
167–184 (2005)
8. R.J. Crossman, P. Coolen-Schrijner, F.P. Coolen, Time-homogeneous birth-death processes
with probability intervals and absorbing state. J. Stat. Theory Pract. 3(1), 103–118 (2009)
9. J. De Bock, Credal networks under epistemic irrelevance: theory and algorithms. Ph.D. thesis,
Ghent University (2015)
10. J. De Bock, The limit behaviour of imprecise continuous-time Markov chains. J. Nonlinear
Sci. 27(1), 159–196 (2017)
11. J. De Bock, Credal networks under epistemic irrelevance. Int. J. Approx. Reason. 85, 107–138
(2017)
12. J. De Bock, G. De Cooman, An efficient algorithm for estimating state sequences in imprecise
hidden Markov models. J. Artif. Intell. Res. 50, 189–233 (2014)
13. G. De Cooman, F. Hermans, Imprecise probability trees: Bridging two theories of imprecise
probability. Artificial Intelligence 172(11), 1400–1427 (2008)
14. G. De Cooman, F. Hermans, E. Quaeghebeur, Imprecise Markov chains and their limit
behavior. Probab. Eng. Inf. Sci. 23(4), 597–635 (2009)
15. G. De Cooman, F. Hermans, A. Antonucci, M. Zaffalon, Epistemic irrelevance in credal nets:
the case of imprecise Markov trees. Int. J. Approx. Reason. 51(9), 1029–1052 (2010)
16. G. De Cooman, J. De Bock, S. Lopatatzidis, A pointwise ergodic theorem for imprecise
Markov chains, in Proceedings of ISIPTA 2015, pp. 107–115 (2015)
177
the goal there is not to represent uncertainty and change these parameters to compute
robust bounds on quantities of interest. Rather, their aim is to optimise the process
evolution towards some operational target.
Finally, we again emphasise that our treatment uses epistemic irrelevance, which
we distinguish from using strong independence. There is, however, an extended
body of literature also on the latter. These alternative models are known as Markov
chains under strong independence, e.g. in [27], as interval Markov chains [20, 29,
42, 43] or as Markov set chains [23–25].
Acknowledgments The author wishes to express his sincere gratitude to Gert de Cooman and
Jasper De Bock, for their helpful comments and suggestions during the writing of this chapter. He
also wants to thank the reviewer, whose comments and suggestions further helped to improve this
work.
References
1. W.J. Anderson, Continuous-Time Markov Chains, An Applications-Oriented Approach.
Springer Series in Statistics (Springer, New York, 1991)
2. A. Antonucci, C.P. de Campos, M. Zaffalon, Probabilistic graphical models, in Introduction to
Imprecise Probabilities, ed. by T. Augustin, F.P.A. Coolen, G. De Cooman, M.C.M. Troffaes
(Wiley, New York, 2014)
3. T. Augustin, F.P.A. Coolen, G. De Cooman, M.C.M. Troffaes, Introduction to Imprecise
Probabilities (Wiley, New York, 2014)
4. A. Benavoli, M. Zaffalon, E. Miranda, Robust filtering through coherent lower previsions.
IEEE Trans. Autom. Control 56(7), 1567–1581 (2011)
5. I. Couso, S. Moral, P. Walley, A survey of concepts of independence for imprecise probabilities.
Risk Decis. Policy 5(2), 165–181 (2000)
6. F.G. Cozman, Credal networks. Artif. Intell. 120, 199–233 (2000)
7. F.G. Cozman, Graphical models for imprecise probabilities. Int. J. Approx. Reason. 39(2–3),
167–184 (2005)
8. R.J. Crossman, P. Coolen-Schrijner, F.P. Coolen, Time-homogeneous birth-death processes
with probability intervals and absorbing state. J. Stat. Theory Pract. 3(1), 103–118 (2009)
9. J. De Bock, Credal networks under epistemic irrelevance: theory and algorithms. Ph.D. thesis,
Ghent University (2015)
10. J. De Bock, The limit behaviour of imprecise continuous-time Markov chains. J. Nonlinear
Sci. 27(1), 159–196 (2017)
11. J. De Bock, Credal networks under epistemic irrelevance. Int. J. Approx. Reason. 85, 107–138
(2017)
12. J. De Bock, G. De Cooman, An efficient algorithm for estimating state sequences in imprecise
hidden Markov models. J. Artif. Intell. Res. 50, 189–233 (2014)
13. G. De Cooman, F. Hermans, Imprecise probability trees: Bridging two theories of imprecise
probability. Artificial Intelligence 172(11), 1400–1427 (2008)
14. G. De Cooman, F. Hermans, E. Quaeghebeur, Imprecise Markov chains and their limit
behavior. Probab. Eng. Inf. Sci. 23(4), 597–635 (2009)
15. G. De Cooman, F. Hermans, A. Antonucci, M. Zaffalon, Epistemic irrelevance in credal nets:
the case of imprecise Markov trees. Int. J. Approx. Reason. 51(9), 1029–1052 (2010)
16. G. De Cooman, J. De Bock, S. Lopatatzidis, A pointwise ergodic theorem for imprecise
Markov chains, in Proceedings of ISIPTA 2015, pp. 107–115 (2015)