Approximations for the likelihood ratio for continuous multi-parameter stochastic processes

Rob Luesink; Arunabha Bagchi

doi:10.1109/CDC.1993.325449

Approximations for the likelihood ratio for continuous multi-parameter stochastic processes

Rob Luesink
, Arunabha Bagchi

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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Abstract

Based on finitely additive white noise theory, one may derive the likelihood ratio for random variables with values in any Hilbert space. This includes stochastic processes, defined on a one- or multi-dimensional continuous-parameter bounded domain. In certain circumstances, the likelihood ratio for continuous processes may be computed directly. In general however, one will have to approximate the likelihood ratio. In this paper approximations for the likelihood ratios for continuous-parameter processes are studied. Starting from a sequence of finite dimensional projection operators in the Hilbert space, strongly converging to identity, the authors show that the likelihood ratios for the projected processes converge to the likelihood ratio for the original process. Discretization of the stochastic process turns out to be one of the possibilities for such approximations. The discretization method is expected to give good results for signals satisfying elliptic PDEs, because discretization of these processes leads to nearest neighbor models, for which the likelihood ratio has been obtained in Luesink (1992).

Original language	English
Title of host publication	Proceedings of the 32nd IEEE Conference on Decision and Control
Subtitle of host publication	December 15-17, 1993, Marriott Rivercenter, San Antonio, Texas, USA
Place of Publication	Piscataway, NJ, USA
Publisher	IEEE
Pages	1559-1563
Number of pages	5
ISBN (Print)	9780780312982
DOIs	https://doi.org/10.1109/CDC.1993.325449
Publication status	Published - 13 Jan 1993
Event	32nd IEEE Conference on Decision and Control, CDC 1993 - San Antonio, United States Duration: 15 Dec 1993 → 17 Dec 1993 Conference number: 32

Publication series

Name	Proceedings IEEE Conference on Decision and Control (CDC
Publisher	IEEE
ISSN (Print)	0191-2216

Conference

Conference	32nd IEEE Conference on Decision and Control, CDC 1993
Abbreviated title	CDC
Country/Territory	United States
City	San Antonio
Period	15/12/93 → 17/12/93

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10.1109/CDC.1993.325449

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Luesink, R., & Bagchi, A. (1993). Approximations for the likelihood ratio for continuous multi-parameter stochastic processes. In Proceedings of the 32nd IEEE Conference on Decision and Control: December 15-17, 1993, Marriott Rivercenter, San Antonio, Texas, USA (pp. 1559-1563). (Proceedings IEEE Conference on Decision and Control (CDC). IEEE. https://doi.org/10.1109/CDC.1993.325449

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title = "Approximations for the likelihood ratio for continuous multi-parameter stochastic processes",

abstract = "Based on finitely additive white noise theory, one may derive the likelihood ratio for random variables with values in any Hilbert space. This includes stochastic processes, defined on a one- or multi-dimensional continuous-parameter bounded domain. In certain circumstances, the likelihood ratio for continuous processes may be computed directly. In general however, one will have to approximate the likelihood ratio. In this paper approximations for the likelihood ratios for continuous-parameter processes are studied. Starting from a sequence of finite dimensional projection operators in the Hilbert space, strongly converging to identity, the authors show that the likelihood ratios for the projected processes converge to the likelihood ratio for the original process. Discretization of the stochastic process turns out to be one of the possibilities for such approximations. The discretization method is expected to give good results for signals satisfying elliptic PDEs, because discretization of these processes leads to nearest neighbor models, for which the likelihood ratio has been obtained in Luesink (1992).",

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note = "32nd IEEE Conference on Decision and Control, CDC 1993, CDC ; Conference date: 15-12-1993 Through 17-12-1993",

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Luesink, R & Bagchi, A 1993, Approximations for the likelihood ratio for continuous multi-parameter stochastic processes. in Proceedings of the 32nd IEEE Conference on Decision and Control: December 15-17, 1993, Marriott Rivercenter, San Antonio, Texas, USA. Proceedings IEEE Conference on Decision and Control (CDC, IEEE, Piscataway, NJ, USA, pp. 1559-1563, 32nd IEEE Conference on Decision and Control, CDC 1993, San Antonio, Texas, United States, 15/12/93. https://doi.org/10.1109/CDC.1993.325449

Approximations for the likelihood ratio for continuous multi-parameter stochastic processes. / Luesink, Rob; Bagchi, Arunabha.
Proceedings of the 32nd IEEE Conference on Decision and Control: December 15-17, 1993, Marriott Rivercenter, San Antonio, Texas, USA. Piscataway, NJ, USA: IEEE, 1993. p. 1559-1563 (Proceedings IEEE Conference on Decision and Control (CDC).

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution › Academic › peer-review

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N2 - Based on finitely additive white noise theory, one may derive the likelihood ratio for random variables with values in any Hilbert space. This includes stochastic processes, defined on a one- or multi-dimensional continuous-parameter bounded domain. In certain circumstances, the likelihood ratio for continuous processes may be computed directly. In general however, one will have to approximate the likelihood ratio. In this paper approximations for the likelihood ratios for continuous-parameter processes are studied. Starting from a sequence of finite dimensional projection operators in the Hilbert space, strongly converging to identity, the authors show that the likelihood ratios for the projected processes converge to the likelihood ratio for the original process. Discretization of the stochastic process turns out to be one of the possibilities for such approximations. The discretization method is expected to give good results for signals satisfying elliptic PDEs, because discretization of these processes leads to nearest neighbor models, for which the likelihood ratio has been obtained in Luesink (1992).

AB - Based on finitely additive white noise theory, one may derive the likelihood ratio for random variables with values in any Hilbert space. This includes stochastic processes, defined on a one- or multi-dimensional continuous-parameter bounded domain. In certain circumstances, the likelihood ratio for continuous processes may be computed directly. In general however, one will have to approximate the likelihood ratio. In this paper approximations for the likelihood ratios for continuous-parameter processes are studied. Starting from a sequence of finite dimensional projection operators in the Hilbert space, strongly converging to identity, the authors show that the likelihood ratios for the projected processes converge to the likelihood ratio for the original process. Discretization of the stochastic process turns out to be one of the possibilities for such approximations. The discretization method is expected to give good results for signals satisfying elliptic PDEs, because discretization of these processes leads to nearest neighbor models, for which the likelihood ratio has been obtained in Luesink (1992).

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Luesink R, Bagchi A. Approximations for the likelihood ratio for continuous multi-parameter stochastic processes. In Proceedings of the 32nd IEEE Conference on Decision and Control: December 15-17, 1993, Marriott Rivercenter, San Antonio, Texas, USA. Piscataway, NJ, USA: IEEE. 1993. p. 1559-1563. (Proceedings IEEE Conference on Decision and Control (CDC). doi: 10.1109/CDC.1993.325449

Approximations for the likelihood ratio for continuous multi-parameter stochastic processes

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