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On computability and disintegration

  • NATHANAEL L. ACKERMAN (a1), CAMERON E. FREER (a2) and DANIEL M. ROY (a3)
Abstract

We show that the disintegration operator on a complete separable metric space along a projection map, restricted to measures for which there is a unique continuous disintegration, is strongly Weihrauch equivalent to the limit operator Lim. When a measure does not have a unique continuous disintegration, we may still obtain a disintegration when some basis of continuity sets has the Vitali covering property with respect to the measure; the disintegration, however, may depend on the choice of sets. We show that, when the basis is computable, the resulting disintegration is strongly Weihrauch reducible to Lim, and further exhibit a single distribution realizing this upper bound.

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Ackerman N.L., Freer C.E. and Roy D.M. (2010). On the computability of conditional probability. Preprint, arXiv:1005.3014.
Ackerman N.L., Freer C.E. and Roy D.M. (2011). Noncomputable conditional distributions. In: Proceedings of the 26th Annual IEEE Symposium on Logic in Computer Science (LICS 2011), IEEE Computer Society 107–116.
Bosserhoff V. (2008). Notions of probabilistic computability on represented spaces. Journal of Universal Computer Science 14 (6) 956995.
Brattka V. (2005). Effective Borel measurability and reducibility of functions. Mathematical Logic Quarterly 51 (1) 1944.
Brattka V. and Gherardi G. (2011a). Effective choice and boundedness principles in computable analysis. Bulletin of Symbolic Logic 17 (1) 73117.
Brattka V. and Gherardi G. (2011b). Weihrauch degrees, omniscience principles and weak computability. Journal of Symbolic Logic 76 (1) 143176.
Brattka V., Gherardi G. and Marcone A. (2012). The Bolzano–Weierstrass Theorem is the jump of Weak Kőnig's Lemma. Annals of Pure Applied Logic 163 (6) 623655.
Brattka V., Hölzl R. and Gherardi G. (2015). Probabilistic computability and choice. Information and Computation 242 249286.
Collins P. (2009). A computable type theory for control systems. In: Proceedings of the 48th IEEE Conference on Decision and Control 5583–5543.
Collins P. (2010). Computable analysis with applications to dynamical systems. Technical Report MAC-1002, Centrum Wiskunde & Informatica.
Escardó M. (2004). Synthetic topology. Electronic Notes in Theoretical Computer Science 87 21156.
Fraser D.A.S., McDunnough P., Naderi A. and Plante A. (1995). On the definition of probability densities and sufficiency of the likelihood map. Probability and Mathematical Statistics 15 301310.
Fraser D.A.S. and Naderi A. (1996). On the definition of conditional probability. In Research Developments in Probability and Statistics, VSP, Utrecht 23–26.
Gács P. (2005). Uniform test of algorithmic randomness over a general space. Theoretical Computer Science 341 (1–3) 91137.
Hoyrup M. and Rojas C. (2009). Computability of probability measures and Martin-Löf randomness over metric spaces. Information and Computation 207 (7) 830847.
Hoyrup M. and Rojas C. (2011). Absolute continuity of measures and preservation of randomness. Preprint, http://www.loria.fr/~hoyrup/abscont.pdf.
Hoyrup M., Rojas C. and Weihrauch K. (2012). Computability of the Radon–Nikodym derivative. Computability 1 (1) 313.
Kallenberg O. (2002). Foundations of Modern Probability, 2nd ed. Springer.
Kolmogorov A.N. (1933). Grundbegriffe der Wahrscheinlichkeitsrechnung, Springer.
Pauly A. (2016). On the topological aspects of the theory of represented spaces. Computability.
Pauly A. and Fouché W. (2014). How constructive is constructing measures? arXiv:1409.3428v1.
Pfanzagl J. (1979). Conditional distributions as derivatives. Annals of Probability 7 (6) 10461050.
Schröder M. (2002a). Admissible Representations for Continuous Computations. Ph.D. Thesis, Fachbereich Informatik, FernUniversität Hagen.
Schröder M. (2002b). Effectivity in spaces with admissible multirepresentations. Mathematical Logic Quarterly 48 (suppl. 1) 7890.
Schröder M. (2007). Admissible representations for probability measures. Mathematical Logic Quarterly 53 (4–5) 431445.
Tjur T. (1975). A Constructive Definition of Conditional Distributions. Preprint 13. Institute of Mathematical Statistics, University of Copenhagen.
Tjur T. (1980). Probability Based on Radon Measures, Wiley Series in Probability and Mathematical Statistics, John Wiley & Sons.
Weihrauch K. (2000). Computable Analysis: An Introduction, Springer.
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Mathematical Structures in Computer Science
  • ISSN: 0960-1295
  • EISSN: 1469-8072
  • URL: /core/journals/mathematical-structures-in-computer-science
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