Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Predictive Constructions Based on Measure-Valued Pólya Urn Processes

Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized <i>k</i>-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP <inline-formula><math xmlns="http://www.w3...

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Autores principales: Sandra Fortini, Sonia Petrone, Hristo Sariev
Formato: article
Lenguaje:EN
Publicado: MDPI AG 2021
Materias:
predictive distributions
random probability measures
reinforced processes
Pólya sequences
urn schemes
Bayesian inference
Mathematics
QA1-939
Acceso en línea:https://doaj.org/article/19707d7563f54698a1f36a918777ae47
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id oai:doaj.org-article:19707d7563f54698a1f36a918777ae47
record_format dspace
institution DOAJ
collection DOAJ
language EN
topic predictive distributions
random probability measures
reinforced processes
Pólya sequences
urn schemes
Bayesian inference
Mathematics
QA1-939
spellingShingle predictive distributions
random probability measures
reinforced processes
Pólya sequences
urn schemes
Bayesian inference
Mathematics
QA1-939
Sandra Fortini
Sonia Petrone
Hristo Sariev
Predictive Constructions Based on Measure-Valued Pólya Urn Processes
description Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized <i>k</i>-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>μ</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> on a Polish space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">X</mi></semantics></math></inline-formula>, the normalized sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>μ</mi><mi>n</mi></msub><mo>/</mo><msub><mi>μ</mi><mi>n</mi></msub><mrow><mo>(</mo><mi mathvariant="double-struck">X</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> agrees with the marginal predictive distributions of some random process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula>. Moreover, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>μ</mi><mi>n</mi></msub><mo>=</mo><msub><mi>μ</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>R</mi><msub><mi>X</mi><mi>n</mi></msub></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>↦</mo><msub><mi>R</mi><mi>x</mi></msub></mrow></semantics></math></inline-formula> is a random transition kernel on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">X</mi></semantics></math></inline-formula>; thus, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>μ</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> represents the contents of an urn, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mi>n</mi></msub></semantics></math></inline-formula> denotes the color of the ball drawn with distribution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>μ</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mi>μ</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi mathvariant="double-struck">X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><msub><mi>X</mi><mi>n</mi></msub></msub></semantics></math></inline-formula>—the subsequent reinforcement. In the case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><msub><mi>X</mi><mi>n</mi></msub></msub><mo>=</mo><msub><mi>W</mi><mi>n</mi></msub><msub><mi>δ</mi><msub><mi>X</mi><mi>n</mi></msub></msub></mrow></semantics></math></inline-formula>, for some non-negative random weights <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mn>1</mn></msub><mo>,</mo><msub><mi>W</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>, the process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.
format article
author Sandra Fortini
Sonia Petrone
Hristo Sariev
author_facet Sandra Fortini
Sonia Petrone
Hristo Sariev
author_sort Sandra Fortini
title Predictive Constructions Based on Measure-Valued Pólya Urn Processes
title_short Predictive Constructions Based on Measure-Valued Pólya Urn Processes
title_full Predictive Constructions Based on Measure-Valued Pólya Urn Processes
title_fullStr Predictive Constructions Based on Measure-Valued Pólya Urn Processes
title_full_unstemmed Predictive Constructions Based on Measure-Valued Pólya Urn Processes
title_sort predictive constructions based on measure-valued pólya urn processes
publisher MDPI AG
publishDate 2021
url https://doaj.org/article/19707d7563f54698a1f36a918777ae47
work_keys_str_mv AT sandrafortini predictiveconstructionsbasedonmeasurevaluedpolyaurnprocesses
AT soniapetrone predictiveconstructionsbasedonmeasurevaluedpolyaurnprocesses
AT hristosariev predictiveconstructionsbasedonmeasurevaluedpolyaurnprocesses
_version_ 1718411365652103168
spelling oai:doaj.org-article:19707d7563f54698a1f36a918777ae472021-11-25T18:16:32ZPredictive Constructions Based on Measure-Valued Pólya Urn Processes10.3390/math92228452227-7390https://doaj.org/article/19707d7563f54698a1f36a918777ae472021-11-01T00:00:00Zhttps://www.mdpi.com/2227-7390/9/22/2845https://doaj.org/toc/2227-7390Measure-valued Pólya urn processes (MVPP) are Markov chains with an additive structure that serve as an extension of the generalized <i>k</i>-color Pólya urn model towards a continuum of possible colors. We prove that, for any MVPP <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>μ</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> on a Polish space <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">X</mi></semantics></math></inline-formula>, the normalized sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>μ</mi><mi>n</mi></msub><mo>/</mo><msub><mi>μ</mi><mi>n</mi></msub><mrow><mo>(</mo><mi mathvariant="double-struck">X</mi><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></semantics></math></inline-formula> agrees with the marginal predictive distributions of some random process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula>. Moreover, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>μ</mi><mi>n</mi></msub><mo>=</mo><msub><mi>μ</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><msub><mi>R</mi><msub><mi>X</mi><mi>n</mi></msub></msub></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>↦</mo><msub><mi>R</mi><mi>x</mi></msub></mrow></semantics></math></inline-formula> is a random transition kernel on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="double-struck">X</mi></semantics></math></inline-formula>; thus, if <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>μ</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> represents the contents of an urn, then <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>X</mi><mi>n</mi></msub></semantics></math></inline-formula> denotes the color of the ball drawn with distribution <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>μ</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>/</mo><msub><mi>μ</mi><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mrow><mo>(</mo><mi mathvariant="double-struck">X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>R</mi><msub><mi>X</mi><mi>n</mi></msub></msub></semantics></math></inline-formula>—the subsequent reinforcement. In the case <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><msub><mi>X</mi><mi>n</mi></msub></msub><mo>=</mo><msub><mi>W</mi><mi>n</mi></msub><msub><mi>δ</mi><msub><mi>X</mi><mi>n</mi></msub></msub></mrow></semantics></math></inline-formula>, for some non-negative random weights <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>W</mi><mn>1</mn></msub><mo>,</mo><msub><mi>W</mi><mn>2</mn></msub><mo>,</mo><mo>…</mo></mrow></semantics></math></inline-formula>, the process <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> is better understood as a randomly reinforced extension of Blackwell and MacQueen’s Pólya sequence. We study the asymptotic properties of the predictive distributions and the empirical frequencies of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mrow><mo>(</mo><msub><mi>X</mi><mi>n</mi></msub><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub></semantics></math></inline-formula> under different assumptions on the weights. We also investigate a generalization of the above models via a randomization of the law of the reinforcement.Sandra FortiniSonia PetroneHristo SarievMDPI AGarticlepredictive distributionsrandom probability measuresreinforced processesPólya sequencesurn schemesBayesian inferenceMathematicsQA1-939ENMathematics, Vol 9, Iss 2845, p 2845 (2021)

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