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...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
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MDPI AG
2021-11-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/9/22/2845 |
| _version_ | 1797509488045981696 |
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| author | Sandra Fortini Sonia Petrone Hristo Sariev |
| author_facet | Sandra Fortini Sonia Petrone Hristo Sariev |
| author_sort | Sandra Fortini |
| collection | DOAJ |
| 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. |
| first_indexed | 2024-03-10T05:18:29Z |
| format | Article |
| id | doaj.art-19707d7563f54698a1f36a918777ae47 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-10T05:18:29Z |
| publishDate | 2021-11-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-19707d7563f54698a1f36a918777ae472023-11-23T00:13:56ZengMDPI AGMathematics2227-73902021-11-01922284510.3390/math9222845Predictive Constructions Based on Measure-Valued Pólya Urn ProcessesSandra Fortini0Sonia Petrone1Hristo Sariev2Department of Decision Sciences, Bocconi University, 20136 Milano, ItalyDepartment of Decision Sciences, Bocconi University, 20136 Milano, ItalyInstitute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, BulgariaMeasure-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.https://www.mdpi.com/2227-7390/9/22/2845predictive distributionsrandom probability measuresreinforced processesPólya sequencesurn schemesBayesian inference |
| spellingShingle | Sandra Fortini Sonia Petrone Hristo Sariev Predictive Constructions Based on Measure-Valued Pólya Urn Processes Mathematics predictive distributions random probability measures reinforced processes Pólya sequences urn schemes Bayesian inference |
| title | 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_short | Predictive Constructions Based on Measure-Valued Pólya Urn Processes |
| title_sort | predictive constructions based on measure valued polya urn processes |
| topic | predictive distributions random probability measures reinforced processes Pólya sequences urn schemes Bayesian inference |
| url | https://www.mdpi.com/2227-7390/9/22/2845 |
| work_keys_str_mv | AT sandrafortini predictiveconstructionsbasedonmeasurevaluedpolyaurnprocesses AT soniapetrone predictiveconstructionsbasedonmeasurevaluedpolyaurnprocesses AT hristosariev predictiveconstructionsbasedonmeasurevaluedpolyaurnprocesses |