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: | , , |
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| Formato: | article |
| Lenguaje: | EN |
| Publicado: |
MDPI AG
2021
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| Materias: | |
| Acceso en línea: | https://doaj.org/article/19707d7563f54698a1f36a918777ae47 |
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| Sumario: | 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. |
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