Bayesian nonparametric inference on a Fréchet class
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866909506005368832 |
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| author | Dreassi, Emanuela Pratelli, Luca Rigo, Pietro |
| author_facet | Dreassi, Emanuela Pratelli, Luca Rigo, Pietro |
| contents | Let $(\mathcal{X},\mathcal{F},μ)$ and $(\mathcal{Y},\mathcal{G},ν)$ be probability spaces and $(Z_n)$ a sequence of random variables with values in $(\mathcal{X}\times\mathcal{Y},\,\mathcal{F}\otimes\mathcal{G})$. Let $Γ(μ,ν)$ be the collection of all probability measures $p$ on $\mathcal{F}\otimes\mathcal{G}$ such that $$p\bigl(A\times\mathcal{Y}\bigr)=μ(A)\quad\text{and}\quad p\bigl(\mathcal{X}\times B\bigr)=ν(B)\quad\text{for all }A\in\mathcal{F}\text{ and }B\in\mathcal{G}.$$ In this paper, we build some probability measures $Π$ on $Γ(μ,ν)$. In addition, for each such $Π$, we assume that $(Z_n)$ is exchangeable with de Finetti's measure $Π$ and we evaluate the conditional distribution $Π(\cdot\mid Z_1,\ldots,Z_n)$. In Bayesian nonparametrics, if $(Z_1,\ldots, Z_n)$ are the available data, $Π$ and $Π(\cdot\mid Z_1,\ldots, Z_n)$ can be regarded as the prior and the posterior, respectively. To support this interpretation, it suffices to think of a problem where the unknown probability distribution of some bivariate phenomenon is constrained to have marginals $μ$ and $ν$. Finally, analogous results are obtained for the set $Γ(μ)$ of those probability measures on $\mathcal{F}\otimes\mathcal{G}$ with marginal $μ$ on $\mathcal{F}$ (but arbitrary marginal on $\mathcal{G}$). That is, we introduce some priors on $Γ(μ)$ and we evaluate the corresponding posteriors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_16192 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Bayesian nonparametric inference on a Fréchet class Dreassi, Emanuela Pratelli, Luca Rigo, Pietro Methodology 60G09, 60G57, 62F15, 62G99 Let $(\mathcal{X},\mathcal{F},μ)$ and $(\mathcal{Y},\mathcal{G},ν)$ be probability spaces and $(Z_n)$ a sequence of random variables with values in $(\mathcal{X}\times\mathcal{Y},\,\mathcal{F}\otimes\mathcal{G})$. Let $Γ(μ,ν)$ be the collection of all probability measures $p$ on $\mathcal{F}\otimes\mathcal{G}$ such that $$p\bigl(A\times\mathcal{Y}\bigr)=μ(A)\quad\text{and}\quad p\bigl(\mathcal{X}\times B\bigr)=ν(B)\quad\text{for all }A\in\mathcal{F}\text{ and }B\in\mathcal{G}.$$ In this paper, we build some probability measures $Π$ on $Γ(μ,ν)$. In addition, for each such $Π$, we assume that $(Z_n)$ is exchangeable with de Finetti's measure $Π$ and we evaluate the conditional distribution $Π(\cdot\mid Z_1,\ldots,Z_n)$. In Bayesian nonparametrics, if $(Z_1,\ldots, Z_n)$ are the available data, $Π$ and $Π(\cdot\mid Z_1,\ldots, Z_n)$ can be regarded as the prior and the posterior, respectively. To support this interpretation, it suffices to think of a problem where the unknown probability distribution of some bivariate phenomenon is constrained to have marginals $μ$ and $ν$. Finally, analogous results are obtained for the set $Γ(μ)$ of those probability measures on $\mathcal{F}\otimes\mathcal{G}$ with marginal $μ$ on $\mathcal{F}$ (but arbitrary marginal on $\mathcal{G}$). That is, we introduce some priors on $Γ(μ)$ and we evaluate the corresponding posteriors. |
| title | Bayesian nonparametric inference on a Fréchet class |
| topic | Methodology 60G09, 60G57, 62F15, 62G99 |
| url | https://arxiv.org/abs/2502.16192 |