Batch learning equals online learning in Bayesian supervised learning
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866917520673341440 |
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| author | Lê, Hông Vân |
| author_facet | Lê, Hông Vân |
| contents | In this paper we study Bayesian supervised learning models proposed by Lê in \cite{Le2025}. We show the existence of Bayesian inversions on universal Bayesian supervised learning models $(\mathcal{P}(\mathcal{Y})^{\mathcal{X}}, μ, \mathrm{Id}_{\mathcal{P}(\mathcal{Y})^{\mathcal{X}}}, \mathcal{P}(\mathcal{Y})^{\mathcal{X}}$ for arbitrary input space $\mathcal{X}$, Souslin label space $\mathcal{Y}$, and prior probability measure $μ\in \mathcal{P}( \mathcal{P}(\mathcal{Y})^{\mathcal{X}})$. Using functoriality of probabilistic morphisms, we prove that sequential and batch Bayesian inversions coincide in supervised learning models with conditionally independent (possibly non-i.i.d.) data \cite{Le2025}. This equivalence holds without domination or discreteness assumptions on sampling operators. We derive a recursive formula for posterior predictive distributions, which reduces to the Kalman filter in Gaussian process regression. For Souslin label spaces $\mathcal{Y}$ and arbitrary input sets $\mathcal{X}$, we characterize probability measures on $\mathcal{P}(\mathcal{Y})^{\mathcal{X}}$ via projective systems, generalizing Orbanz \cite{Orbanz2011}. We revisit MacEachern's Dependent Dirichlet Processes (DDP) \cite{MacEachern2000} using copula-based constructions \cite{BJQ2012} and show how to compute posterior predictive distributions in universal Bayesian supervised models with DDP priors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2510_16892 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Batch learning equals online learning in Bayesian supervised learning Lê, Hông Vân Statistics Theory 62C10, Secondary: 62G05, 62G08 In this paper we study Bayesian supervised learning models proposed by Lê in \cite{Le2025}. We show the existence of Bayesian inversions on universal Bayesian supervised learning models $(\mathcal{P}(\mathcal{Y})^{\mathcal{X}}, μ, \mathrm{Id}_{\mathcal{P}(\mathcal{Y})^{\mathcal{X}}}, \mathcal{P}(\mathcal{Y})^{\mathcal{X}}$ for arbitrary input space $\mathcal{X}$, Souslin label space $\mathcal{Y}$, and prior probability measure $μ\in \mathcal{P}( \mathcal{P}(\mathcal{Y})^{\mathcal{X}})$. Using functoriality of probabilistic morphisms, we prove that sequential and batch Bayesian inversions coincide in supervised learning models with conditionally independent (possibly non-i.i.d.) data \cite{Le2025}. This equivalence holds without domination or discreteness assumptions on sampling operators. We derive a recursive formula for posterior predictive distributions, which reduces to the Kalman filter in Gaussian process regression. For Souslin label spaces $\mathcal{Y}$ and arbitrary input sets $\mathcal{X}$, we characterize probability measures on $\mathcal{P}(\mathcal{Y})^{\mathcal{X}}$ via projective systems, generalizing Orbanz \cite{Orbanz2011}. We revisit MacEachern's Dependent Dirichlet Processes (DDP) \cite{MacEachern2000} using copula-based constructions \cite{BJQ2012} and show how to compute posterior predictive distributions in universal Bayesian supervised models with DDP priors. |
| title | Batch learning equals online learning in Bayesian supervised learning |
| topic | Statistics Theory 62C10, Secondary: 62G05, 62G08 |
| url | https://arxiv.org/abs/2510.16892 |