{"ID":2853189,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.16892","arxiv_id":"2510.16892","title":"Batch learning equals online learning in Bayesian supervised learning","abstract":"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.","short_abstract":"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{...","url_abs":"https://arxiv.org/abs/2510.16892","url_pdf":"https://arxiv.org/pdf/2510.16892v5","authors":"[\"Hông Vân Lê\"]","published":"2025-10-19T15:39:47Z","proceeding":"math.ST","tasks":"[\"math.ST\"]","methods":"[]","has_code":false}