[Merged by Bors] - feat(Probability/Independence): define independence wrt a kernel and a measure#6106
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Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
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Co-authored-by: sgouezel <sebastien.gouezel@univ-rennes1.fr>
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…a measure (#6106) We introduce a new notion of independence with respect to a kernel and a measure. The plan is to eventually express both independence and conditional independence as particular cases of this new notion (see #6098). Two sigma-algebras `m` and `m'` are said to be independent with respect to a kernel `κ` and a measure `μ` if for all `m`-measurable sets `t₁` and `m'`-measurable sets `t₂`, `∀ᵐ a ∂μ, κ a (t₁ ∩ t₂) = κ a t₁ * κ a t₂`. Independence is the special case where `κ` is a constant kernel. Conditional independence can be defined by using the conditional expectation kernel `condexpKernel`. Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
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…a measure (#6106) We introduce a new notion of independence with respect to a kernel and a measure. The plan is to eventually express both independence and conditional independence as particular cases of this new notion (see #6098). Two sigma-algebras `m` and `m'` are said to be independent with respect to a kernel `κ` and a measure `μ` if for all `m`-measurable sets `t₁` and `m'`-measurable sets `t₂`, `∀ᵐ a ∂μ, κ a (t₁ ∩ t₂) = κ a t₁ * κ a t₂`. Independence is the special case where `κ` is a constant kernel. Conditional independence can be defined by using the conditional expectation kernel `condexpKernel`. Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
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…a measure (#6106) We introduce a new notion of independence with respect to a kernel and a measure. The plan is to eventually express both independence and conditional independence as particular cases of this new notion (see #6098). Two sigma-algebras `m` and `m'` are said to be independent with respect to a kernel `κ` and a measure `μ` if for all `m`-measurable sets `t₁` and `m'`-measurable sets `t₂`, `∀ᵐ a ∂μ, κ a (t₁ ∩ t₂) = κ a t₁ * κ a t₂`. Independence is the special case where `κ` is a constant kernel. Conditional independence can be defined by using the conditional expectation kernel `condexpKernel`. Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
kim-em
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…a measure (#6106) We introduce a new notion of independence with respect to a kernel and a measure. The plan is to eventually express both independence and conditional independence as particular cases of this new notion (see #6098). Two sigma-algebras `m` and `m'` are said to be independent with respect to a kernel `κ` and a measure `μ` if for all `m`-measurable sets `t₁` and `m'`-measurable sets `t₂`, `∀ᵐ a ∂μ, κ a (t₁ ∩ t₂) = κ a t₁ * κ a t₂`. Independence is the special case where `κ` is a constant kernel. Conditional independence can be defined by using the conditional expectation kernel `condexpKernel`. Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
kim-em
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Aug 3, 2023
…a measure (#6106) We introduce a new notion of independence with respect to a kernel and a measure. The plan is to eventually express both independence and conditional independence as particular cases of this new notion (see #6098). Two sigma-algebras `m` and `m'` are said to be independent with respect to a kernel `κ` and a measure `μ` if for all `m`-measurable sets `t₁` and `m'`-measurable sets `t₂`, `∀ᵐ a ∂μ, κ a (t₁ ∩ t₂) = κ a t₁ * κ a t₂`. Independence is the special case where `κ` is a constant kernel. Conditional independence can be defined by using the conditional expectation kernel `condexpKernel`. Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
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Aug 14, 2023
…a measure (#6106) We introduce a new notion of independence with respect to a kernel and a measure. The plan is to eventually express both independence and conditional independence as particular cases of this new notion (see #6098). Two sigma-algebras `m` and `m'` are said to be independent with respect to a kernel `κ` and a measure `μ` if for all `m`-measurable sets `t₁` and `m'`-measurable sets `t₂`, `∀ᵐ a ∂μ, κ a (t₁ ∩ t₂) = κ a t₁ * κ a t₂`. Independence is the special case where `κ` is a constant kernel. Conditional independence can be defined by using the conditional expectation kernel `condexpKernel`. Co-authored-by: RemyDegenne <Remydegenne@gmail.com>
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We introduce a new notion of independence with respect to a kernel and a measure. The plan is to eventually express both independence and conditional independence as particular cases of this new notion (see #6098).
Two sigma-algebras
mandm'are said to be independent with respect to a kernelκand a measureμif for allm-measurable setst₁andm'-measurable setst₂,∀ᵐ a ∂μ, κ a (t₁ ∩ t₂) = κ a t₁ * κ a t₂.Independence is the special case where
κis a constant kernel. Conditional independence can be defined by using the conditional expectation kernelcondexpKernel.The file is a copy of Independence/Basic with minor modifications to use the generalized independence definition.