CorollaryStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedprecheck pass

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The VAE evidence equals the ELBO plus reverse KL

Statement

Fix an observation $x$ in a VAE with latent reference measure $μ_{Z}$ . Let $γ_{θ} (\cdot; x)$ be its nonnegative measurable joint-density slice, let $Z_{θ} (x) = \int γ_{θ} (z; x) d μ_{Z} (z) \in (0, \infty)$ , and let $π_{θ} (z ∣ x) = γ_{θ} (z; x) / Z_{θ} (x)$ . Let $q_{ϕ} (\cdot ∣ x)$ be a measurable probability density satisfying $q_{ϕ} = 0$ $μ_{Z}$ -almost everywhere on ${π_{θ} = 0}$ and $\int q_{ϕ} (z ∣ x) (∣ \log γ_{θ} (z; x) ∣ + ∣ \log q_{ϕ} (z ∣ x) ∣) d μ_{Z} (z) < \infty .$ Then the choice-free density-ratio identity is $\log Z_{θ} (x) = {ELBO}_{θ, ϕ} (x) + \int q_{ϕ} (z ∣ x) \log ⁣ (\frac{q_{ϕ} (z ∣ x)}{π_{θ} (z ∣ x)}) d μ_{Z} (z),$ where the ratio integrand is zero when $q_{ϕ} = 0$ . Under the Axiom of Choice, the final integral is the library reverse-KL divergence $D_{K L} (q_{ϕ} μ_{Z} ∥ π_{θ} μ_{Z})$ and is nonnegative; hence ${ELBO}_{θ, ϕ} (x) \leq \log Z_{θ} (x)$ .

Statement references

The VAE density, evidence, posterior, and ELBO definitions.

The generic choice-free density-ratio identity and AC-dependent KL branch.

The Axiom of Choice.

Blei, Kucukelbir, and McAuliffe, Variational Inference: A Review for Statisticians.

Zheng, Hu, and Deng, 10-708 Lecture 13: Variational Inference II.

Facts & Assumptions

Given: A fixed observation $x$ , the VAE joint-density slice, its positive finite evidence, an admissible encoder density, and AC for the library KL branch.

[F1]

Variational autoencoder model, encoder, and ELBO: at each admissible fixed $x$ , $γ_{θ} (\cdot; x)$ is nonnegative measurable, $0 < Z_{θ} (x) = \int γ_{θ} d μ_{Z} < \infty$ , $π_{θ} = γ_{θ} / Z_{θ}$ , and $q_{ϕ}$ is a probability density with the stated support and absolute log-integrability conditions.

[F2]

Evidence lower bound is log evidence minus reverse KL: for a sigma-finite reference measure and such $γ, Z, π, q$ , the density-ratio identity $\log Z = ELBO (q) + \int q \log (q / π) d μ$ holds without choice; under AC the integral is $D_{K L} (q μ ∥ π μ) \geq 0$ .

[A1]

The Axiom of Choice: every family of nonempty sets has a choice function; AC is invoked only for the library KL identification and nonnegativity in [F2].

Proof

1.1givenF1

By [F1], $(μ_{Z}, γ_{θ} (\cdot; x), Z_{θ} (x), π_{θ} (\cdot ∣ x), q_{ϕ} (\cdot ∣ x))$ satisfies the sigma-finiteness, positive finite normalizer, support, and absolute log-integrability hypotheses of the generic decomposition. Its ELBO integral is exactly ${ELBO}_{θ, ϕ} (x)$ by definition.

2.1A1F2step 1.1algebra∎

Apply [F2] to these objects. Its choice-free branch gives the displayed density-ratio identity. Under [A1], its second branch identifies the ratio integral as the nonnegative reverse-KL divergence, and rearranging yields the stated lower bound.

Depends on

Used by

Reparameterized variational training Algorithm
A Gaussian model with a positive inference gap and a collapsed latent Example

Dependency tree · two levels

11 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

David M. Blei, Alp Kucukelbir, and Jon D. McAuliffe, Variational Inference: A Review for Statisticians (standard reference, not scraped)
Ronghuo Zheng, Zhiting Hu, and Yuntian Deng, 10-708 Probabilistic Graphical Models, Lecture 13: Variational Inference II (standard reference, not scraped)