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Gaussian mean-field reverse KL underestimates correlated marginal variance

Statement refuted

Under AC, the claim that maximizing the ELBO over independent Gaussian factors always recovers the target's marginal variances is false, even for a nonsingular centered bivariate Gaussian target.

Facts & Assumptions

Given: Assume AC. Fix $0 < ∣ ρ ∣ < 1$ and put $β = 1 - ρ^{2}$ . On $R^{2}$ with product reference measure $μ = λ_{1} \otimes λ_{1}$ , take the target density $γ_{ρ} (x, y) = \frac{1}{2 π \sqrt{β}} \exp ⁣ (- \frac{x^{2} - 2 ρ x y + y^{2}}{2 β})$ and independent Gaussian variational factors $q (x, y) = g_{m_{1}, v_{1}} (x) g_{m_{2}, v_{2}} (y)$ with $v_{1}, v_{2} > 0$ , where $g_{m, v} (z) = (2 π v)^{- 1 / 2} e^{- (z - m)^{2} / (2 v)}$ .

[F1]

Variational family and evidence lower bound: admissibility requires support on ${π > 0}$ and finite $\int q (∣ \log γ ∣ + ∣ \log q ∣) d μ$ .

[F2]

Variational family and evidence lower bound: $ELBO (q) = \int q (\log γ - \log q) d μ$ .

[A1]

AC implies countable choice by restriction to any countable family of nonempty sets. The Axiom of Choice The Axiom of Countable Choice ( ${A C}_{ω}$ )

[F3]

The standard normal density $ϕ (t) = (2 π)^{- 1 / 2} e^{- t^{2} / 2}$ has Lebesgue mass one. The standard normal density has total mass one

[F4]

For $v > 0$ , $g_{m, v} (t) = v^{- 1 / 2} ϕ ((t - m) / \sqrt{v})$ is the Lebesgue density of $N (m, v)$ . A positive-variance normal law has its location-scale density

[F5]

A variable with law $N (m, v)$ has mean $m$ and variance $v$ . Location-scale normal laws have their stated mean and variance

[F6]

Tonelli applies to nonnegative functions and Fubini applies to absolutely integrable functions on sigma-finite product spaces. Tonelli's theorem for nonnegative measurable functions on a sigma-finite product Fubini's theorem for L^1 functions on a sigma-finite product

[F7]

Integrating against a density measure agrees with integrating the product against the base measure. Integrating against a density agrees with integrating the product

[F8]

By [A1], one-dimensional Lebesgue measure is a measure and each interval $[- n, n]$ has finite measure $2 n$ ; these intervals cover $R$ , so the factor Lebesgue spaces are sigma-finite. Assuming countable choice, $L (R^{n})$ is a sigma-algebra containing every elementary set and $λ_{n}$ is a complete measure extending elementary volume A box in $R^{n}$ with parameters $a_{i} \leq b_{i}$ is Lebesgue measurable of measure $\prod_{i < n} (b_{i} - a_{i})$ , whichever of its faces are included Finite, sigma-finite, and semifinite measures

Counterexample

Given: The hypotheses and explicit densities in Facts & Assumptions.

1.1A1F3F5F6F7F8algebra

Let $s = \sqrt{β} > 0$ and $ϕ (t) = (2 π)^{- 1 / 2} e^{- t^{2} / 2}$ . Direct algebra gives $γ_{ρ} (x, y) = ϕ (x) g_{ρ x, β} (y) = ϕ (y) g_{ρ y, β} (x)$ . The target is nonnegative and measurable, and [F8] supplies the sigma-finite Lebesgue product spaces; Tonelli [F6] and [F3] show that it integrates to one and has both marginals $ϕ$ . By [F5], both marginal means are zero and both marginal variances are one. Since $∣ x y ∣ \leq (x^{2} + y^{2}) / 2$ , the cross moment is absolutely integrable. Fubini [F6], density integration [F7], and [F5] give $E [X Y] = \int x ϕ (x) (ρ x) d x = ρ$ , because the conditional density $g_{ρ x, β}$ has mean $ρ x$ . Thus the covariance is $(\begin{array}{cc} 1 & ρ \\ ρ & 1 \end{array})$ .

2.1F1F4F5F6F7step 1.1algebra

For $m_{j} \in R$ and $v_{j} > 0$ , [F4] makes $g_{m_{j}, v_{j}}$ probability densities with means $m_{j}$ and variances $v_{j}$ . Tonelli [F6] gives $\int q = 1$ . The product density $q$ and target $γ_{ρ}$ are positive everywhere, so the support condition in [F1] holds. Their logarithms are quadratic; the second moments and $∣ x y ∣ \leq (x^{2} + y^{2}) / 2$ make the absolute logarithms integrable under $q$ , so $q$ is admissible. Fubini [F6], density integration [F7], and [F5] give $E_{q} [X^{2}] = m_{1}^{2} + v_{1}$ , $E_{q} [Y^{2}] = m_{2}^{2} + v_{2}$ , and $E_{q} [X Y] = m_{1} m_{2}$ .

3.1F2step 2.1algebra

From the displayed density, $\log γ_{ρ} = - \log (2 π s) - (x^{2} - 2 ρ x y + y^{2}) / (2 β)$ . Also $- E_{g_{m_{j}, v_{j}}} \log g_{m_{j}, v_{j}} = \frac{1}{2} \log (2 π v_{j}) + \frac{1}{2}$ , by the moments in step 2.1. Substituting these expectations into [F2] gives $ELBO (q) = C_{ρ} - \frac{m_{1}^{2} - 2 ρ m_{1} m_{2} + m_{2}^{2} + v_{1} + v_{2}}{2 β} + \frac{1}{2} \log v_{1} + \frac{1}{2} \log v_{2}$ , where $C_{ρ}$ is independent of $m_{1}, m_{2}, v_{1}, v_{2}$ . The mean quadratic equals $(m_{1} - ρ m_{2})^{2} + β m_{2}^{2}$ , so its unique minimizing, hence ELBO-maximizing, means are $m_{1} = m_{2} = 0$ .

4.1givenstep 3.1algebra∎

For each variance, the part depending on $v$ is $g (v) = - v / (2 β) + (1 / 2) \log v$ . Its derivative is $g^{'} (v) = (β - v) / (2 β v)$ , positive for $0 < v < β$ and negative for $v > β$ . Thus the unique maximizer is $v = β$ . Since $0 < ∣ ρ ∣ < 1$ , $0 < β < 1$ , proving strict variance underestimation.

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Sources

David M. Blei, Alp Kucukelbir, and Jon D. McAuliffe, Variational Inference: A Review for Statisticians (2017) (standard reference, not scraped)
Michael I. Jordan, Zoubin Ghahramani, Tommi S. Jaakkola, and Lawrence K. Saul, An Introduction to Variational Methods for Graphical Models (1999) (standard reference, not scraped)