---
title: "Variational autoencoder ELBO"
description: "Balances explaining each observation through latent samples with keeping the learned latent distribution close to the model prior."
canonical_url: "https://fanout.sh/labs/math-decoder/formula/variational-autoencoder-elbo"
md_url: "https://fanout.sh/labs/math-decoder/formula/variational-autoencoder-elbo.md"
last_updated: "2026-08-09"
access: "public"
---

# Variational autoencoder ELBO

Balances explaining each observation through latent samples with keeping the learned latent distribution close to the model prior.

## Public overview

The variational objective for x equals expected log decoder probability of x given z under q phi z given x, minus K L divergence from the approximate posterior to the prior.

Encode x into plausible latent explanations, reward explanations that reconstruct x, and charge for moving too far from the prior.

Subtraction expresses the tradeoff between reconstruction quality and a usable, prior-aligned latent space.

The paper uses reparameterized Monte Carlo samples to differentiate the expectation efficiently despite latent randomness.

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