La borne inférieure de preuve (ELBO) est un concept fondamental dans le domaine de modélisation probabiliste and variational inference. It serves as a crucial fonction objectif that helps in approximating complex posterior distributions, which are often intractable to compute directly.
The ELBO is defined as the logarithm of the evidence (or marginal likelihood) of the observed data, lower-bounded by the divergence de Kullback-Leibler between the approximate posterior distribution and the true posterior distribution. Mathematically, it can be expressed as:
ELBO = E_q[log(p(x|z))] – KL(q(z|x) || p(z))
Dans cette équation :
- E_q[log(p(x|z))] represents the expected log-likelihood of the observed data given the latent variables, weighted by the approximate posterior distribution.
- KL(q(z|x) || p(z)) is the Kullback-Leibler divergence that measures the difference between the approximate posterior q(z|x) and the prior distribution p(z).
The purpose of maximizing the ELBO is to improve the quality of the variational approximation, making it closer to the true posterior distribution. This is essential in many machine learning applications, particularly in Apprentissage profond bayésien et modèles génératifs comme les Autoencodeurs Variationnels (VAEs).
By effectively optimizing the ELBO, practitioners can leverage variational inference to make efficient inferences about hidden variables in complex models, leading to better performance du modèle et des prédictions plus précises.