証拠下限

Evidence Lower Bound（ELBO）は、分野の基本的な概念です 確率モデル and variational inference. It serves as a crucial 目的関数を修正します 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 クルバック・ライブラーダイバージェンス 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))

この式において：

• 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 ベイジアン深層学習 そしてVariational Autoencoders（VAE）のような生成モデル。

By effectively optimizing the ELBO, practitioners can leverage variational inference to make efficient inferences about hidden variables in complex models, leading to better モデルのパフォーマンス より正確な予測を可能にします。