For questions about the Evidence Lower BOund (ELBO) objective function, which is typically optimized in the context of variational auto-encoders or variational Bayesian neural networks.
Questions tagged [evidence-lower-bound]
19 questions
6
votes
1 answer
Why is the evidence equal to the KL divergence plus the loss?
Why is the equation $$\log p_{\theta}(x^1,...,x^N)=D_{KL}(q_{\theta}(z|x^i)||p_{\phi}(z|x^i))+\mathbb{L}(\phi,\theta;x^i)$$ true, where $x^i$ are data points and $z$ are latent variables?
I was reading the original variation autoencoder paper and I…
user8714896
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4
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Why does the variational auto-encoder use the reconstruction loss?
VAE is trained to reduce the following two losses.
KL divergence between inferred latent distribution and Gaussian.
the reconstruction loss
I understand that the first one regularizes VAE to get structured latent space. But why and how does the…
Jun
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4
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2 answers
What's going on in the equation of the variational lower bound?
I don't really understand what this equation is saying or what the purpose of the ELBO is. How does it help us find the true posterior distribution?
Gooby
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3
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What does the approximate posterior on latent variables, $q_\phi(z|x)$, tend to when optimising VAE's
The ELBO objective is described as follows
$$ ELBO(\phi,\theta) = E_{q_\phi(z|x)}[log p_\theta (x|z)] - KL[q_\phi (z|x)||p(z)] $$
This form of ELBO includes a regularisation term in the form of the KL divergence which drives $q_\phi(z|x)…
quest ions
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2
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1 answer
Clarification on the training objective of denoising diffusion models
I'm reading the Denoising Diffusion Probabilistic Models paper (Ho et al. 2020). And I am puzzled about the training objective. I understood (I think) the trick regarding the reparametrization of the variance in terms of the noise:
$$\mu_\theta(x_t,…
user3903647
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2
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2 answers
In variational autoencoders, why do people use MSE for the loss?
In VAEs, we try to maximize the ELBO = $\mathbb{E}_q [\log\ p(x|z)] + D_{KL}(q(z \mid x), p(z))$, but I see that many implement the first term as the MSE of the image and its reconstruction. Here's a paper (section 5) that seems to do that: Don't…
IttayD
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2
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1 answer
How does the implementation of the VAE's objective function equate to ELBO?
For a lot of VAE implementations I've seen in code, it's not really obvious to me how it equates to ELBO.
$$L(X)=H(Q)-H(Q:P(X,Z))=\sum_ZQ(Z)logP(Z,X)-\sum_ZQ(Z)log(Q(Z))$$
The above is the definition of ELBO, where $X$ is some input, $Z$ is a latent…
user8714896
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2
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1 answer
In this VAE formula, why do $p$ and $q$ have the same parameters?
In $$\log p_{\theta}(x^1,...,x^N)=D_{KL}(q_{\theta}(z|x^i)||p_{\phi}(z|x^i))+\mathbb{L}(\phi,\theta;x^i),$$ why does $p(x^1,...,x^N)$ and $q(z|x^i)$ have the same parameter $\theta?$
Given that $p$ is just the probability of the observed data and…
user8714896
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Why does the ELBO come to a steady state and the latent space shrinks?
I'm trying to train a VAE using a graph dataset. However, my latent space shrinks epoch by epoch. Meanwhile, my ELBO plot comes to a steady state after a few epochs.
I tried to play around with parameters and I realized, by increasing the batch size…
Blade
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1
vote
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If we know the joint distribution, can we simply derive the evidence from it?
I'm struggling to understand one specific part of the formalism of the free energy principle.
My understanding is that the free energy principle can be derived from considering statistical dynamics of a system that is coupled with its environment in…
Gustavo
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1
vote
0 answers
Is VAE the same as the E-step of the EM algorithm?
EM(Expectation Maximum)
Target: maximize $p_\theta(x)$
$ p_\theta(x)=\frac{p_\theta(x, z)}{p_\theta(z \mid x)} \\\\$
Take log on both sides:
$ \log p_\theta(x)=\log p_\theta(x, z)-\log p_\theta(z \mid x) \\\\$
Introduce distribution $q_\phi(z)$:
$…
Garfield
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1
vote
0 answers
variational inference but with a weighted loglikelihood
I would like to know if it's correct if I substitute in the ELBO formula
a weighted sum of the loglikelihood
$$\sum E_{q_{\theta}(w)}[w_i \ln{p(y_i|f^{w}(x_i))}]$$
in place of the traditional sum.
My problem is that my dataset comes with the…
Alucard
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1
vote
1 answer
How is the variational lower bound for hard attention derived in Show, Attend and Tell
How is the jump from line 1 to line 2 done in equation 10 of Show, Attend and Tell?
While we're at it, another thing that might be muddying the waters for me is that I'm not clear on what the sum is over. I know that $s$ is indexed as $s_{t,i}$,…
Alexander Soare
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0
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Confusion over taking gradients in Variational Autoencoders (VAE)
I am confused as to when to hold certain parameters constant in a VAE. I will explain with a concrete example.
We can write $\operatorname{ELBO}(\phi, \theta) = \mathbb{E}_{q_{\phi}(z)}\left[\log
\left(p_{\theta}(x| z)\right)\right] -…
Joel
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Why is the variational lower bound is easier to compute than the original marginal distribution?
Why is the ELBO of $p(x)=\int p(x|z)p(z)\mathrm{d}z$ easier to compute/estimate than the expression itself? Can we compute this quantity itself through sampling in the same way? I understanding that aggregating over data means taking log before…
Hanhan Li
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