# Questions tagged [evidence-lower-bound]

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.

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### Are some low dimensional distributions known to be hard to model with VAEs?

I am trying to implement a toy VAE project. My goal is to use a VAE to model the moon dataset from scikit-learn, with an extra constant (but noisy) z-dimension. To this end I use an approximate ...
13 views

### Tensorflow Probability Implementation of Automatic Differentiation Variational Inference with Mixtures

In this paper, the authors suggest using the following loss instead of the traditional ELBO in order to train what basically is a Variational Autoencoder with a Gaussian Mixture Model instead of a ...
24 views

### 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 ...
77 views

### 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 ...
100 views

### In variational autoencoders, why do people use MSE for the loss?

In VAEs, we try to maximize the ELBO $\mathbb(E_q log\ p(x|z) + D_{KL}(q(z|x), p(z))$), but I see that many implement the first term as MSE of the image and it's reconstruction. Is this mathematically ...
62 views

### How does implementation of 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 ...
270 views

### 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 ...
73 views

### 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 ...
288 views

### 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 ...