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In Auto-Encoding Variational Bayes, authors mentioned that they proposed a solution to "Efficient approximate marginal inference of the variable $x$". I read through the paper and appendix, now scratching my head about how we can compute this after optimization of the encoder and decoder: given a particular instance $x^{(i)}$, how to compute (approximately) $p(x^{i})$?

The only thing I saw is the "marginal likelihood estimator" in the appendix D. But in authors' own words, "that produces good estimates of the marginal likelihood as long as the dimensionality of the sampled space is low."

Another way of phrasing my question, what do we really accomplish after the optimization (training VAEs with some data)?

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  • $\begingroup$ The procedure to estimate the marginal likelihood $P(x)$ is right in appendix D, so what is your question? $\endgroup$
    – shimao
    Commented Aug 24, 2018 at 4:01
  • $\begingroup$ @shimao Thanks for the comment. My latent space is relatively high (for example 48 when I want to train VAE with images). How can I efficiently estimate the $p(x^i)$ since the procedure in appendix D is suitable for low dimensionality? $\endgroup$
    – LKS
    Commented Aug 24, 2018 at 9:42
  • $\begingroup$ that may be difficult for VAEs. If you want accurate and fast computation of $p(x)$, I recommend looking into Glow, the latest in a class of models called "normalizing flows". $\endgroup$
    – shimao
    Commented Aug 24, 2018 at 20:44

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