Contributed by Matthew Zak.

- Using the property of a concave function: show that where describes the true conditional probability distribution over x using a latent variable ( equals ) and is an approximation of .
- Show that if and then the variational lower bound (given by second equation) equals a logarithm of true conditional distribution
- Knowing that the difference between variational lower bound and the data distribution is given by , prove the limit of the KL divergence is as approaches .

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I have a small doubt. Should the property of the concave function for the log function be ?

Also, are we supposed to post our answers here for discussions?

Thanks!

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Yes, there is a sum on the right term as well.

And I think that posting your answers is one of the purposes of this Q/A stuff

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This is my answer for Q1.

As , we can write .

As and is a concave function, it is possible to use the property of the concave function (Jensen’s inequality). Thus,

,

,

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