**T**he latest ABC One World webinar was a talk by Yixin Wang about the posterior collapse of auto-encoders, of which I was completely unaware. It is essentially an *identifiability* issue with auto-encoders, where the latent variable z at the source of the VAE does not impact the likelihood, assumed to be an exponential family with parameter depending on z and on θ, through possibly a neural network construct. The *variational* part comes from the parameter being estimated as θ⁰, via a variational approximation.

*“….the problem of posterior collapse mainly arises from the model and the data, rather than from inference or optimization…”*

The collapse means that the posterior for the latent satisfies p(z|θ⁰,x)=p(z), which is not a standard property since θ⁰=θ⁰(x). Which Yixin Wang, David Blei and John Cunningham show is equivalent to p(x|θ⁰,z)=p(x|θ⁰), i.e. z being unidentifiable. The above quote is then both correct and incorrect in that the choice of the inference approach, i.e. of the estimator θ⁰=θ⁰(x) has an impact on whether or not p(z|θ⁰,x)=p(z) holds. As acknowledged by the authors when describing “*methods modify the optimization objectives or algorithms of VAE to avoid parameter values θ at which the latent variable is non-identifiable*“. They later build a resolution for identifiable VAEs by imposing that the conditional p(x|θ,z) is injective in z for all values of θ. Resulting in a neural network with Brenier maps.

From a Bayesian perspective, I have difficulties to connect to the issue, the folk lore being that selecting a proper prior is a sufficient fix for avoiding non-identifiability, but more fundamentally I wonder at the relevance of inferring about the latent z’s and hence worrying about their identifiability or lack thereof.

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