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Probability density function estimation with weighted samples is the mainfoundation of all adaptive importance sampling algorithms. Classically, atarget distribution is approximated either by a non-parametric model or withina parametric family. However, these models suffer from the curse ofdimensionality or from their lack of flexibility. In this contribution, wesuggest to use as the approximating model a distribution parameterised by avariational autoencoder. We extend the existing framework to the case ofweighted samples by introducing a new objective function. The flexibility ofthe obtained family of distributions makes it as expressive as a non-parametricmodel, and despite the very high number of parameters to estimate, this familyis much more efficient in high dimension than the classical Gaussian orGaussian mixture families. Moreover, in order to add flexibility to the modeland to be able to learn multimodal distributions, we consider a learnable priordistribution for the variational autoencoder latent variables. We alsointroduce a new pre-training procedure for the variational autoencoder to findgood starting weights of the neural networks to prevent as much as possible theposterior collapse phenomenon to happen. At last, we explicit how the resultingdistribution can be combined with importance sampling, and we exploit theproposed procedure in existing adaptive importance sampling algorithms to drawpoints from a target distribution and to estimate a rare event probability inhigh dimension on two multimodal problems.

Variational autoencoder with weighted samples for high-dimensional non-parametric adaptive importance sampling