Likelihood-Free Variational Autoencoders

Variational Autoencoders (VAEs) typically rely on a probabilistic decoder with a predefined likelihood, most commonly an isotropic Gaussian, to model the data conditional on latent variables. While convenient for optimization, this choice often leads to likelihood misspecification, resulting in blurry reconstructions and poor data fidelity, especially for high-dimensional data such as images. In this work, we propose \textit{EnVAE}, a novel likelihood-free generative framework that has a deterministic decoder and employs the energy score -- a proper scoring rule -- to build the reconstruction loss. This enables likelihood-free inference without requiring explicit parametric density functions. To address the computational inefficiency of the energy score, we introduce a fast variant, \textit{FEnVAE}, based on the local smoothness of the decoder and the sharpness of the posterior distribution of latent variables. This yields an efficient single-sample training objective that integrates seamlessly into existing VAE pipelines with minimal overhead. Empirical results on standard benchmarks demonstrate that \textit{EnVAE} achieves superior reconstruction and generation quality compared to likelihood-based baselines. Our framework offers a general, scalable, and statistically principled alternative for flexible and nonparametric distribution learning in generative modeling.