Bayesian neural networks (BNNs) have been widely applied to solve inverse problems (e.g., deconvolution), offering flexibility between prediction fidelity and uncertainty quantification. In this work, we focus on BNNs with heavy-tailed distributed weights, which aim to preserve the sharpness of the underlying unknown state.

Numerically, we consider a one-dimensional deconvolution problem to illustrate the effectiveness of heavy-tailed BNNs. Gaussian and Laplace distributed priors are also studied to show the limitations of smooth and sparse promoting priors. Across all considered priors, we evaluated three posterior approximation methods: deep ensembles, Markov Chain Monte Carlo (MCMC) sampling and, our primary interest, the mean-field variational approximation with Cauchy-distributed factors.

The results suggest that the Cauchy mean-field approximation introduces a sharp-promoting bias, making it a suitable modeling choice for signals with discontinuous structure. By explicitly imposing a heavy-tailed structure on the posterior distribution, the recovered signal exhibits sharp transitions that closely resemble the edges of the underlying hidden state.

Furthermore, we find that the Cauchy mean-field approximation achieves competitive edge-preserving reconstructions even with a significantly reduced network capacity, provided the network is large enough to sample a sufficient number of extreme weights. This makes it a computationally appealing alternative to full Bayesian inference methods such as MCMC.