New convex framework makes neural likelihood training provably optimal

Bayesian inverse problems are notoriously difficult when the underlying physical model is unknown or computationally prohibitive. Traditional Markov chain Monte Carlo methods struggle in high-dimensional settings, prompting a shift toward neural likelihood approximation, where a neural network learns the likelihood function directly from data. However, existing approaches often rely on restrictive model classes and lack theoretical guarantees on convergence or optimality. The new framework from Schneider, Helin, and Taghizadeh addresses this gap by reformulating the training objective using un-normalized potentials. By folding the normalization constant into the loss function, they prove the resulting optimization problem is strictly convex, ensuring a unique global minimum. They further show that as the amount of training data grows, the empirical minimizer converges to the true likelihood—a key theoretical result absent in prior work.

Numerical experiments on a deblurring problem and a nonlinear partial differential equation (PDE)-based imaging task demonstrate the framework's effectiveness. The method achieves accurate posterior approximations with improved stability and data efficiency compared to standard neural likelihood surrogates. This work strengthens the theoretical foundation of neural likelihood methods, making them more viable for critical applications in scientific computing, medical imaging, and geophysics where uncertainty quantification is paramount. By eliminating the need for explicit parametric models, the framework opens the door to tackling previously intractable inverse problems with limited data.