Sigma-point expectation propagation

Nonlinear Bayesian inference problems often involve black-box forward models with strong curvature, for which Jacobians are inconvenient or unavailable. At the same time, priors are frequently non-Gaussian (e.g., Laplace or Studentt) to promote sparsity and robustness. Expectation propagation (EP) can in principle handle such priors, but in nonlinear settings it typically relies on first-order Taylor linearization of the forward map to obtain likelihood moments, leading to biased approximations at low signal-to-noise ratio (SNR) and coupling the algorithm to hand-crafted Jacobians. We propose sigma-point expectation propagation (SP-EP), a Jacobian-free EP framework for nonlinear, non-Gaussian inference. SP-EP embeds a sigmapoint rule, instantiated here by the unscented transform, into the EP updates: for each latent variable we build an extrinsic likelihood by propagating a Gaussian cavity distribution over the remaining variables through the nonlinear measurement, then perform one-dimensional moment matching under the true prior. This yields a plug-and-play likelihood–moment module that leaves the EP shell unchanged. We further provide a simple damping and covariance-jitter recipe that ensures robustness. Simulations on a nonlinear three-dimensional model with Laplace and Student-t priors show that SP-EP consistently improves posterior-mean RMSE over a Taylor-based EP baseline when benchmarked against a Metropolis–Hastings MMSE reference across a range of SNRs.