Analytical form of the posterior distribution of 3D pose given 2D pose

Derive the analytical form of the posterior distribution of 3D joint positions given 2D joint observations in monocular 3D pose estimation, p(X^3D | X^2D), to enable direct computation of the Bayesian posterior expectation (the MMSE estimator) used for aggregation of multiple 3D pose hypotheses.

Background

The Reprojection-based Posterior Expectation Aggregation (RPEA) module aims to approximate the minimum mean squared error (MMSE) estimator by computing the posterior expectation of the 3D pose conditioned on observed 2D keypoints. This estimator requires integrating 3D poses against the posterior distribution p(X^3D | X^2D).

Because the analytical form of this posterior is not available, the paper substitutes a pseudo-likelihood based on the 2D reprojection loss to weight sampled hypotheses. Determining the analytical posterior would allow exact evaluation of the posterior expectation and remove reliance on heuristic likelihood proxies.