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We present the marginal unbiased score expansion (MUSE) method, an algorithmfor generic high-dimensional hierarchical Bayesian inference. MUSE performsapproximate marginalization over arbitrary non-Gaussian latent parameterspaces, yielding Gaussianized asymptotically unbiased and near-optimalconstraints on global parameters of interest. It is computationally muchcheaper than exact alternatives like Hamiltonian Monte Carlo (HMC), excellingon funnel problems which challenge HMC, and does not require anyproblem-specific user supervision like other approximate methods such asVariational Inference or many Simulation-Based Inference methods. MUSE makespossible the first joint Bayesian estimation of the delensed Cosmic MicrowaveBackground (CMB) power spectrum and gravitational lensing potential powerspectrum, demonstrated here on a simulated data set as large as the upcomingSouth Pole Telescope 3G 1500 deg$^2$ survey, corresponding to a latentdimensionality of ${\sim}\,6$ million and of order 100 global bandpowerparameters. On a subset of the problem where an exact but more expensive HMCsolution is feasible, we verify that MUSE yields nearly optimal results. Wealso demonstrate that existing spectrum-based forecasting tools which ignorepixel-masking underestimate predicted error bars by only ${\sim}\,10\%$. Thismethod is a promising path forward for fast lensing and delensing analyseswhich will be necessary for future CMB experiments such as SPT-3G, SimonsObservatory, or CMB-S4, and can complement or supersede existing HMCapproaches. The success of MUSE on this challenging problem strengthens itscase as a generic procedure for a broad class of high-dimensional inferenceproblems.

MUSE: Marginal Unbiased Score Expansion and Application to CMB Lensing