Posterior Sampling from the Spiked Models via Diffusion Processes

• DOI: 10.48550/arxiv.2304.11449
• Publication Date: 2023-01-01
• Authors: Andrea Montanari; Yuchen Wu
• Keywords: FOS: Mathematics; Mathematics - Statistics Theory; Statistics Theory (math.ST)

Sampling from the posterior is a key technical problem in Bayesian statistics. Rigorous guarantees are difficult to obtain for Markov Chain Monte Carlo algorithms of common use. In this paper, we study an alternative class based on diffusion processes. The constructed such way that, at its final time, it approximates target distribution. stochastic differential equation that defines process discretized (using Euler scheme) provide efficient sampling algorithm. Our construction notion observation and related idea localization. Namely, describes sample conditioned increasing information. An overlapping family processes was derived machine learning literature via time-reversal. We apply method high-dimensional symmetric spiked model. observe rank-one matrix ${\boldsymbol \theta}{\boldsymbol \theta}^{\sf T}$ corrupted by Gaussian noise, want \theta}$ posterior. algorithm makes use oracle computes expectation given data additional process. implementation using approximate message passing. thus develop first with approximation guarantees.