The expectation maximization (EM) algorithm is a widespread method for
empirical Bayesian inference, but its expectation step (E-step) is often
intractable. Employing a stochastic approximation scheme with Markov chain
Monte Carlo (MCMC) can circumvent this issue, resulting in an algorithm known
as MCMC-SAEM. While theoretical guarantees for MCMC-SAEM have previously been
established, these results are restricted to the case where asymptotically
unbiased MCMC algorithms are used. In practice, MCMC-SAEM is often run with
asymptotically biased MCMC, for which the consequences are theoretically less
understood. In this work, we fill this gap by analyzing the asymptotics and
non-asymptotics of SAEM with biased MCMC steps, particularly the effect of
bias. We also provide numerical experiments comparing the Metropolis-adjusted
Langevin algorithm (MALA), which is asymptotically unbiased, and the unadjusted
Langevin algorithm (ULA), which is asymptotically biased, on synthetic and real
datasets. Experimental results show that ULA is more stable with respect to the
choice of Langevin stepsize and can sometimes result in faster convergence.

使用异步偏置的 MCMC 步骤分析了 SAEM 的渐近性和非渐近性，特别是偏置的影响，并在合成和真实数据集上进行了与渐近无偏算法 (MALA) 和渐近有偏算法 (ULA) 的数值实验比较。实验结果表明，ULA 在 Langevin 步长选择方面更稳定，并且有时可以实现更快的收敛。