MCMC vs Variational Inference: Two Paths to Approximate Bayesian Posteriors

In complex probabilistic models, computing the exact posterior distribution over hidden variables is often analytically intractable due to high-dimensional latent spaces and intricate variable interactions. Approximate inference offers two principal solutions: Markov Chain Monte Carlo (MCMC) and Variational Inference (VI). MCMC constructs a Markov chain whose stationary distribution matches the target posterior, then uses collected samples to estimate posterior quantities empirically. Variational Inference instead reframes the problem as optimization, searching within a tractable family of distributions for the one closest to the true posterior. While MCMC is flexible, it can be computationally costly and slow to converge, whereas VI trades some accuracy for greater scalability through optimization.