Uncertainty quantification (UQ) using Markov Chain Monte Carlo (MCMC) methods is a powerful approach in Bayesian inference. However, there are scenarios where MCMC may not be the best choice or can fail. Below is an example illustrating a case where MCMC struggles with uncertainty quantification due to multimodality and poor mixing.

Example: MCMC Struggling with a Multimodal Distribution

Consider a Bayesian inference problem where the posterior distribution is highly multimodal. If the MCMC sampler does not mix well between the modes, it can give misleading results.

Problem Setup

We define a posterior distribution as a mixture of two Gaussians:

$$ n(x)=0.5 \cdot N (-3.1)+0.5 \cdot N (3.1) $$

This distribution has two well-separated peaks, making it hard for some MCMC algorithms to explore both modes effectively.

🧠Python Implementation

We will use the Metropolis-Hastings algorithm, which can struggle in this scenario.

https://gist.github.com/viadean/b91a1f32ae352cc04322980c37968ca9

Issues with MCMC in This Case

1. Poor Mixing: If the proposal distribution is too narrow, the sampler gets stuck in one mode and rarely jumps to the other.
2. Mode Hopping Difficulty: If the modes are far apart, standard Metropolis-Hastings can take a long time to discover and switch between them.
3. Bias in Sampling: If the chain starts near one mode, it may disproportionately sample from that mode, underrepresenting the true uncertainty.

Better Approaches

To improve uncertainty quantification in such cases, one could:

• Use Hamiltonian Monte Carlo (HMC), which efficiently explores high-dimensional spaces.
• Employ Parallel Tempering to help chains jump between modes.
• Use Variational Inference (VI) as an alternative to MCMC.