Chapter 18: Langevin Dynamics and Sampling. In the chapter mind map this icon labels Metropolis-Adjusted Langevin Algo (MALA). The discussion below is excerpted and lightly edited from § Metropolis-Adjusted Langevin Algorithm (MALA) in Mathematics for AI and Machine Learning.

To illustrate how MCMC works, consider using the Metropolis-Hastings algorithm to sample from a 1D bimodal distribution.

Use a Gaussian proposal distribution centered at the current sample:

$q(x' \mid x) = \mathcal{N}(x'; x, \sigma^2)$$ with $\sigma = 0.5$.

1. Initialize: Start with an initial sample $x_0 = 0$ 2. Iterate: For each step $t$: a. Generate proposal: Sample $x' \sim q(x' \mid x_t)$ b. Compute acceptance ratio: c. Accept or reject: Sample $u \sim \text{Uniform}(0, 1)$ - If $u \leq r$, accept the proposal: $x_{t+1} = x'$ - Otherwise, reject the proposal: $x_{t+1} = x_t$

• Burn-in period: Discard the first 1,000 samples to allow the chain to reach stationarity
• Sample distribution: The remaining 9,000 samples form a distribution that closely matches the target bimodal distribution
• Exploration: The chain successfully explores both modes of the distribution

While Metropolis-Hastings works well for this example, Langevin dynamics would typically converge faster by using gradient information to move toward high-probability regions more efficiently.