###### tags: `one-offs` `markov chains` `expository` `monte carlo` `sampling`

# Nested Structure in MCMC Algorithms

**Overview**: In this note, I describe some aspects of hierarchical structure in MCMC algorithms, and how they can be of theoretical and practical relevance.

## The Spectral Telescope and Convergence Theory

I recently enjoyed reading a paper which presented the idea of the 'spectral telescope' for the $\mathrm{L}^{2}$ convergence analysis of Markov chains. The construction applies to Markov chains $P$ which can be embedded into a sequence $\left\{ P_{k}\right\} _{0\leqslant k\leqslant K}$ of Markov chains (not necessarily practically implementable!) which have certain properties. In particular, assume that:

1. All of the $\left\{ P_{k}\right\}$ admit the same invariant measure.

2. $P_{0}=P$, and $P_{K}$ corresponds to independent sampling from the invariant measure.

3. For $0<k\leqslant K$, $P_{k-1}$ can be viewed as an approximation to $P_{k}$.

In practice, some more refined and specific conditions are needed, but this is close to the essence. The idea is then roughly that if for each $k$, one iteration of $P_{k-1}$ is 'as good as' $\varepsilon_{k}$ iterations of $P_{k}$, then it follows that $P = P_{0}$ is 'as good as' $\varepsilon_{1:K} := \varepsilon_{1} \cdot \cdots \cdot \varepsilon_{K}$ iterations of $P_{K}$ (this is the 'telescope' alluded to in the title), and so $P$ should decorrelate in roughly $\varepsilon_{1:K}^{-1}$ steps. These statements can be made rigorous without too much pain, and this elegance is one of the great triumphs of the paper.

In terms of applications, the paper shows that this precise structure is exhibited by the random-scan Gibbs sampler, for which the interpolating kernels correspond to random-scan *block* Gibbs samplers of varying size. That is, using blocks of size $(k - 1)$ is a randomised approximation to using blocks of size $k$, and so on.

One major curiosity of mine going forward is whether a similar nested / recursive structure is readily available in other MCMC algorithms of interest. Some cursory guesses:

1. The Multigrid Monte Carlo algorithm of Goodman-Sokal (discussed [here](https://hackmd.io/@sp-monte-carlo/SyTOBh9gt)) has an explicit recursive structure, whilst still making global moves.

2. The Multilevel Delayed Acceptance algorithms pioneered by Christen-Fox, refined by Teckentrup et al. and subsequently by Dodwell et al. also have a recursive structure, which behaves slightly differently to the MGMC algorithm (more Monte Carlo-themed than Multigrid-themed, let's say).

I'm sure that there will be other examples as well, and I look forward to seeing them understood better and better.