Further extensions to Gibbs and Metropolis#

Part 3 · Stage 9 · ⛓️ MCMC: Gibbs, Metropolis & HMC · Lesson 077 of 144 · intermediate

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Important

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Beyond the two basics#

Gibbs needs conditionals; Metropolis needs a tuned proposal. A family of extensions removes one requirement or the other, usually by the same device: enlarge the space with auxiliary variables, so that a hard move in \(\theta\) becomes an easy move in \((\theta, u)\).

Slice sampling#

Introduce a height \(u\) beneath the unnormalised density and sample uniformly from the area under the curve. Given \(\theta\), draw \(u \sim \mathrm{Uniform}(0, q(\theta))\); given \(u\), draw \(\theta\) uniformly from the slice \(\{\theta : q(\theta) \ge u\}\). Marginally, \(\theta\) has the target distribution.

\[p(\theta, u) \propto \mathbf{1}\{0 < u < q(\theta)\} \quad \Longrightarrow \quad p(\theta) \propto q(\theta).\]

The appeal is that it is self-tuning: no step size, no acceptance rate. The slice is found by stepping out from the current point and shrinking on rejection. Slice sampling is the workhorse for one-dimensional conditionals inside a larger Gibbs sweep, and it handles awkward univariate shapes that would defeat a fixed proposal.

Reversible jump#

When the number of parameters is itself unknown — how many mixture components, which predictors belong in the regression — the posterior lives on a union of spaces of different dimension. Reversible jump MCMC moves between them, proposing births and deaths of parameters and correcting the acceptance ratio with a Jacobian for the dimension change. It is powerful, notoriously fiddly to tune, and largely superseded in applied work by two alternatives: continuous model expansion (Stage 6) and the nonparametric priors of Stage 16, which let complexity grow without discrete jumps.

Other auxiliary-variable tricks#

Data augmentation — latent variables that restore conjugacy: a \(t\) as a scale-mixture of normals, probit regression given latent utilities.
Simulated tempering / parallel tempering — run chains at several “temperatures”, flattening the posterior so a chain can cross low-probability valleys between modes, and swap states between them.
Adaptive MCMC — learn the proposal covariance online, with the adaptation vanishing over time so that ergodicity is preserved.

# PyMC assigns samplers per variable: NUTS for continuous, specialised steppers otherwise
import pymc as pm
with model:
    idata = pm.sample(step=[pm.NUTS([mu, tau]), pm.BinaryGibbsMetropolis([z])])

Where they matter now#

Gradient-based sampling handles continuous parameters better than any of these. Their enduring role is the complement: discrete unknowns (which HMC cannot touch, since it needs derivatives), multimodal posteriors (where tempering still helps), and univariate conditionals inside composite samplers. Modern practice marginalises discrete parameters analytically where possible, and reaches for these tools where it is not.

Hint