Q18. Equilibrium and independence

Topic: Markov chains, Gibbs sampling, and Metropolis–Hastings · more questions on this topic
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[4 marks]

A Gibbs sampler targets a bivariate posterior in which the two parameters are strongly correlated. Its initial state is drawn exactly from the target posterior. A student states:

“The chain starts at equilibrium, so its successive sampled pairs can be treated as independent posterior observations.”

Explain what starting at equilibrium guarantees. Does independence follow? Justify your answer by referring to how Gibbs updates are generated.

Worked answer (click to show)

Definitions. An invariant or equilibrium distribution is preserved by the chain’s transition. Starting from it makes the chain stationary: every iteration has the same marginal target distribution. This is weaker than independence across iterations. Here one Gibbs sweep updates each coordinate once from its exact full conditional, using the newest available conditioning value.

To see invariance directly, start with a pair having density p(\theta_1,\theta_2\mid{\boldsymbol{x}}). Updating \theta_1 while leaving \theta_2 unchanged gives density p(\theta_2\mid{\boldsymbol{x}})\, p(\theta_1'\mid\theta_2,{\boldsymbol{x}}) =p(\theta_1',\theta_2\mid{\boldsymbol{x}}). The second conditional update reconstructs the same joint density in the same way. Therefore the full sweep preserves the target. Starting at equilibrium guarantees stationarity. If (\theta_1^{(0)},\theta_2^{(0)}) \sim p(\theta_1,\theta_2\mid{\boldsymbol{x}}), then \boxed{ (\theta_1^{(t)},\theta_2^{(t)}) \sim p(\theta_1,\theta_2\mid{\boldsymbol{x}}) \quad\text{for every }t\geq0. } Thus no convergence from an incorrectly distributed initial state is needed.

Independence does not follow. A systematic Gibbs update has the form \theta_1^{(t+1)} \sim p(\theta_1\mid\theta_2^{(t)},{\boldsymbol{x}}), followed by \theta_2^{(t+1)} \sim p(\theta_2\mid\theta_1^{(t+1)},{\boldsymbol{x}}). The new pair is generated using components of the current pair, so successive states are generally serially dependent. This dependence can be particularly strong when the target parameters are strongly correlated. Starting at equilibrium gives the correct marginal distribution at every iteration, but it does not turn the Markov chain into independent posterior draws.