Given the current state $x=(x_1,\ldots,x_d)$, a <Random-scan Gibbs sampler> chooses $I$ uniformly from $\{1,\ldots,d\}$, leaves $x_{-I}$ unchanged, and samples the new coordinate from the complete conditional distribution
$$
X_I'\sim\pi(dx_I\mid x_{-I}).
$$
Its transition kernel is
$$
K(x,dy)=\frac1d\sum_{i=1}^d
\pi(dy_i\mid x_{-i})\,\delta_{x_{-i}}(dy_{-i}),
$$
and it leaves $\pi$ invariant.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 216 / 2 / a / Solution

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A random-scan Gibbs sampler chooses one coordinate, commonly uniformly at random, and redraws it from its complete conditional distribution while leaving all other coordinates fixed. Each coordinate update preserves the target distribution.


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