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A Gibbs sampler updates one coordinate by drawing from its full conditional distribution, leaving the others unchanged and using their current values. Each coordinate kernel $K_i$ preserves the target joint law $P$: integrate first over the unchanged coordinates, then over the old coordinate, and finally draw the new coordinate from the same conditional distribution. The resulting joint law is again $P$. In the discrete notation, if two states share the remaining coordinates,which is symmetric in $x_i,x_i^{\prime }$. Thus a coordinate update satisfies detailed balance. A systematic full sweep has kernel $K_1\cdots K_k$ and remains invariant because each factor preserves $P$. A fixed-probability random-scan mixture also preserves $P$; the mixture is reversible, whereas a systematic composition need not be.

In the autologistic binary-image model, count each neighboring pair once. Changing one pixel from zero to one raises $n_1$ by one and $n_2$ by the number $z_{ij}$ of neighboring ones. The ratio of its two conditional weights is $\exp (\alpha +\beta z_{ij})$. ThereforeFor single-site Gibbs sampling, select a pixel, compute its current neighbor sum and replace it by a draw from a Bernoulli distribution with this probability. One may use random scans or complete ordered sweeps. The normalizing constant is not needed. For finite $\alpha ,\beta$, every conditional probability lies strictly between zero and one; the finite chain is irreducible and has self-transitions, so its stationary law is unique and iterating converges to it.

For checkerboard Gibbs sampling, no edge joins two black pixels or two white pixels. Conditional on the whites, the black-pixel law factorizes:Draw all black pixels independently with these probabilities, then draw all white pixels independently conditional on the newly drawn blacks. Each color block is a full conditional update, so the two-block sweep preserves the joint target. Pixels of the same color can be updated together because their conditional distribution factorizes.

For the noisy image, the log likelihood contributes $-\frac{1}{2}{\sum }_v(y_v-x_v{)}^2$. Since $x_v^2=x_v$, dropping terms independent of $x$ givesThe Gaussian-noise posterior for an autologistic image therefore has conditional probabilitiesUse this probability in both single-pixel and sequential color-block schemes. The independent observation factors change only the local field, so conditional independence within each color is retained. The same invariance proof then targets the posterior distribution rather than the prior.