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 Incoming links: Full conditional distribution

Tempered Gibbs sampler Created 2026-10-03 Updated 2026-11-05

Tempered Gibbs sampling combines modified full conditional distributions with state-dependent coordinate selection and importance sampling. For a positive target density

f (x)

, choose strictly positive normalized densities

g_{i} (x_{i} ∣ x_{- i})

with compatible support, and define

p_{i} (x) = \frac{g _{i} ( x _{i} ∣ x _{- i} )}{f ( x _{i} ∣ x _{- i} )}, Z (x) = \frac{1}{d} \sum_{i} p_{i} (x) .

Select coordinate

i

proportionally to

p_{i} (x)

and redraw it from

g_{i}

. The invariant density is

f (x) Z (x)

, not generally

f (x)

. For a move changing only coordinate

i

, stationary transition weights are proportional to

f (x_{- i}) g_{i} (x_{i} ∣ x_{- i}) g_{i} (y_{i} ∣ x_{- i})

, symmetric in old and new values. Integrating also shows

\int f Z = 1

1/ Z (x)

recover expectations under

f

. Flattening a conditional can encourage longer moves, but omitting the weights changes the estimand. The

g_{i}

need not be full conditionals of a common second joint distribution.^[ref]