Griddy-Gibbs Sampler

Definition

The griddy-Gibbs sampler is a variant of the Gibbs sampler for cases where the full conditional posterior densities are known only up to a constant and cannot be sampled directly. Instead of drawing from a closed-form conditional, it evaluates each univariate full conditional on a grid of points, numerically integrates it to obtain the (approximate) cumulative distribution function, and inverts that CDF at a uniform draw. Introduced by Ritter and Tanner (1992) and applied to GARCH models by Bauwens and Lubrano (1998).

Key Ideas

Only pointwise evaluation is needed. The method requires evaluating the unnormalized conditional density $k(\theta_i \mid \theta_{-i}, y)$ at grid points — which is available for many models (GARCH, Student-t, nonlinear regressions) where the conditional is not a standard distribution.
Numerical inversion. On a grid $\theta_i^{(1)}<\dots<\theta_i^{(G)}$ , approximate the CDF by cumulative numerical integration (e.g. the trapezoidal rule), normalize, draw $u\sim\text{Uniform}(0,1)$ , and invert by interpolation to get $\theta_i$ .
Automatic, no tuning. Unlike importance sampling or Metropolis–Hastings, there is no proposal/importance density to design and no acceptance rate to tune, so the sampler is close to "automatic" — its main inputs are the grid range and resolution.
Cost scales with dimension. Each coordinate requires a fresh grid evaluation every sweep, so the per-iteration cost grows with the number of parameters and the grid size; the method is best for low-to-moderate dimensional blocks.

How It Works

Cycle over coordinates $i=1,\dots,p$ . For coordinate $i$ given the current values of the others:

Choose a grid over a range covering the effective support of $\theta_i \mid \theta_{-i}, y$ .
Evaluate the unnormalized conditional $k(\cdot)$ at each grid point.
Numerically integrate to form the approximate CDF $\Phi_i$ and normalize.
Draw $u\sim\text{Uniform}(0,1)$ and set $\theta_i = \Phi_i^{-1}(u)$ by interpolation.

Repeat to obtain a Markov chain whose stationary distribution is the joint posterior. Grid ranges are typically adapted from pilot runs or set generously and refined.

Why It Matters

It made Bayesian GARCH estimation feasible without designing a good global approximation to the posterior, and yields smooth marginal density estimates.
It is a general-purpose fallback whenever a full conditional is one-dimensional and cheap to evaluate but not a recognizable distribution.

Open Questions