Why another sampler is needed
Section of Markov chains, Gibbs sampling, and Metropolis–Hastings, part of Bayesian statistics
Why another sampler is needed
| Type | Section |
| Taught in | Markov chains, Gibbs sampling, and Metropolis–Hastings |
| Part | Bayesian statistics |
Gibbs sampling is wonderfully direct when every full conditional is a distribution from which we can simulate. Often it is not. The logit-normal coin model in the logit-normal Metropolis–Hastings exercise, for example, has posterior kernel
f(\theta) =\theta^{x-1}(1-\theta)^{n-x-1} \exp\!\left[-\frac12\{\operatorname{logit}(\theta)-1\}^2\right], \qquad 0<\theta<1.
This is not a Beta kernel, nor another standard density from which we can draw directly. We can nevertheless evaluate f(\theta) at any admissible value. Writing
\pi(\theta)=\frac{f(\theta)}{C}, \qquad C=\int_0^1 f(t)\,\mathrm{d} t,
makes the computational problem precise: we want draws from \pi using pointwise evaluations of f, without first calculating the intractable normalising constant C.
The Metropolis–Hastings strategy is to construct a Markov chain whose stationary distribution is \pi. After warm-up, its dependent states can be used to estimate posterior quantities. The design problem is therefore not to sample from \pi in one step, but to devise a transition rule that preserves \pi.
Used in
See also
Markov chain · Stationarity and detailed balance · Detailed balance implies stationarity · Systematic-scan Gibbs sampler · Warm-up is not a convergence proof · Metropolis–Hastings update · Metropolis–Hastings invariance · Markov chains and equilibrium