Gibbs sampler for a normal mean and variance
Section of Markov chains, Gibbs sampling, and Metropolis–Hastings, part of Bayesian statistics
Gibbs sampler for a normal mean and variance
| Type | Section |
| Taught in | Markov chains, Gibbs sampling, and Metropolis–Hastings |
| Part | Bayesian statistics |
Example 1 (Normal mean and variance) Question. For the joint normal-model posterior in Equation (4) in Conjugate analysis for normal models, describe a systematic Gibbs sampler and state how retained draws estimate the posterior mean of \mu.
Solution. For the joint posterior in Equation (4) in Conjugate analysis for normal models, alternate \begin{align*} \mu^{(t+1)}&\sim p(\mu\mid\sigma^{2(t)},{\boldsymbol{x}}) \quad\text{using (5)},\\ \sigma^{2(t+1)}&\sim p(\sigma^2\mid\mu^{(t+1)},{\boldsymbol{x}}) \quad\text{using (6)}. \end{align*} After discarding a warm-up segment, Monte Carlo estimates use the retained draws. For example,
\widehat{\mathbb{E}}(\mu\mid{\boldsymbol{x}}) =\frac1L\sum_{t=B+1}^{B+L}\mu^{(t)},
where B is the number of discarded iterations. Because the draws are dependent, L correlated draws contain less information than L independent draws. Autocorrelation, trace plots, multiple chains, and effective sample size are therefore part of a serious analysis.
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