Bivariate exponential example
Section of Markov chains, Gibbs sampling, and Metropolis–Hastings, part of Bayesian statistics
Bivariate exponential example
| Type | Section |
| Taught in | Markov chains, Gibbs sampling, and Metropolis–Hastings |
| Part | Bayesian statistics |
Example 1 (Bivariate exponential target) The target is
\pi(x_1,x_2)\propto e^{-x_1x_2-x_1-x_2}, \qquad x_1,x_2\geq0.
Question. Derive both full conditional distributions and use them to specify a systematic-scan Gibbs sampler.
Solution. To obtain X_1\mid X_2=x_2, hold x_2 fixed: \begin{align*} \pi(x_1\mid x_2) &\propto\pi(x_1,x_2)\\ &\propto e^{-x_1x_2-x_1-x_2}\\ &\propto e^{-x_1(x_2+1)}, \qquad x_1\geq0. \end{align*} The omitted factor e^{-x_2} is constant in x_1. Normalising gives
\pi(x_1\mid x_2)=(x_2+1)e^{-(x_2+1)x_1},
so
X_1\mid X_2=x_2\sim\operatorname{Exp}(x_2+1).
By symmetry,
X_2\mid X_1=x_1\sim\operatorname{Exp}(x_1+1).
Starting from any nonnegative pair (x_1^{(0)},x_2^{(0)}), the sampler is \begin{align*} x_1^{(t+1)}&\sim\operatorname{Exp}(x_2^{(t)}+1),\\ x_2^{(t+1)}&\sim\operatorname{Exp}(x_1^{(t+1)}+1). \end{align*}
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