Slice sampling

In mathematics and physics, Slice sampling is a type of Markov chain Monte Carlo sampling algorithm based on the observation that to sample a random variable one can sample uniformly from the region under the graph of its density function.

Implementation

To sample a random variable X with density $f(x)$ we introduce an auxiliary variable $Y$ and iterate as follows: Given a sample x we choose y uniformly at random from the interval $[0,f(x)]$; given y we choose x uniformly at random from the set $f^{-1}[0,y]$. The sample of x is obtained by ignoring the y values.

Example

To sample from the normal distribution $N(0,1)$ we first choose an initial x -- say 0. After each sample of x we choose y uniformly at random from $[0,e^{-x^{2}/2}/{\sqrt {2\pi }}]$; after each y sample we choose x uniformly at random from $[-\alpha ,\alpha ]$ where $\alpha ={\sqrt {-2\log(y{\sqrt {2\pi }})}}$.

See also

• Sampling (statistics)
• Markov chain Monte Carlo

References

• Radford M. Neal, "Slice Sampling". The Annals of Statistics, 31(3):705-767, 2003.