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 $f^\{-1\}\; [y,\; f(x)]$. 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 $[0,\; e^\{-x^2/2\}/sqrt\{2pi\}]$; after each "y" sample we choose "x" uniformly at random from $[-alpha,\; alpha]$ where $alpha\; =\; sqrt\{-2ln(ysqrt\{2pi\})\}$.

An implementation in the Macsyma language is:

slice(x):=block( [y,alpha] , y:random( exp(-x^2/2.0)/sqrt(2.0*dfloat(%pi))), alpha:sqrt(-2.0*ln(y*sqrt(2.0*dfloat(%pi)))), x:signum(random())*random(alpha));

See also

* Sampling (statistics)
* Markov chain Monte Carlo

References

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