# Slice Sampling: Simple Definition & Steps

Slice sampling (Neal, 2003) is a Markov Chain Monte Carlo (MCMC) method for sampling from a probability distribution. The basic idea is that any distribution can be sampled from by selecting uniformly spaced points under a probability distribution curve using a MCMC algorithm; multivariate distributions can be sampled by treating each variable separately.

Although Neal stated that slice sampling is “easily implemented for univariate distributions,” other authors (e.g. Royle, 2013) have noted that it can be a complicated procedure to implement. It’s probably more accurate to say that it’s easier to implement than other, similar MCMC methods like Gibbs sampling. Plus, it needs less tuning than the Metropolis–Hastings algorithm (Huzurbazar, 2004), which is more sensitive to parameters from proposed distributions. As slice sampling adaptively chooses the magnitude of changes made in prior steps, it is more efficient than basic Metropolis algorithms.

## Basic Steps

The basic method is as follows (Bandyopadhyay & Bhattacharya, 2014):

1. Generate a uniformly distributed random number x.
2. Select a random curve f(x).
3. Choose a starting point x0, where f(x0) > 0.
4. Sample a y-value, 0< y < f(x0).
5. Slice the curve horizontally at y.
6. Sample a point (x,y) from within the line segment you just created.
7. Repeat the steps.

Randomly select a slice that falls on or below the x-value from the previous iteration. This makes sure that the slices will be chosen with probabilities that are proportional to their segment lengths on the curve.

## References

Bandyopadhyay, S. & Bhattacharya, R. (2014). Discrete and Continuous Simulation: Theory and Practice. CRC Press.
Huzurbazar, A. (2004). Flowgraph Models for Multistate Time-to-Event Data. John Wiley & Sons.
Neal (2003). Slice Sampling. Annals of Statistics. Volume 31, Number 3, 705-767.
Retrieved March 11, 2018 from: https://projecteuclid.org/euclid.aos/1056562461
Royle, J. et. al (2013). Spatial Capture-Recapture. Academic Press.

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