Slice sampling: Difference between revisions

Revision as of 07:57, 27 November 2015

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==Motivation==

To visualize this motivation, imagine printing out a simple bell curve and throwing darts at it. Assume that the darts are uniformly distributed around the board. Now take off all of the darts that are outside the curve (i.e. perform [[rejection sampling]]). The x-positions of the remaining darts will be distributed according to the bell curve. This is because there is the most room for the darts to land where curve is highest and thus the probability density is greatest.

Suppose you want to sample some random variable ''X'' with distribution f(x). Suppose that the following is the graph of f(x). The height of f(x) corresponds to the likelihood at that point.

[[File:Some probability distribution.png|350px|alt=alt text]]

If you were to uniformly sample ''X'', each value would have the same likelihood of being sampled, and your distribution would be of the form f(x)=y for some ''y'' value instead of some non-uniform function f(x). Instead of the original black line, your new distribution would look more like the blue line.

[[File:A horizontally sliced distribution.png|350px|alt=alt text]]

In order to sample ''X'' in a manner which will retain the distribution f(x), some sampling technique must be used which takes into account the varied likelihoods for each range of f(x).

==Method==

Slice sampling, in its simplest form, samples uniformly from underneath the curve f(x) without the need to reject any points, as follows:

#Choose a starting value x₀ for which f(x₀)>0.

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Note that, in contrast to many available methods for generating random numbers from non-uniform distributions, random variates generated directly by this approach will exhibit serial statistical dependence. This is because to draw the next sample, we define the slice based on the value of f(x) for the current sample.

==Motivation==

Suppose you want to sample some random variable ''X'' with distribution f(x). Suppose that the following is the graph of f(x). The height of f(x) corresponds to the likelihood at that point.

[[File:Some probability distribution.png|350px|alt=alt text]]

If you were to uniformly sample ''X'', each value would have the same likelihood of being sampled, and your distribution would be of the form f(x)=y for some ''y'' value instead of some non-uniform function f(x). Instead of the original black line, your new distribution would look more like the blue line.

[[File:A horizontally sliced distribution.png|350px|alt=alt text]]

In order to sample ''X'' in a manner which will retain the distribution f(x), some sampling technique must be used which takes into account the varied likelihoods for each range of f(x).

==Compared to Other Methods==