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'''Slice sampling''' is a type of [[Markov chain Monte Carlo]] [[algorithm]] for [[pseudo-random number sampling]], i.e. for drawing random samples from a statistical distribution. The method is based on the observation that to sample a [[random variable]] one can sample uniformly from the region under the graph of its density function.<ref>Damlen, P., Wakefield, J., & Walker, S. (1999). Gibbs sampling for Bayesian non‐conjugate and hierarchical models by using auxiliary variables. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61(2), 331-344.Chicago</ref><ref name="radford03">{{cite journal
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|first=Radford M. |last=Neal
|title=Slice Sampling
|journal=[[Annals of Statistics]]
  |volume=31 |issue=3 |pages=705–767 |year=2003
|doi=10.1214/aos/1056562461
|mr=1994729 | zbl = 1051.65007
}}</ref><ref name="bishop06">{{cite book
  | last = Bishop
  | first = Christopher
  | title = Pattern Recognition and Machine Learning
  | publisher = [[Springer Science+Business Media|Springer]]
  | year = 2006
  | chapter = 11.4: Slice sampling
  | isbn = 0387310738
}}</ref>


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.
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Slice sampling, in its simplest form, samples uniformly from underneath the curve f(x) without the need to reject any points, as follows:
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#Choose a starting value x<sub>0</sub> for which f(x<sub>0</sub>)>0.
#Sample a y value uniformly between 0 and f(x<sub>0</sub>).
#Draw a horizontal line across the curve at this y position.
#Sample a point (x,y) from the line segments within the curve.
#Repeat from step 2 using the new x value.


The motivation here is that one way to sample a point uniformly from within an arbitrary curve is first to draw thin uniform-height horizontal slices across the whole curve. Then, we can sample a point within the curve by randomly selecting a slice that falls at or below the curve at the x-position from the previous iteration, then randomly picking an x-position somewhere along the slice. By using the x-position from the previous iteration of the algorithm, in the long run we select slices with probabilities proportional to the lengths of their segments within the curve.
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Generally, the trickiest part of this algorithm is finding the bounds of the horizontal slice, which involves inverting the function describing the distribution being sampled from.  This is especially problematic for multi-modal distributions, where the slice may consist of multiple discontiguous parts.  It is often possible to use a form of rejection sampling to overcome this, where we sample from a larger slice that is known to include the desired slice in question, and then discard points outside of the desired slice.
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Note also that this algorithm can be used to sample from the area under ''any'' curve, regardless of whether the function integrates to 1. In fact, scaling a function by a constant has no effect on the sampled x-positions. This means that the algorithm can be used to sample from a distribution whose [[probability density function]] is only known up to a constant (i.e. whose [[normalizing constant]] is unknown), which is common in [[computational statistics]].
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==Implementation==
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Slice sampling gets its name from the first step: defining a ''slice'' by sampling from an auxiliary variable <math>Y</math>. This variable is sampled from <math>[0, f(x)]</math>, where <math>f(x)</math> is either the probability density function (pdf) of ''X'' or is at least proportional to its pdf. This defines a slice of ''X'' where <math>f(x) > Y</math>. In other words, we are now looking at a region of ''X'' where the probability density is at least <math>Y</math>. Then the next value of ''X'' is sampled uniformly from this slice. A new value of <math>Y</math> is sampled, then ''X'', and so on. This can be visualized as alternatively sampling the y-position and then the x-position of points under pdf, thus the ''X''s are from the desired distribution. The <math>Y</math> values have no particular consequences or interpretations outside of their usefulness for the procedure.
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If both the pdf and its inverse are available, and the distribution is unimodal, then finding the slice and sampling from it are simple. If not, a stepping-out procedure can be used to find a region whose endpoints fall outside the slice. Then, a sample can be drawn from the slice using [[rejection sampling]]. Various procedures for this are described in detail by Neal.<ref name="radford03"/>
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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.
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==Motivation==
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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|]]
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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.
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[[File:A horizontally sliced distribution.png|350px|alt=alt text|]]
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Slice sampling is a Markov chain method and as such serves the same purpose as Gibbs sampling and Metropolis. Unlike Metropolis, there is no need to manually tune the candidate function or candidate standard deviation.
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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. In other words, not all points have the same independent likelihood of selection.
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[[File:A horizontally and vertically sliced distribution.png|thumb|350px|alt=alt text|For a given sample x, a value for y is chosen from [0, f(x)], which defines a "slice" of the distribution (shown by the solid horizontal line). In this case, there are two slices separated by an area outside the range of the distribution.]]
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)];
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*given ''y'' we choose ''x'' uniformly at random from the set <math>f^{-1}[y, +\infty)</math>.
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Our auxiliary variable ''Y'' represents a horizontal "slice" of the distribution. The rest of each iteration is dedicated to sampling an ''x'' value from the slice which is representative of the density of the region being considered.
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*First, a width parameter ''w'' is used to define the area containing the given ''x'' value. Each endpoint of this area is tested to see if it lies outside the given slice. If not, the region is extended in the appropriate direction(s) by ''w'' until the end both endpoints lie outside the slice.
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[[Image:summary of slice sampling.png|thumb|center|500px|alt=alt text|Finding a sample given a set of slices (the slices are represented here as blue lines and correspond to the solid line slices in the previous graph of f(x) ). a) A width parameter ''w'' is set. b) A region of width ''w'' is identified around a given point <math>x_0</math>. c) The region is expanded by ''w'' until both endpoints are outside of the considered slice. d) <math>x_1</math> is selected uniformly from the region. e) Since <math>x_1</math> lies outside the considered slice, the region's left bound is adjusted to <math>x_1</math>. f) Another uniform sample <math>x</math> is taken and accepted as the sample since it lies within the considered slice.]]
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===Treating each variable independently===
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Single variable slice sampling can be used in the multivariate case by sampling each variable in turn repeatedly, as in Gibbs sampling. To do so requires that we can compute, for each component <math>x_i</math> a function that is proportional to <math>p(x_i|x_0...x_n)</math>.
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To prevent random walk behavior, overrelaxation methods can be used to update each variable in turn. Overrelaxation chooses a new value on the opposite side of the mode from the current value, as opposed to choosing a new independent value from the distribution as done in Gibbs.
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This method adapts the univariate algorithm to the multivariate case by substituting a hyperrectangle for the one-dimensional ''w'' region used in the original. The hyperrectangle ''H'' is initialized to a random position over the slice. ''H'' is then shrunken as points from it are rejected.
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Reflective slice sampling is a technique to suppress random walk behavior in which the successive candidate samples of distribution f(x) are kept within the bounds of the slice by "reflecting" the direction of sampling inward toward the slice once the boundary has been hit.


In this graphical representation of reflective sampling, the shape indicates the bounds of a sampling slice. The dots indicate start and stopping points of a sampling walk. When the samples hit the bounds of the slice, the direction of sampling is "reflected" back into the slice.
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[[File:An example of reflection sampling.png|350px|alt=alt text]]
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Let us consider a single variable example. Suppose our true distribution <math>g(x)\sim N(0,5)</math>. So:
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<math>f(x) = \frac{1}{\sqrt{2\pi*5^2}} * e^{ -\frac{(x-0)^2}{2*5^2} }</math>
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*We first draw a uniform random value ''y'' from the range of f(x) in order to define our slice(es). Suppose y=0.01.
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*Next, we set our width parameter ''w'' which we will use to expand our region of consideration. Suppose w=2.
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*Now, each endpoint of this area is tested to see if it lies outside the given slice. Our right bound lies outside our slice, but the left value does not. We expand the left bound by adding ''w'' to it until it extends past the limit of the slice. After this process, the new bounds of our region of interest are (-4,3).
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*Next, we take a uniform sample within (-4,3). Suppose this sample yields x=-3.9. Though this sample is within our region of interest, it does not lie within our slice, so we modify the left bound of our region of interest to this point. Now we take a uniform sample from (-3.9,3). This time our sample yields x=1, which is within our slice, and thus is our accepted sample. Had our new ''x'' not been within our slice, we would continue the shrinking/resampling process until a valid ''x'' within bounds is found.
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==Another Example==
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To sample from the [[normal distribution]] <math>N(0,1)</math> we first choose an initial ''x'' -- say 0. After each sample of ''x'' we choose ''y'' uniformly at random from <math>(0, e^{-x^2/2}/\sqrt{2\pi}]</math>, which is bounded the pdf of <math>N(0,1)</math>. After each ''y'' sample we choose ''x'' uniformly at random from <math>[-\alpha, \alpha]</math> where <math>\alpha = \sqrt{-2\ln(y\sqrt{2\pi})}</math>. This is the slice where <math>f(x) > y</math>.
</ul>
 
An implementation in the [[Macsyma]] language is:
 
<source lang="smalltalk">
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)
);
</source>
 
==See also==
 
* [[Markov chain Monte Carlo]]
 
==References==
 
<references/>
 
==External links==
* http://www.probability.ca/jeff/java/slice.html
 
[[Category:Markov chain Monte Carlo]]
[[Category:Non-uniform random numbers]]

Latest revision as of 01:23, 19 October 2014

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