sampleuv

func IID ¶

func IID(batch []float64, d distuv.Rander)

IID generates a set of independently and identically distributed samples from the input distribution.

func Importance ¶

func Importance(batch, weights []float64, target distuv.LogProber, proposal distuv.RandLogProber)

Importance sampling generates len(batch) samples from the proposal distribution, and stores the locations and importance sampling weights in place.

Importance sampling is a variance reduction technique where samples are generated from a proposal distribution, q(x), instead of the target distribution p(x). This allows relatively unlikely samples in p(x) to be generated more frequently.

The importance sampling weight at x is given by p(x)/q(x). To reduce variance, a good proposal distribution will bound this sampling weight. This implies the support of q(x) should be at least as broad as p(x), and q(x) should be "fatter tailed" than p(x).

If weights is nil, the weights are not stored. The length of weights must equal the length of batch, otherwise Importance will panic.

func LatinHypercube ¶

func LatinHypercube(batch []float64, q distuv.Quantiler, src *rand.Rand)

LatinHypercube generates len(batch) samples using Latin hypercube sampling from the given distribution. If src != nil, it will be used to generate random numbers, otherwise rand.Float64 will be used.

Latin hypercube sampling divides the cumulative distribution function into equally spaced bins and guarantees that one sample is generated per bin. Within each bin, the location is randomly sampled. The distuv.UnitUniform variable can be used for easy generation from the unit interval.

func MetropolisHastings ¶

func MetropolisHastings(batch []float64, initial float64, target distuv.LogProber, proposal MHProposal, src *rand.Rand)

MetropolisHastings generates len(batch) samples using the Metropolis Hastings algorithm (http://en.wikipedia.org/wiki/Metropolis%E2%80%93Hastings_algorithm), with the given target and proposal distributions, starting at the intial location and storing the results in-place into samples. If src != nil, it will be used to generate random numbers, otherwise rand.Float64 will be used.

Metropolis-Hastings is a Markov-chain Monte Carlo algorithm that generates samples according to the distribution specified by target by using the Markov chain implicitly defined by the proposal distribution. At each iteration, a proposal point is generated randomly from the current location. This proposal point is accepted with probability

p = min(1, (target(new) * proposal(current|new)) / (target(current) * proposal(new|current)))

If the new location is accepted, it is stored into batch and becomes the new current location. If it is rejected, the current location remains and is stored into samples. Thus, a location is stored into batch at every iteration.

The samples in Metropolis Hastings are correlated with one another through the Markov chain. As a result, the initial value can have a significant influence on the early samples, and so, typically, the first samples generated by the chain are ignored. This is known as "burn-in", and can be accomplished with slicing. The best choice for burn-in length will depend on the sampling and target distributions.

Many choose to have a sampling "rate" where a number of samples are ignored in between each kept sample. This helps decorrelate the samples from one another, but also reduces the number of available samples. A sampling rate can be implemented with successive calls to MetropolisHastings.

package main

import "github.com/gonum/stat/distuv"

type ProposalDist struct {
	Sigma float64
}

func (p ProposalDist) ConditionalRand(y float64) float64 {
	return distuv.Normal{Mu: y, Sigma: p.Sigma}.Rand()
}

func (p ProposalDist) ConditionalLogProb(x, y float64) float64 {
	return distuv.Normal{Mu: y, Sigma: p.Sigma}.LogProb(x)
}

func main() {
	n := 1000    // The number of samples to generate.
	burnin := 50 // Number of samples to ignore at the start.
	var initial float64
	// target is the distribution from which we would like to sample.
	target := distuv.Weibull{K: 5, Lambda: 0.5}
	// proposal is the proposal distribution. Here, we are choosing
	// a tight Gaussian distribution around the current location. In
	// typical problems, if Sigma is too small, it takes a lot of samples
	// to move around the distribution. If Sigma is too large, it can be hard
	// to find acceptable samples.
	proposal := ProposalDist{Sigma: 0.2}

	samples := make([]float64, n+burnin)
	MetropolisHastings(samples, initial, target, proposal, nil)

	// Remove the initial samples through slicing.
	samples = samples[burnin:]
}

Output:

package main

import "github.com/gonum/stat/distuv"

func max(a, b int) int {
	if a < b {
		return b
	}
	return a
}

func main() {
	// See Burnin example for a description of these quantities.
	n := 1000
	burnin := 300
	var initial float64
	target := distuv.Weibull{K: 5, Lambda: 0.5}
	proposal := ProposalDist{Sigma: 0.2}

	// Successive samples are correlated with one another through the
	// Markov Chain defined by the proposal distribution. To get less
	// correlated samples, one may use a sampling rate, in which only
	// one sample from every few is accepted from the chain. This can
	// be accomplished through a for loop.
	rate := 50

	tmp := make([]float64, max(rate, burnin))

	// First deal with burnin.
	tmp = tmp[:burnin]
	MetropolisHastings(tmp, initial, target, proposal, nil)
	// The final sample in tmp in the final point in the chain.
	// Use it as the new initial location.
	initial = tmp[len(tmp)-1]

	// Now, generate samples by using one every rate samples.
	tmp = tmp[:rate]
	samples := make([]float64, n)
	samples[0] = initial
	for i := 1; i < len(samples); i++ {
		MetropolisHastings(tmp, initial, target, proposal, nil)
		initial = tmp[len(tmp)-1]
		samples[i] = initial
	}
}

Output:

func Rejection ¶

func Rejection(batch []float64, target distuv.LogProber, proposal distuv.RandLogProber, c float64, src *rand.Rand) (nProposed int, ok bool)

Rejection generates len(batch) samples using the rejection sampling algorithm and stores them in place into samples. Sampling continues until batch is filled. Rejection returns the total number of proposed locations and a boolean indicating if the rejection sampling assumption is violated (see details below). If the returned boolean is false, all elements of samples are set to NaN. If src is not nil, it will be used to generate random numbers, otherwise rand.Float64 will be used.

Rejection sampling generates points from the target distribution by using the proposal distribution. At each step of the algorithm, the proposed point is accepted with probability

p = target(x) / (proposal(x) * c)

where target(x) is the probability of the point according to the target distribution and proposal(x) is the probability according to the proposal distribution. The constant c must be chosen such that target(x) < proposal(x) * c for all x. The expected number of proposed samples is len(samples) * c.

Target may return the true (log of) the probablity of the location, or it may return a value that is proportional to the probability (logprob + constant). This is useful for cases where the probability distribution is only known up to a normalization constant.