// 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.
func Rejection(batch []float64, target distuv.LogProber, proposal distuv.RandLogProber, c float64, src *rand.Rand) (nProposed int, ok bool) {
if c < 1 {
panic("rejection: acceptance constant must be greater than 1")
}
f64 := rand.Float64
if src != nil {
f64 = src.Float64
}
var idx int
for {
nProposed++
v := proposal.Rand()
qx := proposal.LogProb(v)
px := target.LogProb(v)
accept := math.Exp(px-qx) / c
if accept > 1 {
// Invalidate the whole result and return a failure.
for i := range batch {
batch[i] = math.NaN()
}
return nProposed, false
}
if accept > f64() {
batch[idx] = v
idx++
if idx == len(batch) {
break
}
}
}
return nProposed, true
}
// 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 Importance(batch, weights []float64, target distuv.LogProber, proposal distuv.RandLogProber) {
if len(batch) != len(weights) {
panic(badLengthMismatch)
}
for i := range batch {
v := proposal.Rand()
batch[i] = v
weights[i] = math.Exp(target.LogProb(v) - proposal.LogProb(v))
}
}
// 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.
func MetropolisHastings(batch []float64, initial float64, target distuv.LogProber, proposal MHProposal, src *rand.Rand) {
f64 := rand.Float64
if src != nil {
f64 = src.Float64
}
current := initial
currentLogProb := target.LogProb(initial)
for i := range batch {
proposed := proposal.ConditionalRand(current)
proposedLogProb := target.LogProb(proposed)
probTo := proposal.ConditionalLogProb(proposed, current)
probBack := proposal.ConditionalLogProb(current, proposed)
accept := math.Exp(proposedLogProb + probBack - probTo - currentLogProb)
if accept > f64() {
current = proposed
currentLogProb = proposedLogProb
}
batch[i] = current
}
}