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// Copyright ©2016 The Gonum Authors. All rights reserved.
// Use of this source code is governed by a BSD-style
// license that can be found in the LICENSE file.
package samplemv
import (
"math"
"golang.org/x/exp/rand"
"gonum.org/v1/gonum/mat"
"gonum.org/v1/gonum/stat/distmv"
)
var _ Sampler = MetropolisHastingser{}
// MHProposal defines a proposal distribution for Metropolis Hastings.
type MHProposal interface {
// ConditionalLogProb returns the probability of the first argument
// conditioned on being at the second argument.
// p(x|y)
// ConditionalLogProb panics if the input slices are not the same length.
ConditionalLogProb(x, y []float64) (prob float64)
// ConditionalRand generates a new random location conditioned being at the
// location y. If the first argument is nil, a new slice is allocated and
// returned. Otherwise, the random location is stored in-place into the first
// argument, and ConditionalRand will panic if the input slice lengths differ.
ConditionalRand(x, y []float64) []float64
}
// MetropolisHastingser is a type for generating 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 location
// specified by Initial. 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 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 becomes the new current location.
// If it is rejected, the current location remains. This is the sample stored in
// batch, ignoring BurnIn and Rate (discussed below).
//
// 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 the number of samples ignored
// at the beginning is specified by BurnIn. The proper BurnIn value will depend
// on the mixing time of the Markov chain defined by the target and proposal
// 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.
// This value is specified by Rate. If Rate is 0 it is defaulted to 1 (keep
// every sample).
//
// The initial value is NOT changed during calls to Sample.
type MetropolisHastingser struct {
Initial []float64
Target distmv.LogProber
Proposal MHProposal
Src rand.Source
BurnIn int
Rate int
}
// Sample generates rows(batch) samples using the Metropolis Hastings sample
// generation method. The initial location is NOT updated during the call to Sample.
//
// The number of columns in batch must equal len(m.Initial), otherwise Sample
// will panic.
func (m MetropolisHastingser) Sample(batch *mat.Dense) {
burnIn := m.BurnIn
rate := m.Rate
if rate == 0 {
rate = 1
}
r, c := batch.Dims()
if len(m.Initial) != c {
panic("metropolishastings: length mismatch")
}
// Use the optimal size for the temporary memory to allow the fewest calls
// to MetropolisHastings. The case where tmp shadows samples must be
// aligned with the logic after burn-in so that tmp does not shadow samples
// during the rate portion.
tmp := batch
if rate > r {
tmp = mat.NewDense(rate, c, nil)
}
rTmp, _ := tmp.Dims()
// Perform burn-in.
remaining := burnIn
initial := make([]float64, c)
copy(initial, m.Initial)
for remaining != 0 {
newSamp := min(rTmp, remaining)
metropolisHastings(tmp.Slice(0, newSamp, 0, c).(*mat.Dense), initial, m.Target, m.Proposal, m.Src)
copy(initial, tmp.RawRowView(newSamp-1))
remaining -= newSamp
}
if rate == 1 {
metropolisHastings(batch, initial, m.Target, m.Proposal, m.Src)
return
}
if rTmp <= r {
tmp = mat.NewDense(rate, c, nil)
}
// Take a single sample from the chain.
metropolisHastings(batch.Slice(0, 1, 0, c).(*mat.Dense), initial, m.Target, m.Proposal, m.Src)
copy(initial, batch.RawRowView(0))
// For all of the other samples, first generate Rate samples and then actually
// accept the last one.
for i := 1; i < r; i++ {
metropolisHastings(tmp, initial, m.Target, m.Proposal, m.Src)
v := tmp.RawRowView(rate - 1)
batch.SetRow(i, v)
copy(initial, v)
}
}
func metropolisHastings(batch *mat.Dense, initial []float64, target distmv.LogProber, proposal MHProposal, src rand.Source) {
f64 := rand.Float64
if src != nil {
f64 = rand.New(src).Float64
}
if len(initial) == 0 {
panic("metropolishastings: zero length initial")
}
r, _ := batch.Dims()
current := make([]float64, len(initial))
copy(current, initial)
proposed := make([]float64, len(initial))
currentLogProb := target.LogProb(initial)
for i := 0; i < r; i++ {
proposal.ConditionalRand(proposed, 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() {
copy(current, proposed)
currentLogProb = proposedLogProb
}
batch.SetRow(i, current)
}
}
// ProposalNormal is a sampling distribution for Metropolis-Hastings. It has a
// fixed covariance matrix and changes the mean based on the current sampling
// location.
type ProposalNormal struct {
normal *distmv.Normal
}
// NewProposalNormal constructs a new ProposalNormal for use as a proposal
// distribution for Metropolis-Hastings. ProposalNormal is a multivariate normal
// distribution (implemented by distmv.Normal) where the covariance matrix is fixed
// and the mean of the distribution changes.
//
// NewProposalNormal returns {nil, false} if the covariance matrix is not positive-definite.
func NewProposalNormal(sigma *mat.SymDense, src rand.Source) (*ProposalNormal, bool) {
mu := make([]float64, sigma.Symmetric())
normal, ok := distmv.NewNormal(mu, sigma, src)
if !ok {
return nil, false
}
p := &ProposalNormal{
normal: normal,
}
return p, true
}
// ConditionalLogProb returns the probability of the first argument conditioned on
// being at the second argument.
// p(x|y)
// ConditionalLogProb panics if the input slices are not the same length or
// are not equal to the dimension of the covariance matrix.
func (p *ProposalNormal) ConditionalLogProb(x, y []float64) (prob float64) {
// Either SetMean or LogProb will panic if the slice lengths are innaccurate.
p.normal.SetMean(y)
return p.normal.LogProb(x)
}
// ConditionalRand generates a new random location conditioned being at the
// location y. If the first argument is nil, a new slice is allocated and
// returned. Otherwise, the random location is stored in-place into the first
// argument, and ConditionalRand will panic if the input slice lengths differ or
// if they are not equal to the dimension of the covariance matrix.
func (p *ProposalNormal) ConditionalRand(x, y []float64) []float64 {
if x == nil {
x = make([]float64, p.normal.Dim())
}
if len(x) != len(y) {
panic(errLengthMismatch)
}
p.normal.SetMean(y)
p.normal.Rand(x)
return x
}
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