Bayesian inference on mixtures

IV Workshop Bayesian Nonparametrics, Roma, 12 Giugno 2004 1

Bayesian Inference on Mixtures

Christian P. Robert

Universit e Paris Dauphine

Joint work with

JEAN-MICHEL MARIN, KERRIE MENGERSEN AND JUDITH ROUSSEAU

IV Workshop Bayesian Nonparametrics, Roma, 12 Giugno 2004 2

What’s new?!

• Density approximation & consistency

• Scarsity phenomenon

• Label switching & Bayesian inference

• Nonconvergence of the Gibbs sampler & population Monte Carlo

• Comparison of RJMCM with B& D

Intro/Inference/Algorithms/Beyond fixed k 3

1 Mixtures

Convex combination of “usual” densities (e.g., exponential family)

pif(x|θi) ,k∑

pi = 1 k > 1 ,

−1 0 1 2 30.

40 1 2 3 4 5

0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0

0 2 4 6 8 10

0 2 4 6

−2 0 2 4 6 8 10

0 5 10 15

Normal mixture densities for K = 2, 5, 25, 50

Likelihood

L(θ, p|x) =n∏

pjf (xi|θj)

c© Computable in O (nk) time

Intro:Misg/Inference/Algorithms/Beyond fixed k 6

Missing data representation

Demarginalisation

pif(x|θi) =∫

f(x|θ, z) f(z|p) dz

X|Z = z ∼ f(x|θz), Z ∼Mk(1; p1, ..., pk)

Missing “data” z1, . . . , zn that may be or may not be meaningful

[Auxiliary variables]

Nonparametric re-interpretation

Approximation of unknown distributions

E.g., Nadaraya–Watson kernel

kn(x|x) =1

ϕ (x; xi, hn)

Bernstein polynomials

Bounded continuous densities on [0, 1] approximated by Beta mixtures

(αk,βk)∈N2+

pk Be(αk, βk) αk, βk ∈ N∗

[Consistency]

Associated predictive is then

fn(x|x) =∞∑

Eπ[ωkj |x] Be(j, k + 1− j)P(K = k|x) .

[Petrone and Wasserman, 2002]

0.0 0.2 0.4 0.6 0.8 1.0

811,0.1,0.9

0.0 0.2 0.4 0.6 0.8 1.0

31,0.6,0.3

0.0 0.2 0.4 0.6 0.8 1.0

5,0.8,0.9

0.0 0.2 0.4 0.6 0.8 1.0

54,0.8,2.6

0.0 0.2 0.4 0.6 0.8 1.0

22,1.2,1.6

0.0 0.2 0.4 0.6 0.8 1.0

45,2.9,1.8

0.0 0.2 0.4 0.6 0.8 1.0

7,4.9,3.3

0.0 0.2 0.4 0.6 0.8 1.0

67,5.1,9.3

0.0 0.2 0.4 0.6 0.8 1.00

91,19.1,17.5

Realisations from the Bernstein prior

0.0 0.2 0.4 0.6 0.8 1.0

0.0 0.2 0.4 0.6 0.8 1.00.

Realisations from a more general prior

Intro:Constancy/Inference/Algorithms/Beyond fixed k 11

Density estimation

[CPR & Rousseau, 2000–04]

Reparameterisation of a Beta mixture

p0U(0, 1) + (1− p0)K∑

pkB(αkεk, αk(1− εk))∑

pk = 1 ,

with density fψ

Can approximate most distributions g on [0, 1]

Assumptions

– g is piecewise continuous on {x ; g(x) < M} for all M ’s

–∫

g(x) log g(x) d x <∞

Prior distributions

– π(K) has a light tail

P (K ≥ tn/ log n) ≤ exp−rn

– p0 ∼ Be(a0, b0), a0 < 1, b0 > 1

– pk ∝ ωk and ωk ∼ Be(1, k)

– location-scale “hole” prior

(αk, εk) ∼ {1− exp [−{β1(αk − 2)c3 + β2(εk − .5)c4}]}exp

[−τ0αc0k /2− τ1/{α2c1

k εc1k (1− εk)c1}]

Consistency results

Hellinger neighbourhood

Aε(f0) = {f, d(f, f0) ≤ ε}

Then, for all ε > 0,

π[Aε(g)|x1:n]→ 1, as n→∞, g a.s.

Eπ [d(g, fψ)|x1:n]→ 0, g a.s.

Extension to general parametric distributions by the cdf transform Fθ(x)

2 [B] Inference

Difficulties:

• identifiability

• label switching

• loss function

• ordering constraints

• prior determination

Intro/Inference:Identifability/Algorithms/Beyond fixed k 15

Central (non)identifiability issue

∑kj=1 pjf(y|θj) is invariant to relabelling of the components

Consequence

((pj , θj))1≤i≤k

only known up to a permutation τ ∈ Sk

Example 1. Two component normal mixture

p N (µ1, 1) + (1− p)N (µ2, 1)

where p 6= 0.5 is known

The parameters µ1 and µ2 are identifiable

Bimodal likelihood [ 500 observations and (µ1, µ2, p) = (0, 2.5, 0.7)]

−1 0 1 2 3 4

Influence of p on the modes

−2 0 2 4

p=0.75

−2 0 2 4

p=0.85

Intro/Inference:Com’ics/Algorithms/Beyond fixed k 19

Combinatorics

For a normal mixture,

pϕ(x; µ1, σ1) + (1− p)ϕ(x; µ2, σ2)

under the pseudo-conjugate priors (i = 1, 2)

µi|σi ∼ N (ζi, σ2i /λi), σ−2

i ∼ G a(νi/2, s2i /2), p ∼ Be(α, β) ,

the posterior is

π (θ, p|x) ∝n∏

{pϕ(xj ; µ1, σ1) + (1− p)ϕ(xj ; µ2, σ2)}π (θ, p) .

Computation: complexity O (2n)

Missing variables (2)

Auxiliary variables z = (z1, . . . , zn) ∈ Z associated with observations

x = (x1, . . . , xn)

For (n1, . . . , nk), where n1 + . . . + nk = n,

Izi=1 = n1, . . . ,n∑

Izi=k = nk

which consists of all allocations with the given allocation vector (n1, . . . , nk) (and

j corresponding lexicographic order).

Number of nonnegative integer solutions of this decomposition of n

n + k − 1n

Partition

Z = ∪ri=1Zi[Number of partition sets of order O(nk−1)]

Posterior decomposition

π(θ, p|x)

z∈Zi

ω (z)π(θ, p|x, z

with ω (z) posterior probability of allocation z.

Corresponding representation of posterior expectation of(θ, p

z∈Zi

ω (z)Eπ[θ, p|x, z

Very sensible from an inferential point of view:

1. consider each possible allocation z of the dataset,

2. allocates a posterior probability ω (z) to this allocation, and

3. constructs a posterior distribution for the parameters conditional on this

allocation.

All possible allocations: complexity O (kn)

Posterior

For a given permutation/allocation (kt), conditional posterior distribution

π(θ|(kt)) = N(

ξ1(kt),σ2

n1 + `

)× IG((ν1 + `)/2, s1(kt)/2)

ξ2(kt),σ2

n2 + n− `

)× IG((ν2 + n− `)/2, s2(kt)/2)

×Be(α + `, β + n− `)

x1(kt) = 1`

∑`t=1 xkt , s1(kt) =

∑`t=1(xkt − x1(kt))2,

x2(kt) = 1n−`

∑nt=`+1 xkt , s2(kt) =

∑nt=`+1(xkt − x2(kt))2

ξ1(kt) =n1ξ1 + `x1(kt)

n1 + `, ξ2(kt) =

n2ξ2 + (n− `)x2(kt)n2 + n− `

s1(kt) = s21 + s2

1(kt) +n1`

n1 + `(ξ1 − x1(kt))2,

s2(kt) = s22 + s2

2(kt) +n2(n− `)n2 + n− `

(ξ2 − x2(kt))2,

posterior updates of the hyperparameters

Intro/Inference:Scarcity/Algorithms/Beyond fixed k 26

Scarcity

Frustrating barrier:

Almost all posterior probabilities ω (z) are zero

Example 2. Galaxy dataset with k = 4 components, Set of allocations with the

partition sizes (n1, n2, n3, n4) = (7, 34, 38, 3) with probability 0.59 and

(n1, n2, n3, n4) = (7, 30, 27, 18) with probability 0.32, and no other size group

getting a probability above 0.01.

Example 3. Normal mean mixture

For a same normal prior,

µ1, µ2 ∼ N (0, 10)

posterior weight associated with a z such that

Izi=1 = l

ω (z) ∝√

(l + 1/4)(n− l + 1/4) pl(1− p)n−l,

Thus posterior distribution of z only depends on l and repartition of the partition size

follows a distribution close to a binomial B(n, p) distribution.

For two different normal priors on the means,

µ1 ∼ N (0, 4) , µ2 ∼ N (2, 4) ,

posterior weight of z is

ω (z) ∝√

(l + 1/4)(n− l + 1/4) pl(1− p)n−l×exp

{−[(l + 1/4)s1 (z) + l{x1 (z)}2/4]/2}×

exp{−[(n− l + 1/4)s2 (z) + (n− l){x2 (z)− 2}2/4]/2

x1 (z) =1l

Izi=1xi, x2 (z) =1

n− l

Izi=2xi

s1 (z) =n∑

Izi=1 (xi − x1 (z))2 , s2 (z) =n∑

Izi=2 (xi − x2 (z))2 .

Computation of exact weight of all partition sizes l impossible

Monte Carlo experiment by drawing z’s at random.

Example 4. Sample of 45 points simulated when p = 0.7, µ1 = 0 and µ2 = 2.5leads to

l = 23 as the most likely partition, with a weight approximated by 0.962For l = 27, weight approximated by 4.56 10−11.

log(ω(kt))

−750 −700 −650 −600 −550

log(ω(kt))

−750 −700 −650 −600 −550

Ten highest log-weights ω (z) (up to an additive constant)

0 10 20 30 40

−700

−650

−600

−550

Most likely allocation z for a simulated dataset of 45 observations

−2 −1 0 1 2 3 4

0.00.1

0.20.3

0.40.5

Caution! We simulated 450, 000 permutations, to be compared with a

total of 245 permutations!

Intro/Inference: Priors/Algorithms/Beyond fixed k 34

Prior selection

Basic difficulty: if exchangeable prior used on

θ = (θ1, . . . , θk)

all marginals on the θi’s are identical

Posterior expectation of θ1 identical to posterior expectation of θ2!

Identifiability constraints

Prior restriction by identifiability constraint on the mixture parameters, for instance

by ordering the means [or the variances or the weights]

Not so innocuous!

• truncation unrelated to the topology of the posterior distribution

• may induce a posterior expectation in a low probability region

• modifies the prior modelling

−4 −3 −2 −1 0

0.00.2

0.40.6

θ(10)

−1.0 −0.5 0.0 0.5 1.0

0.00.2

0.40.6

0.81.0

1.21.4

θ(19)

1 2 3 4

0.00.2

0.40.6

• with many components, ordering in terms of one type of parameter is unrealistic

• poor estimation (posterior mean)

-2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

Gibbs sampling

-2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

-2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

-2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

Random walk

-2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

-2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

-2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

Langevin

-2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

-2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

-2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

Tempered random walk

-2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

-2 -1 0 1 2 3

0.00.1

0.20.3

0.40.5

• poor exploration (MCMC)

Improper priors??

Independent improper priors,

π (θ) =k∏

πi(θi) ,

cannot be used since, if ∫πi(θi)dθi =∞

then for every n, ∫π(θ, p|x)dθdp =∞

Still, some improper priors can be used when the impropriety is on a common

(location/scale) parameter

[CPR & Titterington, 1998]

Intro/Inference: Loss/Algorithms/Beyond fixed k 38

Loss functions

Once a sample can be produced from the unconstrained posterior distribution, an

ordering constraint can be imposed ex post

[Stephens, 1997]

Good for MCMC exploration

Again, difficult assesment of the true effect of the ordering constraints...

order p1 p2 p3 θ1 θ2 θ3 σ1 σ2 σ3

p 0.231 0.311 0.458 0.321 -0.55 2.28 0.41 0.471 0.303

θ 0.297 0.246 0.457 -1.1 0.83 2.33 0.357 0.543 0.284

σ 0.375 0.331 0.294 1.59 0.083 0.379 0.266 0.34 0.579

true 0.22 0.43 0.35 1.1 2.4 -0.95 0.3 0.2 0.5

−4 −2 0 2 4

0.00.1

0.20.3

0.40.5

Pivotal quantity

For a permutation τ ∈ Sk, corresponding permutation of the parameter

τ(θ, p) ={(θτ(1), . . . , θτ(k)), (pτ(1), . . . , pτ(k))

does not modify the value of the likelihood (& posterior under exchangeability).

Label switching phenomenon

Reordering scheme:

Based on a simulated sample of size M ,

(i) compute the pivot (θ, p)(i∗) such that

i∗ = arg maxi=1,...,M

π((θ, p)(i)|x)

Monte Carlo approximation of the MAP estimator of (θ, p).

(ii) For i ∈ {1, . . . ,M}:1. Compute

τi = arg minτ∈Sk

d(τ((θ, p)(i)), (θ, p)(i

2. Set (θ, p)(i) = τi((θ, p)(i)).

Step (ii) chooses the reordering the closest to the MAP estimator

After reordering, the Monte Carlo posterior expectation is

(θi)(j)/M .

Probabilistic alternative

[Jasra, Holmes & Stephens, 2004]

Also put a prior on permutations σ ∈ Sk

Defines a specific model M based on a preliminary estimate (e.g., by relabelling)

Computes

θj =1N

n∑t=1

σ∈Sk

θ(t)σ(j)p(σ|θ(t), M)

3 Computations

Intro/Inference/Algorithms: Gibbs/Beyond fixed k 45

3.1 Gibbs sampling

Same idea as the EM algorithm: take advantage of the missing data representation

General Gibbs sampling for mixture models

0. Initialization: choose p(0) and θ(0) arbitrarily

1. Step t. For t = 1, . . .

1.1 Generate z(t)i (i = 1, . . . , n) from (j = 1, . . . , k)

P(z(t)i = j|p(t−1)

j , θ(t−1)j , xi

)∝ p

(t−1)j f

(xi|θ(t−1)

1.2 Generate p(t) from π(p|z(t)),

1.3 Generate θ(t) from π(θ|z(t), x).

Trapping states

Gibbs sampling may lead to trapping states , concentrated local modes that require

an enormous number of iterations to escape from, e.g., components with a small

number of allocated observations and very small variance

[Diebolt & CPR, 1990]

Also, most MCMC samplers fail to reproduce the permutation invariance of the

posterior distribution, that is, do not visit the k! replications of a given mode.

[Celeux, Hurn & CPR, 2000]

Example 5. Mean normal mixture

0. Initialization. Choose µ(0)1 and µ

(0)2 ,

1. Step t. For t = 1, . . .

1.1 Generate z(t)i (i = 1, . . . , n) from

P(z(t)i = 1

)= 1−P

(z(t)i = 2

)∝ p exp

(xi − µ

(t−1)1

1.2 Compute n(t)j =

Iz(t)i =j

and (sxj )(t) =

Iz(t)i =j

1.3 Generate µ(t)j (j = 1, 2) from N

(λδ + (sxj )

λ + n(t)j

−1 0 1 2 3 4

But...

−1 0 1 2 3 4

Intro/Inference/Algorithms: HM/Beyond fixed k 50

3.2 Metropolis–Hastings

Missing data structure is not necessary for MCMC implementation: the mixture

likelihood is available in closed form and computable in O(kn) time:

Step t. For t = 1, . . .

1.1 Generate (θ, p) from q(θ, p|θ(t−1), p(t−1)

1.2 Compute

r =f(x|θ, p)π(θ, p)q(θ(t−1), p(t−1)|θ, p)

f(x|θ(t−1), p(t−1))π(θ(t−1), p(t−1))q(θ, p|θ(t−1), p(t−1)),

1.3 Generate u ∼ U[0,1]

If r < u then (θ(t), p(t)) = (θ, p)

else (θ(t), p(t)) = (θ(t−1), p(t−1)).

Proposal

Use of random walk inefficient for constrained parameters like the weights and the

variances.

Reparameterisation:

For the weights p, overparameterise the model as

pj = wj

/ k∑

wl , wj > 0

[Cappe, Ryden & CPR]

The wj ’s are not identifiable, but this is not a problem.

Proposed move on the wj ’s is

log(wj) = log(w(t−1)j ) + uj , uj ∼ N (0, ζ2)

Gaussian random walk proposal

µ1 ∼ N(µ

(t−1)1 , ζ2

)and µ2 ∼ N

(t−1)2 , ζ2

associated with

0. Initialization. Choose µ(0)1 and µ

1. Step t. For t = 1, . . .

1.1 Generate µj (j = 1, 2) from N(µ

(t−1)j , ζ2

1.2 Compute

r =f (x|µ1, µ2, )π (µ1, µ2)

f(x|µ(t−1)

1 , µ(t−1)2

(t−1)1 , µ

(t−1)2

1.3 Generate u ∼ U[0,1]

If r < u then(µ

(t)1 , µ

)= (µ1, µ2)

else(µ

(t)1 , µ

(t−1)1 , µ

(t−1)2

−1 0 1 2 3 4

Intro/Inference/Algorithms: PMC/Beyond fixed k 56

3.3 Population Monte Carlo

Idea Apply dynamic importance sampling to simulate a sequence of iid samples

x(t) = (x(t)1 , . . . , x(t)

n )iid≈ π(x)

where t is a simulation iteration index (at sample level)

Dependent importance sampling

The importance distribution of the sample x(t)

qt(x(t)|x(t−1))

can depend on the previous sample x(t−1) in any possible way as long as marginal

distributions

qit(x) =∫

qt(x(t)) dx(t)−i

can be expressed to build importance weights

%it =π(x(t)

qit(x(t)i )

Special case

qt(x(t)|x(t−1)) =n∏

qit(x(t)i |x(t−1))

[Independent proposals]

In that case,

var(It

var(%(t)i h(x(t)

Population Monte Carlo (PMC)

Use previous sample (x(t)) marginaly distributed from π

E[%ith(X(t)

[∫π(x(t)

qit(x(t)i )

h(x(t)i )qit(x

(t)i ) dx

]= E [Eπ [h(X)]]

to improve on approximation of π

Resampling

Over iterations (in t), weights may degenerate:

%1 ' 1

while %2, . . . , %n negligible

Use instead Rubin’s (1987) systematic resampling: at each iteration resample the

x(t)i ’s according to their weight %

(t)i and reset the weights to 1 (preserves

“unbiasedness”/increases variance)

PMC for mixtures

Proposal distributions qit that simulate (θ(i)(t), p

(i)(t)) and associated importance

weight

ρ(i)(t) =

f(x|θ(i)

(t), p(i)(t)

(θ(i)(t), p

(i)(t)

(θ(i)(t), p

(i)(t)

) , i = 1, . . . ,M

Approximations of the form

ρ(i)(t)∑M

l=1 ρ(l)(t)

h(θ(i)(t), p

(i)(t)

give (almost) unbiased estimators of Eπx [h(θ, p)],

0. Initialization. Choose θ(1)(0), . . . , θ

(M)(0) and p

(1)(0), . . . , p

(M)(0)

1. Step t. For t = 1, . . . , T

1.1 For i = 1, . . . ,M

1.1.1 Generate(θ(i)(t), p

(i)(t)

)from qit (θ, p),

1.1.2 Compute

ρ(i) = f(x|θ(i)

(t), p(i)(t)

(θ(i)(t), p

(i)(t)

(θ(i)(t), p

(i)(t)

1.2 Compute ω(i) = ρ(i)

/ M∑

ρ(l),

1.3 Resample M values with replacement from the(θ(i)(t), p

(i)(t)

)’s using

the weights ω(i)

Implementation without the Gibbs augmentation step, using normal random walk

proposals based on the previous sample of (µ1, µ2)’s as in Metropolis–Hastings.

Selection of a “proper” scale:

bypassed by the adaptivity of the PMC algorithm

Several proposals associated with a range of variances vk, k = 1, . . . , K .

At each step, new variances can be selected proportionally to the performances of

the scales vk on the previous iterations, for instance, proportional to its

non-degeneracy rate

Step t. For t = 1, . . . , T

1.1 For i = 1, . . . , M

1.1.1 Generate k from M (1; r1, . . . , rK),

1.1.2 Generate (µj)(i)(t) (j = 1, 2) from N

((µj)

(i)(t−1) , vk

1.1.4 Compute

ρ(i) =f

(x|(µ1)

(i)(t), (µ2)

(i)(t)

((µ1)

(i)(t), (µ2)

(i)(t)

ϕ((µj)

(i)(t); (µ1)

(i)(t−1), vl

1.2 Compute ω(i) = ρ(i)/ M∑

ρ(l),

1.3 Resample the (µ1)(i)(t), (µ2)

(i)(t)’s using the weights ω(i)

1.4 Update the rl’s: rl is proportional to the number of (µ1)(i)(t), (µ2)

(i)(t)’s with

variance vl resampled.

−1 0 1 2 3 4

4 Unknown number of components

When k number of components is unknown, there are several models

with corresponding parameter sets

in competition.

Intro/Inference/Algorithms/Beyond fixed k: RJ 67

Reversible jump MCMC

Reversibility constraint put on dimension-changing moves that bridge the sets Θk /

the models Mk

[Green, 1995]

Local reversibility for each pair (k1, k2) of possible values of k: supplement Θk1

and Θk2 with adequate artificial spaces in order to create a bijection between them:

Basic steps

Choice of probabilities

πij∑

πij = 1

of jumping to modelMkj while in modelMki

θ(k1) is completed by a simulation u1 ∼ g1(u1) into (θ(k1), u1) and θ(k2) by

u2 ∼ g2(u2) into (θ(k2), u2)

(θ(k2), u2) = Tk1→k2(θ(k1), u1),

Green reversible jump algorithm

0. At iteration t, if x(t) = (m, θ(m)),

1. Select model Mn with probability πmn,

2. Generate umn ∼ ϕmn(u),

3. Set (θ(n), vnm) = Tm→n(θ(m), umn),

4. Take x(t+1) = (n, θ(n)) with probability

π(n, θ(n))π(m, θ(m))

πnmϕnm(vnm)πmnϕmn(umn)

∣∣∣∣∂Tm→n(θ(m), umn)

∂(θ(m), umn)

∣∣∣∣ , 1)

and take x(t+1) = x(t) otherwise.

Example 8. For a normal mixture

Mk :k∑

pjkN (µjk, σ2jk) ,

restriction to moves from Mk to neighbouring models Mk+1 and Mk−1.

[Richardson & Green, 1997]

Birth and death steps

birth adds a new normal component generated from the prior

death removes one of the k components at random.

Birth acceptance probability

π(k+1)k

πk(k+1)

(k + 1)!k!

πk+1(θk+1)πk(θk) (k + 1)ϕk(k+1)(uk(k+1))

= min(

π(k+1)k

πk(k+1)

%(k + 1)%(k)

`k+1(θk+1) (1− pk+1)k−1

`k(θk), 1

where %(k) is the prior probability of model Mk

Proposal that can work well in some settings, but can also be inefficient (i.e. high

rejection rate), if the prior is vague.

Alternative: devise more local jumps between models,

(i). split

pjk = pj(k+1) + p(j+1)(k+1)

pjkµjk = pj(k+1)µj(k+1) + p(j+1)(k+1)µ(j+1)(k+1)

pjkσ2jk = pj(k+1)σ

2j(k+1) + p(j+1)(k+1)σ

2(j+1)(k+1)

(ii). merge (reverse)

Histogram and rawplot of 100, 000 k’s produced by RJMCMC

Histogram of k

1 2 3 4 5

0e+00 2e+04 4e+04 6e+04 8e+04 1e+05

Rawplot of k

Normalised enzyme dataset

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Intro/Inference/Algorithms/Beyond fixed k: b&d 75

Birth and Death processes

Use of an alternative methodology based on a Birth–&-Death (point) process

Idea: Create a Markov chain in continuous time, i.e. a Markov jump process,

moving between models Mk, by births (to increase the dimension), deaths (to

decrease the dimension), and other moves

[Preston, 1976; Ripley, 1977; Stephens, 1999]

Time till next modification (jump) is exponentially distributed with rate depending on

current state

Remember: if ξ1, . . . , ξv are exponentially distributed, ξi ∼ Exp(λi),

min ξi ∼ Exp

Difference with MH-MCMC : Whenever a jump occurs, the corresponding move is

always accepted. Acceptance probabilities replaced with holding times.

Balance condition

Sufficient to have detailed balance

L(θ)π(θ)q(θ, θ′) = L(θ′)π(θ′)q(θ′, θ) for all θ, θ′

for π(θ) ∝ L(θ)π(θ) to be stationary.

Here q(θ, θ′) rate of moving from state θ to θ′.

Possibility to add split/merge and fixed-k processes if balance condition satisfied.

[Cappe, Ryden & CPR, 2002]

Case of mixtures

Representation as a (marked) point process

Φ ={{pj , (µj , σj)}

Birth rate λ0 (constant) and proposal from the prior

Death rate δj(Φ) for removal of component j

Overall death ratek∑

δj(Φ) = δ(Φ)

Balance condition

(k + 1) d(Φ ∪ {p, (µ, σ)}) L(Φ ∪ {p, (µ, σ)}) = λ0L(Φ)π(k)

π(k + 1)

d(Φ \ {pj , (µj , σj)}) = δj(Φ)

Stephen’s original algorithm:

For v = 0, 1, · · · , Vt← v

Run till t > v + 1

1. Compute δj(Φ) =L(Φ|Φj)

L(Φ)λ0λ1

2. δ(Φ)←k∑

δj(Φj), ξ ← λ0 + δ(Φ), u ∼ U([0, 1])

3. t← t− u log(u)

4. With probability δ(Φ)/ξ

Remove component j with probability δj(Φ)/δ(Φ)

k ← k − 1

p` ← p`/(1− pj) (` 6= j)

Otherwise,

Add component j from the prior π(µj , σj)pj ∼ Be(γ, kγ)

p` ← p`(1− pj) (` 6= j)

k ← k + 1

5. Run I MCMC(k, β, p)

Rescaling time

In discrete-time RJMCMC, let the time unit be 1/N , put

βk = λk/N and δk = 1− λk/N

As N →∞, each birth proposal will be accepted, and having k components births

occur according to a Poisson process with rate λk

while component (w, φ) dies with rate

limN→∞

Nδk+1 × 1k + 1

×min(A−1, 1)

= limN→∞

k + 1× likelihood ratio−1 × βk

δk+1× b(w, φ)

(1− w)k−1

= likelihood ratio−1 × λkk + 1

× b(w, φ)(1− w)k−1

“RJMCMC→BDMCMC”

Even closer to RJMCM

Exponential (random) sampling is not necessary, nor is continuous time!

Estimator of

I =∫

g(θ)π(θ)dθ

g(θ(τi))

where {θ(t)} continuous time MCMC process and τ1, . . . , τN sampling instants.

New notations:

1. Tn time of the n-th jump of {θ(t)} with T0 = 0

2. {θn} jump chain of states visited by {θ(t)}3. λ(θ) total rate of {θ(t)} leaving state θ

Then holding time Tn − Tn−1 of {θ(t)} in its n-th state θn exponential rv with rate

λ(θn)

Rao–Blackwellisation

If sampling interval goes to 0, limiting case

I∞ =1

N∑n=1

g(θn−1)(Tn − Tn−1)

Rao–Blackwellisation argument: replace I∞ with

N∑n=1

g(θn−1)

λ(θn−1)=

N∑n=1

E[Tn − Tn−1 | θn−1] g(θn−1) .

Conclusion: Only simulate jumps and store average holding times!

Completely remove continuous time feature

Example 9. Galaxy dataset

Comparison of RJMCMC and CTMCMC in the Galaxy dataset

[Cappe & al., 2002]

Experiment:

• Same proposals (same Ccode)

• Moves proposed in equal proportions by both samplers (setting the probability

PF of proposing a fixed k move in RJMCMC equal to the rate ηF at which

fixed k moves are proposed in CTMCMC, and likewise PB = ηB for the birth

moves)

• Rao–Blackwellisation

• Number of jumps (number of visited configurations) in CTMCMC == number of

iterations of RJMCMC

Results:

• If one algorithm performs poorly, so does the other. (For RJMCMC

manifested as small A’s—birth proposals are rarely accepted—while for

BDMCMC manifested as large δ’s—new components are indeed born but die

again quickly.)

• No significant difference between samplers for birth and death only

• CTMCMC slightly better than RJMCMC with split-and-combine moves

• Marginal advantage in accuracy for split-and-combine addition

• For split-and-combine moves, computation time associated with one step of

continuous time simulation is about 5 times longer than for reversible jump

simulation.

Box plot for the estimated posterior on k obtained from 200 independent runs:

RJMCMC (top) and BDMCMC (bottom). The number of iterations varies from 5 000

(left), to 50 000 (middle) and 500 000 (right).

2 4 6 8 10 12 14 0

0.3CT (500 000 it.)

2 4 6 8 10 12 14 0

0.3RJ (500 000 it.)

2 4 6 8 10 12 14 0

0.3CT (50 000 it.)

2 4 6 8 10 12 14 0

0.3RJ (50 000 it.)

2 4 6 8 10 12 14 0

0.3CT (5 000 it.)

2 4 6 8 10 12 14 0

0.3RJ (5 000 it.)

Same for the estimated posterior on k obtained from 500 independent runs: Top

RJMCMC and bottom, CTMCMC. The number of iterations varies from 5 000 (left

plots) to 50 000 (right plots).

2 4 6 8 10 12 14 0

0.3CT (50 000 it.)

2 4 6 8 10 12 14 0

0.3RJ (50 000 it.)

2 4 6 8 10 12 14 0

0.3CT (5 000 it.)

2 4 6 8 10 12 14 0

0.3RJ (5 000 it.)

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