Seminaire ihp

Éric Barat

Positron Emissiontomography

NPB Model for 4DPET

Random probabilitymeasure

Spatial PET model

Space-time PETmodel

Conclusion

Bayesian Nonparametric Modelling forSpace-Time Emission Tomography

Éric Barat1joint work with Mame Diarra Fall

1Laboratory of Modelling, Simulation and SystemsCEA–LIST

October 25, 2010Séminaire Parisien de Statistiques

NPB Modelling for Space-Time Emission Tomography October 25, 2010 1 / 46

Éric Barat

Positron EmissiontomographyPhysics

Idealized PET

Usual approach

Limitations

NPB Model for 4DPET


Spatial PET model

Space-time PETmodel

Conclusion

Positron Emission Tomography (PET)Exploiting physical properties of positron annihilation.

PET: a molecular imaging modality.

Physical basis:Radioactively taggedmolecule injected to thepatient (glucose + 18F).(β+, β−) annihilation →two-photons emission inopposite directions.Event: detectors recordphotons pair coincidence.

Mathematical modelling.Image reconstruction(2D/3D).3D + time: space-timeactivity distribution →metabolic imaging.


Éric Barat


Idealized PET

Usual approach

Limitations

NPB Model for 4DPET


Spatial PET model

Space-time PETmodel

Conclusion

Positron Emission Tomography (PET)An idealized 2D problem.

uφ

D2 l

D1

ρu =D1D22

A+

Figure: Patient, detector circle and parameterized line of response.

Ideal detector circle and Radon transformDenote by g(x1, x2) the activity pdf in brain space.All (φ, u) are exactly observable → pdf in the detector spacef (φ, u) = 1

2ρu

∫ ρu−ρu

g(u cosφ− t sinφ, u sinφ+ t cosφ)dt.The integral above is the so-called Radon transform of g .Reconstruction = inversion of the mapping of densities.


Éric Barat


Idealized PET

Usual approach

Limitations

NPB Model for 4DPET


Spatial PET model

Space-time PETmodel

Conclusion

Positron Emission Tomography (PET)Optimality of linear reconstruction in idealized system.

Linear estimators for inversion of the Radon transform (2D)Known as filtered backprojection (FPB), see Natterer (1986).

gn(x) =1n

n∑i=1

Kη(〈~φi , x〉 − ui )

with n = # events, ~φ = (cos(φ), sin(φ))′, x = (x1, x2)′ and

Kη(t) ∝∫ 1

η

0r cos(t r)dr Fourier−→ FKη (ν) ∝ |ν| 1|ν|≤ 1

η

Kη is a band-limited filter (≈ kernel with window η).See Johnstone and Silverman (1990), Cavalier (2000) forrates of convergence and efficiency of linear estimators inidealized situation (mildly ill-posed problem).Smoothness p: direct data: rD ≈ ( log n

n )p

p+1 ; indirect: rI ≈ ( 1n )

pp+2 .


Éric Barat


Idealized PET

Usual approach

Limitations

NPB Model for 4DPET


Spatial PET model

Space-time PETmodel

Conclusion

Positron Emission Tomography (PET)Real life systems.

uφ

D2 l

D1

ρu =D1D22

A+

Figure: Finite size detectors ring.

Physical limitations → data more incomplete (wrt Radon)Detectors geometry → only a limited number of distinctdetectors pair coordinates values y = {D1,D2} are observed.Finite pmf in detector space → more strongly ill-posed.Results on rates may not hold for linear estimation of acontinuous distribution from a finite number of projections.


Éric Barat


Idealized PET

Usual approach

Limitations

NPB Model for 4DPET


Spatial PET model

Space-time PETmodel

Conclusion

Positron Emission Tomography (PET)Usual algorithms: parametric models and discrete-discrete reconstruction.

Brain space discretization: voxels (pixels) basis function

g(x) =∑K

k=1 gk1(x∈vk )|v | where |v | = volume of voxels vk .

fy =∫X p(y|x)g(x)dx =

∑Kk=1 ay,kgk .

e.g. Radon: p(y|x) ∝ δ(〈~φy, x〉 − uy).Denote ny=# events recorded in y: ny|g ∼ Poisson(fy)

→ Parametric Poisson inverse problem framework.

Penalized likelihood estimators of g = {g1, . . . , gK}g = argmax

g>0(logL(g|{ny}) + λΨ(g)).

Expectation-Maximization algorithm.ML estimator (λ = 0): Shepp and Vardi (1982).MAP estimator: exp(λΨ(g)) = prior on g, e.g. Gibbs field,see Green (1990).


Éric Barat


Idealized PET

Usual approach

Limitations

NPB Model for 4DPET


Spatial PET model

Space-time PETmodel

Conclusion

Positron Emission Tomography (PET)Parametric approach limitations.

Voxel size impact on reconstructionRegularization parameter.Trade-off between reconstruction noise and resolution.

Finite model limitationsDo we really trust in a discretized brain structure ?How do we choose the voxel size ?Can we give an interpretation to models with several millions(3D) or billions (4D) of parameters ?Do Gibbs fields correspond to biological structures prior ?→ model selection and averaging ?

Nonparametric discrete to continuous reconstruction ?Solutions in the whole space of probability measuresM(X ) ?Regularization ?


Éric Barat


NPB Model for 4DPET


Spatial PET model

Space-time PETmodel

Conclusion

Nonparametric Bayesian Model for 4D PETA more general framework

Nonparametric Bayesian Poisson inverse problem framework

G ∼ G

F (·, t) =

∫XP (·|x) G (dx, t)

Yi ,Ti |Fiid∼ F , for i = 1, . . . , n

(1)

G(·): G-distributed random probability measure (RPM), defined on(X × T , σ(X )⊗ σ(T )).Objective: estimate the posterior distribution of G(·) from the observedF -distributed dataset (Y,T)′ = {(Y1,T1), . . . , (Yn,Tn)}.P(·|x): given probability distribution, indexed by x, defined on (Y, σ(Y)).

Emission Tomography context X ⊆ R3, T ⊆ R

+.

Yi : index of the tube of response (TOR) and Ti : arrival time of the i thobserved event.Radon: P (y = l |x) ∝ δ(〈~φl , x〉 − ul )


Éric Barat


NPB Model for 4DPET


Spatial PET model

Space-time PETmodel

Conclusion

Nonparametric Bayesian Model for 4D PETPro & contras.

Relevant pointsDoes not require any adhoc discretization.

Classical dynamic PET: collection of spatial reconstructionson separate time frames.EM with 4D discretization (≈ 109 parameters).

Smoothness prior on continuous distribution → regularization.Gives access to posterior uncertainty.

e.g. highest probability density (HPD) interval of activityconcentration for any region of interest (R ⊂ X ).Key point in dose reduction (few events).

DifficultiesHow to elicit prior for G(·) (overM(X )) ?How to infer on infinite dimensional objects ?


Éric Barat


NPB Model for 4DPET

Random probabilitymeasureDirichlet process

Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Dirichlet ProcessFerguson (1973)

DefinitionG0 be a probability measure over (X , σ(X )) and α ∈ R+?.A Dirichlet process is the distribution of a random measure Gover (X , σ(X )) s.t., for any finite partition (B1, . . . ,Br ) of X ,

(G (B1) , . . . ,G (Br )) ∼ Dir (αG0 (B1) , . . . , αG0 (Br ))

G0 is the mean distribution, α the concentration parameter.We write G ∼ DP (α,G0).

Representations of Dirichlet processesChinese restaurant (prior over partitions).Pólya urns (DP arises here as the De Finetti measure of theexchangeable sequence).Stick-breaking representation (constructive).


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Dirichlet ProcessSome properties

PropertiesSample discrete random distributions.Expectation and variance :

∀B ⊆ X , E [G(B)] = G0 (B)

V (G(B)) =G0 (B) (1− G0 (B))

1 + α

Therefore, if α is large G is concentrated around G0.

Conjugacy : Consider G ∼ DP (α,G0) and X1, . . . ,Xni.i.d.∼ G .

G (·) |X1, . . . ,Xn ∼ DP(α + n, αG0 (·) +

∑ni=1 δXi (·)

α + n

)

Predictive distribution: Xn+1|X1, . . . ,Xn ∼αG0(·)+

∑ni=1δXi (·)

α+n


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Chinese Restaurant ProcessA worthy allegory for partition prior construction.

. . .

1 P(X1)

Figure: Assignment probability for customer 1.

Sequentially generating from a CRPFirst customer sits at table 1 and order θ1 ∼ G0.Customer n + 1 sits at:

Table k with probability nkn+α

with nk the number ofcustomers at table k.A new table K + 1 with probability α

n+αand order θK+1 ∼ G0

Xn = X1, . . . ,Xn take on K ≤ n distinct values θ1, . . . , θK .This defines a partition of {1, . . . , n} into K clusters, s.t. ibelongs to cluster k iff Xi = θk .


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


. . .θ1

Figure: Table draw for customer 1.







Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


. . .

11+α

α1+α

P(X2|X1)

θ1








Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


. . .

11+α

α1+α

θ1

P(X2|X1)








Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


. . .

22+α

α2+α

θ1

P(X3|X2)








Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


. . .

22+α

α2+α

θ1

P(X3|X2)

θ2








Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


. . .

23+α

13+α

θ1

P(X4|X3)

θ2

α3+α








Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


. . .

23+α

13+α

θ1

P(X4|X3)

θ2

α3+α

θ3








Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


. . .

24+α

14+α

θ1

P(X5|X4)

θ2

14+α

θ3

α4+α








Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


. . .

24+α

14+α

θ1

P(X5|X4)

θ2 θ3

α4+α

14+α








Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


. . .

35+α

15+α

θ1

P(X6|X5)

θ2 θ3

α5+α

15+α








Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Chinese Restaurant ProcessClustering behaviour (α = 30).

The CRP exhibits the clustering property of the DP.Expected number of clusters K = O(α log n).Rich-gets-richer effect → Reinforcement (small number oflarge clusters).E.g.: Ewens sampling formula, species sampling.


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Blackwell-MacQueen UrnExchangeabiliy and de Finetti measure

Blackwell-MacQueen Urn

Xn|X1, . . . ,Xn−1 ∼αG0 (·) +

∑n−1i=1 δXi (·)

α + n − 1

→ P(X1, . . . ,Xn) =∏n

i=1 P(Xi |X1, . . . ,Xi−1)

Infinitely exchangeable sequenceFor any n and permutation σ, P(X1, . . . ,Xn) = P(Xσ(1), . . . ,Xσ(n))

de Finetti theorem

For any exchangeable X1,X2 . . ., it exists RPM G s.t. Xn|Giid∼ G and

P(X1, . . . ,Xn) =

∫M(X )

n∏i=1

G(Xi )dP(G)

For exchangeable Blackwell-MacQueen urn sequences,G ∼ DP(α,G0)


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Exchangeable Random PartitionsKingman (1975), Pitman (1995,1996,2006).

Exchangeable Partition Probability FunctionXn exchangeable sequence from CRP. (A1, . . . ,AK ) partitionof {1, . . . , n} in the order of appearance, nj = #Aj for all j .

P(∩K

j=1 (Xl = θj for all l ∈ Aj))

= pα(n1, . . . , nK |K )

and pα is a symmetric function s.t.

pα(n1, . . . , nK |K ) =αK ∏K

j=1(nj − 1)!

[1 + α]n−1

with [x ]m =∏m

j=1(x + j − 1)

Ewens sampling formula: ml = #{j : nj = l} (∑n

l=1 l ml = n)

pα(m1, . . . ,mn) =n!

[1 + α]n−1

n∏l=1

αml

lmlml !


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Stick-Breaking RepresentationConstructive definition, Sethuraman (1994)

Stick-breaking representation.

θ = (θ1, θ2, . . .)iid∼ G0

V = (V1,V2, . . .)iid∼ Beta (1, α)

p = (p1, p2, . . .), s.t. p1 = V1 and pk = Vk∏k−1

i=1 (1− Vi ).Then,

G (·) =∞∑

k=1pk δθk (·)

is a DP (α,G0)-distributed random probability distribution.We say that: p ∼ GEM(α).Almost sure truncation, Ishwaran and James (2001):PN (·) =

∑Nk=1 pk δθk (·) with VN = 1 converges a.s. to a

DP (αG0) random probability measure.


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Stick-Breaking RepresentationExample of construction

0 0.5 1

Stick-breaking

0

0.25

0.5

0.75

1

DP

weights

−3 −2 −1 0 1 2 3

k = 0

Figure: Dirichlet process GEM construction (α = 3, G0 = N (0, 1)).


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


0 0.5 1

Stick-breaking

0

0.25

0.5

0.75

1

DP

weights

−3 −2 −1 0 1 2 3

k = 1



Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


0 0.5 1

Stick-breaking

0

0.25

0.5

0.75

1

DP

weights

−3 −2 −1 0 1 2 3

k = 2



Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


0 0.5 1

Stick-breaking

0

0.25

0.5

0.75

1

DP

weights

−3 −2 −1 0 1 2 3

k = 3



Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


0 0.5 1

Stick-breaking

0

0.25

0.5

0.75

1

DP

weights

−3 −2 −1 0 1 2 3

k = 4



Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


0 0.5 1

Stick-breaking

0

0.25

0.5

0.75

1

DP

weights

−3 −2 −1 0 1 2 3

k = 5



Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


0 0.5 1

Stick-breaking

0

0.25

0.5

0.75

1

DP

weights

−3 −2 −1 0 1 2 3

k = 6



Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion


0 0.5 1

Stick-breaking

0

0.25

0.5

0.75

1

DP

weights

−3 −2 −1 0 1 2 3

k → ∞



Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Stick-Breaking RepresentationRelation to CRP and exchangeable partitions

Invariance by Size-Biased Permutation (ISBP)Let p ∼ GEM(α). Generate p = (p1, p2, . . .) as follows:

i P(p1 = pk |p) = pk and for j ≥ 1,ii P(pj+1 = pk |p1, . . . , pj , p) = pk

1−p1−...−pj1(pk 6= p1, . . . , pj )

Then, p ∼ GEM(α).The limiting relative frequencies of clusters of a CRP randompartition are GEM(α)-distributed (ISBP ↔ EPPF).

General stick-breaking modelsRemark: if Vk ∼ Beta(ak , bk), → p is not ISBP in general.Relaxing exchangeability (Sethuraman (1994); Ishwaran andJames (2001)): clusters labels are explicitly definedindependently from any sequence sampling.6= CRP: X1 = θ1 by construction (cluster: equivalence class).


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Posterior Distribution and Exchangeability6= different behaviours between representations

Posterior distribution for stick-breaking weightsVk |Xn ∼ Beta(1 + nk , α+

∑∞l=k+1 nl ) (nk = #{i : Xi = θk}).

→ Symmetry of active clusters is lost in G(·)|Xn !

Posterior distribution for limiting frequencies from CRPUpdate Vk

i Vk |Xn ∼ Beta(nk , α +∑∞

l=k+1 nl ) for k ≤ Kii Vk |Xn ∼ Beta(1, α) for k > K

→ exchangeable posterior for G(·)|Xn:

G(·)|Xn =K∑

k=1p?k δθk (·) + p?K+1G?(·)

wherep? ∼ Dirichlet(n1, . . . , nK , α) and G?(·) ∼ DP(αG0)

Consequences on mixing properties of posterior sampling ?NPB Modelling for Space-Time Emission Tomography October 25, 2010 19 / 46

Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Pitman-Yor ProcessA two-parameters extension of DP, Pitman and Yor (1997)

Constructive representation

θ = (θ1, θ2, . . .)iid∼ G0

for all k, Vk ∼ Beta (1− d , α + k d)

p = (p1, p2, . . .), s.t. p1 = V1 and pk = Vk∏k−1

i=1 (1− Vi ).Then,

G (·) =∞∑

k=1pk δθk (·)

is a PY (d , α,G0)-distributed RPM where d ∈ [0, 1[ andα > −d . We note: p ∼ GEM(d , α)

Extended CRP representation

P(Xn+1 = θk |Xn) = nk−dα+n for k ≤ K .

P(Xn+1 = θK+1|Xn) = α+K dα+n .


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Pitman-Yor ProcessClustering properties

α = 30, d = 0.3

α = 30, d = 0

PropertiesExpected number of clusters K = O(αnd ) (Zipf’s law).


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Pitman-Yor ProcessPosterior distribution.

Posterior distribution based on extended CRP weightsUpdate Vk

i Vk |Xn ∼ Beta(nk − d , α + k d +∑∞

l=k+1 nl ) for k ≤ Kii Vk |Xn ∼ Beta(1− d , α + k d) for k > K

→ exchangeable posterior for G(·)|Xn:

G(·)|Xn =K∑

k=1p?k δθk (·) + p?K+1G?(·)

where

p? ∼ Dirichlet(n1 − d , . . . , nK − d , α + K d)

G?(·) ∼ PY(d , α + d K ,G0)

GEM(d , α) is the maximal family of ISBP distributions.


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Pitman-Yor Process Mixtures (PYM)Continuous data modelling

Discreteness of PY(d , α,G0) generated measuresCannot be used for probability density functions estimation !→ Hierarchical mixture model with continuous distribution φ.

Hierarchical data model

Yi |Xi ∼ φ(Yi |Xi )

Xi ∼ G(·)G ∼ PY(d , α,G0)

Data distribution

y |G ∼∫

Θ

φ(y |θ)G(dθ) =∞∑

k=1pk φ(y |θk)

E.g.: PYM of Normals with G0 taken as Normal-InverseWishart (NIW), s.t. θk = (µk ,Σk).


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Posterior Sampling of PYMSpecific random schemes

How to infer on infinite dimensional objects in a real world(and in a decent time) ?Sampling from the posterior: specific MCMC techniques.

Integrate out the random distribution: Escobar (1995),Mac-Eachern (1998), Neal (2000).

side-step infiniteness by marginalization, only the allocationto occupied clusters (finite number) is sampled (Pólya Urnscheme).Collapsing → good mixing properties.Gives only access to sequences generated from the RPM.

Almost sure truncation: Ishwaran and James (2001).Easy implementation.

Slice sampling: Walker (2007), Kalli (2009).Conditional approach: inference retains whole distribution.Use of auxiliary variables: only a finite pool of atoms areinvolved at each iteration, without truncation.Gives access to posterior of any functional of the RPM(mean, variance, credible intervals, etc.).

Variational techniques: Blei (2006).


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Posterior Sampling of PYMSampler profiling

Drawback of marginalization approachInference for the RPM posterior is based only on posteriorsampled values of Xi : ok for posterior means but cumbersomefor distribution of RPM functionals (credible intervals).Not easy in non conjugate case (G0 vs. φ).

Computational considerationsAllocation of data to clusters when sampling from mixtures.Huge datasets (n ≈ 107) → allocation time turns out toseverely dominate the computation cost.Need for a parallelizable allocation → blocking.

Which PY representation ?Retaining whole RPM samples while maintainingexchangeability and avoiding truncation ?


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Posterior Sampling of PYMProposed Gibbs sampler

Combination of:

Use of auxiliary variables.Use of a dependent (thresholded) slicing function.Use of exchangeable PY posterior from weighted CRP.

Joint density

Let u = u1, u2, . . . , un uniform auxiliary variables. Joint density forany (Yi , ui ), for some positive sequence (ξk ):

f (Yi , ui |p, θ) =

∞∑k=1

U (ui |0, ξk ) pk φ (Yi |θk ) (2)

where U (·|a, b) is the uniform distribution over ]a, b].We propose a dependent (ξk ), s.t. for all k,

ξk = min (pk , ζ)

where ζ ∈ ]0, 1], independent of pk .


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Posterior Sampling of PYMProposed Gibbs sampler (2)

Joint density (cont.)

f (Yi , ui |p, θ) =1 (ζ > ui )

ζ

∑pk>ζ

pk φ (Yi |θk ) +∑pk≤ζ

1 (pk > ui )φ (Yi |θk )

where both sums are finite since #{j : pj > ε} <∞ for any ε > 0.

Sampling from the posterior

Let C s.t for i ≤ n, Ci = k iff Yi = θk . Jointly sample (u, p|C):(p1, p2, . . . , pK , rK |C) ∼ Dirichlet (n1 − d , n2 − d , . . . , nK − d , α+ K d)

i ≤ n, (ui |p1, p2, . . . , pK ,C) ∼ U(

ui |0,min(

pCi , ζ))

, set u? = min(u)

For K < k ≤ k? = min(

k : 1−∑k

l=1 pl < u?), pk ∼ GEM(d , α+ k d)

Then, for k ≤ k?, sample (θk |C,Y), and

(Ci |p, θ,Y, u) ∼∑k?

j=1 wj,i , δj and wj,i ∝ 1 (pj > ui )max (pj , ζ)φ (Yi |θj )

Re-label (p, θ) in the order of appearance of clusters in allocation.


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Mixing Properties of Gibbs SamplerIntegrated Autocorrelation Times on galaxy data

1 Proposed Gibbs sampler2 Efficient Slice sampler, Kalli (2009)3 Neal (2000), algorithm 8, m = 2 (collapsed sampler).4 Ishwaran and James (2001).5 “Pitman posterior” with IJ truncation for G?.

-0.2

0

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0.8

1

0 50 100 150 200 250 300

# Clusters: autocorrelation time

Exc.+thres: 3.983 (0.928) iat = 14.322 (0.503) T = 2.56e-04Slice: 3.980 (0.914) iat = 60.286 (2.117) T = 2.45e-04

Trunc.: 3.994 (0.927) iat = 35.785 (1.256) T = 2.96e-04Neal8 (2): 3.987 (0.927) iat = 8.750 (0.307) T = 2.85e-04

Trunc Unlab.: 3.980 (0.925) iat = 14.175 (0.498) T = 2.87e-04

Figure: IAT for K (active clusters).


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Mixing Properties of Gibbs SamplerIntegrated Autocorrelation Times on galaxy data


-0.2

0

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0.8

1

0 10 20 30 40 50 60 70 80

Deviance: autocorrelation time




Figure: IAT for deviance.


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Mixing Properties of Gibbs SamplerIntegrated Autocorrelation Times on leptokurtic data, n = 1000


0

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0 50 100 150 200 250 300

# Clusters: autocorrelation time




Figure: IAT for K (active clusters).


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Mixing Properties of Gibbs SamplerIntegrated Autocorrelation Times on leptokurtic data, n = 1000


0

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0 10 20 30 40 50 60 70 80 90 100

Deviance: autocorrelation time

Exc.+thres: 2341.660 (12.175) iat = 8.743 (0.178) T = 7.76e-04Slice: 2341.660 (12.341) iat = 13.659 (0.277) T = 6.98e-04Trunc.: 2341.636 (12.231) iat = 9.188 (0.187) T = 1.62e-03

Neal8 (2): 2341.674 (12.309) iat = 8.025 (0.163) T = 1.23e-03Trunc Unlab.: 2341.666 (12.197) iat = 8.578 (0.174) T = 1.41e-03

Figure: IAT for deviance.


Éric Barat


NPB Model for 4DPET


Chinese restaurant

Stick-breaking

Pitman-Yor process

Mixtures of PY

Posterior sampling

Consistency of DPM

Spatial PET model

Space-time PETmodel

Conclusion

Consistency Results on Dirichlet MixturesFrequentist validation of Bayesian estimates.

An issue in Bayesian nonparametrics: Diaconis andFreedman, 1986

Depend on mixing RPM and kernel density.Ghosal et al. (1999): consistency (weak, strong) of DPM ofnormals (1D).Ghosal and Van der Vaart (2007): convergence rates forDPM of normals ≈ n− 2

5 (log n)45 (twice differentiable pdf,

equivalent as kernel estimators).Wu and Ghosal (2010): L1-consistency of DPM ofmultivariate normals (with general covariance matrix).Density deconvolution ?

Tokdar et al. (2009): consistency of (not Bayesian) recursiveestimator (Newton, 2002) in density deconvolution (inrelation with NPB).


Éric Barat


NPB Model for 4DPET


Spatial PET modelModel

Inference

Application

Space-time PETmodel

Conclusion

Spatial Model for PET DataPYM of latent emission locations

Spatial hierarchical model

Yi |Xiind∼ P (Yi |Xi )

Xi |Ziind∼ N (Xi |Zi )

Zi |Hiid∼ H

H ∼ PY (d , α,NIW)

(3)

RemarksTomography: Only Yi is observed, thus Xi (the emission location)is introduced as latent variable.In EM approach, latent variables are the number of emissions fromvoxel v which are recorded in line of response l .Compared to NPB density estimation, PET reconstruction mainlyinvolves a sampling step from conditional (Xi |Yi , p, θ).Spatial distribution: G(·) =

∫ΘN (·|θ)H(dθ) =

∑∞k=1 pk N (·|θk ).


Éric Barat


NPB Model for 4DPET



Inference

Application

Space-time PETmodel

Conclusion

Spatial Model for PET DataInference by Gibbs sampling

Sampling from the posterior

Let introduce C = C1, C2, . . . , Cn, the classification of emissionsto PY components s.t. Zi = θCi for all i < n.Let u = u1, u2, . . . , un uniform auxiliary variables, cf. (2).Successively draw samples from the following conditionals

Annihilation location : (X|Y, p, θ, u)

PYM component parameters : (θ|C,X)

Emission allocation to PY atoms : (C|p, θ,X, u)

PY weights & auxiliary variables : (p, u|C)

Sampling X|Y, p, θ, u: Metropolis (independent MH) within Gibbs

(Xi |Yi , p, θ, u)∝∼ P(Yi |Xi ) G(Xi |p, θ, u)

P(Yi |Xi ) accounts for physical and geometrical properties of PETsystem → no hope for conjugacy...Candidate: X?

i |Yi , p, θ, u∝∼ N (X?

i |µYi ,ΣYi ) G(X?i |p, θ, u)


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NPB Model for 4DPET



Inference

Application

Space-time PETmodel

Conclusion

Spatial Model for PET DataGibbs sampler in action

Iteration k , (p, u|C), (θ|C,X)


Éric Barat


NPB Model for 4DPET



Inference

Application

Space-time PETmodel

Conclusion


Event Yi


Éric Barat


NPB Model for 4DPET



Inference

Application

Space-time PETmodel

Conclusion


Back-projection Xi |Yi , p, u, θ


Éric Barat


NPB Model for 4DPET



Inference

Application

Space-time PETmodel

Conclusion


Event Yi+1


Éric Barat


NPB Model for 4DPET



Inference

Application

Space-time PETmodel

Conclusion


Back-projection Xi+1|Yi+1, p, u, θ


Éric Barat


NPB Model for 4DPET



Inference

Application

Space-time PETmodel

Conclusion


Back-projections X|Y, p, u, θ


Éric Barat


NPB Model for 4DPET



Inference

Application

Space-time PETmodel

Conclusion


Cluster allocations C|θ, p, u, X


Éric Barat


NPB Model for 4DPET



Inference

Application

Space-time PETmodel

Conclusion


Iteration k + 1, (p, u|C), (θ|C, X)


Éric Barat


NPB Model for 4DPET



Inference

Application

Space-time PETmodel

Conclusion

Spatial Model for PET DataApplication

Data generationRealistic digital 3D brain phantom.n = 107 events.Geometrical and physical model of system.Truncated data in axial dimension.

Algorithm parametrizationDirichlet case: α = 1000, d = 0NIW: Wishart centred on isotropic 3D normal withσ = 2.5mm and dof = 4.→ K ≈ 4000, k? ≈ 10000 during iterations at equilibrium.Comparison with EM approach: MAP using Gibbs prior and“log cosh” energy function (Green, 1990) with parametersβ = .25 and δ = 10.


Éric Barat


NPB Model for 4DPET



Inference

Application

Space-time PETmodel

Conclusion

Spatial Model for PET DataResults

phant

EM

NPB


Éric Barat


NPB Model for 4DPET



Inference

Application

Space-time PETmodel

Conclusion

Spatial Model for PET DataResults: reconstruction uncertainty

1) 2) 0

2

4

6

8

NP

B95%

CI

Act

ivit

y

−100 −50 0 50 100

x-axis (mm)

Figure: 1) NPB conditional standard deviation, 3D isosurfaces; 2) 95%HPD on a profile: 97.5% (red), 2.5% (blue), median (green) andphantom profile (black).


Éric Barat


NPB Model for 4DPET


Spatial PET model

Space-time PETmodelPólya Tree

Model

Inference

Application

Conclusion

Space-Time Model for PET DataTissue kinetics: time dependency

Modelling metabolic activityBiokinetic: tissuedependent.Functional volume (FV):spatial region characterizedby a particular kinetic.Radioactive decay.

Separable space-time activity distribution

G (x, t) =∞∑

k=1pk N (x|θk) Qk (t)

Kinetics RPMEach event Yi is time stamped (Ti).Continuous measure with compact support (right truncation).


Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion

Pólya Tree ProcessDefinition

DefinitionLet E = {0, 1}, Em = E × · · · × E and E? = ∪∞m=0Em.Let πm = {Bε : ε ∈ Em} be a partition of T and Π = ∪∞m=0πm.A probability distribution Q on T has a Pólya tree distributionPT(Π,A) if there are nonnegative numbers A = {αε : ε ∈ E?}and r.v. W = {Wε : ε ∈ E?} s.t.

W is a sequence of independent random variables,for all ε in E?, Wε ∼ Beta(αε0, αε1), andfor all integer m and ε = ε1 · · · εm in Em,

Q(Bε1···εm ) =m∏

j=1εj =0

Wε1···εj−1 ×m∏

j=1εj =1

(1−Wε1···εj−1)

Note that for ε ∈ E?, Wε0 = Q (Bε0|Bε)


Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion

Some Properties of Pólya Tree ProcessesLavine (1992), Mauldin and Sudderth (1992)

PropertiesPólya trees are tail free processes.Dirichlet processes are Pólya trees s.t. αε0 = αε1 = αε/2PT(Π,A) can generate absolutely continuous distributions.Conjugacy : posterior of PT(Π,A) after observationsX = (X1, . . . ,Xn) is the Pólya tree PT(Π,AX) with

αTε = αε + nεnε = # {i ∈ {1, . . . , n} : Ti ∈ Bε}

Predictive density (conditional mean)

Pr (Tn+1 ∈ Bε1···εm |T) =m∏

k=1

αε1···εk + nε1···εk

αε1···εk−10 + αε1···εk−11 + nε1···εk−1


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NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion

Pólya Tree ConstructionDyadic partition

0

0.25

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Density

−3 −2 −1 0 1 2 3

m = 0

Figure: Pólya tree sequence construction (normal mean).


Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion


0

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−3 −2 −1 0 1 2 3

m = 1

Figure: Pólya tree sequence construction (A = {αm = 3m}).


Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion


0

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−3 −2 −1 0 1 2 3

m = 2



Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion


0

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−3 −2 −1 0 1 2 3

m = 3



Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion


0

0.25

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−3 −2 −1 0 1 2 3

m = 4



Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion


0

0.25

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−3 −2 −1 0 1 2 3

m = 5



Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion


0

0.25

0.5

0.75

1

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−3 −2 −1 0 1 2 3

m = 6



Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion


0

0.25

0.5

0.75

1

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−3 −2 −1 0 1 2 3

m → ∞



Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion

Pólya Tree Mixtures (PTM)Discontinuities mitigation

DiscontinuitiesContinuous RPM but discontinuities at partition endpoints.Mitigation of partitions endpoints discontinuities ?

Pólya tree mixturesPartitions and parameters depend on an r.v. Ψ

(G |Ψ) ∼ PT(ΠΨ,AΨ

)Ψ ∼ µ (Ψ)

E.g. Shifted PT.Uniform partition Πu.ΠΨ random shift of Πu.Adapt AΨ s.t. mean distribution given Ψ remains uniformand invariant for all Ψ.Easy and efficient with finite PT (αm =∞ for m > M).


Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion

Space-Time Model for PET DataDependent PYM of Pólya Trees

Space-Time hierarchical model

Yi |Xiind∼ P (Yi |Xi )

Xi ,Ti |Zi ,Qiind∼ N (Xi |Zi )× Qi (Ti )

Zi ,Qi |Hiid∼ H

H|K0 ∼ PY (d , α,NIW ×K0)

K0 ∼ PY (h, β,PT (A,Q0))

(4)

With H =∑∞

k=1 pk δθk ,Qk

where Q are i.i.d. K0.

K0 =∑∞

j=1 πj δQ?j with π ∼ GEM(h, β), Q? are i.i.d. PT(A,Q0),a Pólya tree with parameters A and mean Q0.K0: PY process with PT process as base distribution → nestedRPM (cf. nested DP, Rodriguez et al., 2008).Distinct θk may share the same Q?

j (K0 is discrete) → partialHierarchical PY (Teh, 2006); (diffuse NIW ×K0).


Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion

Space-Time Model for PET DataInference

Additional latent variables

Allocation variable: Dk = j iff Qk = Q?j (kinetics clustering).

Auxiliary variables v for slice sampling of K0.

Posterior computationsGibbs sampling of additional conditionals is straightforward.

Functional volumes distributionFor all j (label of K0 atoms),

FVj (x) =∑

k: Qk =Q?j

pk N (x|θk)

Nonparametrics issue: labels are permanently re-ordered →only the FV-distribution is accessible → need for postidentification of classes.


Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion

Space-Time Model for PET DataApplication and results

Data generation5 functional volumes: blood pool, gray matter, white matter,cerebellum, tumors.Blood fraction in tissues (between 5% and 10%).n = 107 events (≈ 1

10 usual dose for 4D PET).Spatial model unchanged.

ResultsPoint-wise PT kinetics value distribution (temporal marginal).

Space-time distribution.


Éric Barat


NPB Model for 4DPET


Spatial PET model


Model

Inference

Application

Conclusion

Space-Time Model for PET DataFunctional volumes estimation

Clinical interpretation (kinetics discovery)Construct metabolic parameter from kinetics distribution ?Post selection of groups → coming back to parametric...


Éric Barat


NPB Model for 4DPET


Spatial PET model

Space-time PETmodel

Conclusion

Conclusion and Perspectives

Some observations...Suitable framework for 4D PET: really nonparametric(K ≈ 4000...).Alternative approach for Poisson inverse problems (Antoniadis(2006)).Flexible nonparametric data modelling (hierarchies,dependencies, etc.).Posterior distribution of any RPM functional is accessible(clinical requirements).Efficient sampling schemes.

...and perspectivesConsistency and rates results ?Prior refinement: fragmentation/coagulation, kernel choice ?General indirect regression problems.


Éric Barat


NPB Model for 4DPET


Spatial PET model

Space-time PETmodel

Conclusion

For Further Reading.

T. Ferguson.Ann. Statist. 1, 209–230 (1973).

C. Antoniak.Ann. Statist. 2, 1152–1174 (1974).

H. Ishwaran and L. F. James.J. Am. Stat. Assoc. 96, 161–173 (2001).

P. Müller and F. A. Quintana.Statist. Sci. 19, 95–110 (2004).

Y. W. Teh et al..J. Am. Stat. Assoc. 101, 1566–1581 (2006).

J. Pitman.Combinatorial Stochastic Processes, Springer, 2006.

N. Hjort et al..Bayesian Nonparametrics, Cambridge, 2010.


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