Posterior concentration and adaptation of the mixing measure in Dirichlet process mixtures

Filippo Ascolani

arxiv: 2606.29109 · v1 · pith:BOKEYLDV · submitted 2026-06-27 · math.ST · stat.ME· stat.TH

Posterior concentration and adaptation of the mixing measure in Dirichlet process mixtures

Filippo Ascolani This is my paper

Reviewed by Pith T0 review T1 audit T2 compute T3 formal T4 kernel 2026-06-30 07:58 UTCgrok-4.3pith:BOKEYLDV record.json open to challenge →

classification math.ST stat.MEstat.TH

keywords Dirichlet process mixturesposterior adaptationmixing measureWasserstein distancestick-breaking representationclustering behaviourtruncation approximationswell-specified regime

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The pith

Dirichlet process mixture posteriors adapt to the true finite number of components K, with mass on extra stick-breaking weights vanishing at rate n^{-1/2}.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that when data are generated exactly by a finite mixture of location densities with K components, the posterior distribution of an infinite Dirichlet process mixture automatically adapts to this finite structure. The total posterior mass placed on all stick-breaking weights after the K-th one contracts at rate n^{-1/2}, up to factors that grow slower than any positive power of n. This adaptation produces a nearly optimal contraction rate for the mixing measure itself under Wasserstein distance. A phase transition is identified in which any finer approximation to the mixing measure demands a number of components that grows only logarithmically in sample size, which in turn governs the posterior clustering behavior.

Core claim

In the well-specified regime where the data are generated by a finite mixture of location densities with K components, the posterior on the Dirichlet process mixing measure is adaptive to K: the cumulative mass assigned to weights of the stick-breaking representation beyond the K-th one vanishes as n^{-1/2}, up to terms growing slower than any polynomial. This also implies a nearly optimal posterior contraction rate for the mixing measure in Wasserstein distance. A remarkable phase transition underlies this result: approximating the mixing measure to any precision finer than n^{-1/2} requires a number of components growing logarithmically with the sample size. This has a profound impact on t

What carries the argument

The stick-breaking representation of the Dirichlet process mixing measure, whose successive weights receive posterior mass that adapts to the true finite K by driving the tail mass to zero at the stated rate.

If this is right

The mixing measure contracts at a nearly optimal rate in Wasserstein distance.
The number of clusters grows logarithmically with n, yet the proportion of observations outside the K largest clusters vanishes at a polynomial rate.
Truncation approximations using at least K components recover the optimal contraction rates for both the density and the mixing measure.
O(log n) components are both necessary and sufficient to reproduce the clustering properties of the exact posterior.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The adaptation result implies that Dirichlet process mixtures can serve as automatic substitutes for finite-mixture models in well-specified problems without sacrificing asymptotic accuracy.
The identified phase transition between n^{-1/2} precision and logarithmic component count may guide the design of truncation levels in scalable implementations that target different inferential targets such as density estimation versus cluster recovery.

Load-bearing premise

The observations are generated exactly by a finite mixture of location densities with a fixed number of components.

What would settle it

Simulate data from a known two-component Gaussian mixture and check whether the summed posterior mass on the third and all subsequent stick-breaking weights fails to decay at rate n^{-1/2} up to sub-polynomial factors.

read the original abstract

We study the asymptotic properties of the posterior on the latent space for infinite mixtures driven by a Dirichlet process, both in terms of mixing measure and clustering behaviour. In the well-specified regime, where the data are generated by a finite mixture of location densities, we show that the posterior is adaptive to the true number of components $K$: indeed the cumulative mass assigned to weights of the stick-breaking representation beyond the $K$-th one vanishes as $n^{-1/2}$, up to terms growing slower than any polynomial. This also implies a nearly optimal posterior contraction rate for the mixing measure in Wasserstein distance. A remarkable phase transition underlies this result: approximating the mixing measure to any precision finer than $n^{-1/2}$ requires a number of components growing logarithmically with the sample size. We show that this has a profound impact on the clustering behaviour: the number of clusters grows logarithmically, as in the prior case, but the proportion of observations outside the $K$ largest clusters vanishes polynomially fast. Finally, we turn these results into posterior guarantees for truncation-based approximations: while any truncation with at least $K$ elements recovers the optimal contraction rates for both density and mixing measure, $\mathcal{O}(\log n)$ components are both necessary and sufficient to reproduce the clustering of the exact posterior.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

DP mixture posteriors adapt to true finite K at n^{-1/2} tail rate, with log n phase transition for truncation and clustering.

read the letter

The main thing to know is that this paper gives an explicit adaptation rate of n^{-1/2} for the tail of the stick-breaking weights in Dirichlet process mixtures when the truth is a finite mixture. This leads to nearly optimal contraction for the mixing measure in Wasserstein distance.

It does a good job spelling out the phase transition: to get approximation better than that rate you need a number of components that grows like log n. This has consequences for clustering, where the number of clusters is logarithmic but the misassignment rate decays polynomially. They also provide guidelines for truncation that recover the rates with O(log n) terms.

The strength here is turning abstract concentration into usable truncation rules and clustering guarantees. It builds on existing literature on posterior concentration for these models in a direct way.

The results are limited to the well-specified case with location densities, which is stated clearly but means they don't cover misspecification. The proofs are not visible here, but the stress test found no inconsistencies or circularity.

This paper is for specialists in Bayesian nonparametrics interested in rates and practical approximations. A serious editor should send it to peer review because the adaptation and truncation results are concrete enough to be worth checking in detail.

Referee Report

0 major / 0 minor

Summary. The manuscript studies the asymptotic properties of the posterior on the latent space for infinite mixtures driven by a Dirichlet process, both in terms of mixing measure and clustering behaviour. In the well-specified regime where the data are generated by a finite mixture of K location densities, it claims that the posterior adapts to the true number of components K: the cumulative mass assigned to weights of the stick-breaking representation beyond the K-th one vanishes as n^{-1/2}, up to terms growing slower than any polynomial. This implies a nearly optimal posterior contraction rate for the mixing measure in Wasserstein distance. The paper identifies a phase transition in approximation precision requiring a logarithmic number of components, with consequences for clustering (logarithmic growth in number of clusters but polynomial vanishing of observations outside the K largest) and derives guarantees for truncation-based approximations.

Significance. If the central claims hold, the work provides a precise characterization of posterior adaptation and clustering in well-specified Dirichlet process mixtures, including a phase-transition argument that explains the logarithmic cluster growth and supplies concrete truncation guidelines. The adaptation result for stick-breaking weights and the resulting Wasserstein contraction rate constitute a substantive advance in the theoretical analysis of Bayesian nonparametric mixtures.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary, significance assessment, and recommendation to accept. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives asymptotic posterior concentration rates for the mixing measure and stick-breaking weights in the well-specified finite-mixture case, along with clustering behavior and truncation guarantees. These are presented as mathematical results obtained via phase-transition arguments relating approximation precision, truncation level, and cluster proportions. No load-bearing step reduces by definition or construction to a fitted parameter, self-referential prediction, or unverified self-citation chain; the central claims rest on independent analytic arguments under explicitly stated assumptions. This is the typical non-circular outcome for a theoretical asymptotic analysis paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard background assumptions in Bayesian nonparametric asymptotics; no free parameters or invented entities are introduced in the abstract.

axioms (2)

domain assumption Data are generated i.i.d. from a finite mixture of location densities
This defines the well-specified regime invoked for the adaptation result.
domain assumption The mixing measure is given a Dirichlet process prior
Standard modeling choice for the infinite mixture studied.

pith-pipeline@v0.9.1-grok · 5760 in / 1382 out tokens · 35524 ms · 2026-06-30T07:58:36.796243+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

55 extracted references · 4 canonical work pages · 1 internal anchor

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Alamichel, L., D. Bystrova, J. Arbel, and G. Kon Kam King (2024). Bayesian mixture models (in) consistency for the number of clusters.Scandinavian Journal of Statistics 51(4), 1619–1660

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Lijoi, G

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2023
[3]

Bariletto, N., D. Le, A. Rinaldo, and N. Ho (2026). Convergence Rates for Latent Mix- ing Measures in Infinite Homoscedastic Location-Scale Mixture Models.arXiv preprint arXiv:2605.06845

work page internal anchor Pith review Pith/arXiv arXiv 2026
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Lijoi, L

Barrios, E., A. Lijoi, L. E. Nieto-Barajas, and I. Prünster (2013). Modeling with Normalized Random Measure Mixture Models.Statistical Science 28(3), 313–334

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Chen, J. (1995). Optimal rate of convergence for finite mixture models.Annals of statistics 23(1), 221–233

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Connor, R. J. and J. E. Mosimann (1969). Concepts of independence for proportions with a generalization of the Dirichlet distribution.Journal of the American Statistical Association 64(325), 194–206

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[7]

Favaro, A

De Blasi, P ., S. Favaro, A. Lijoi, R. H. Mena, I. Prünster, and M. Ruggiero (2013). Are Gibbs- type priors the most natural generalization of the Dirichlet process?IEEE transactions on pattern analysis and machine intelligence 37(2), 212–229

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Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems.The Annals of statistics, 209–230

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Gassiat, E. and R. Van Handel (2014). The local geometry of finite mixtures.Transactions of the American Mathematical Society 366(2), 1047–1072

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Genovese, C. R. and L. Wasserman (2000). Rates of convergence for the Gaussian mixture sieve.The Annals of Statistics 28(4), 1105–1127. 19

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Ghosal, S., J. K. Ghosh, and A. W. van der Vaart (2000). Convergence rates of posterior distributions.Annals of Statistics, 500–531

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Ghosal, S. and A. W. Van Der Vaart (2001). Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities.Annals of Statistics, 1233– 1263

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[13]

Ghosal, S. and A. W. van der Vaart (2007). Posterior convergence rates of Dirichlet mixtures at smooth densities.Ann. Stat. 35, 697–723

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Ghosal, S. and A. W. van der Vaart (2017).Fundamentals of Nonparametric Bayesian Inference. Cambridge University Press

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Ghosh, J. K. and R. Ramamoorthi (2003).Bayesian nonparametrics. Springer

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Gormley, I. C., T. B. Murphy, and A. E. Raftery (2023). Model-based clustering.Annual Review of Statistics and Its Application 10(1), 573–595

2023
[17]

Ho, and X

Guha, A., N. Ho, and X. Nguyen (2021). On posterior contraction of parameters and inter- pretability in Bayesian mixture modeling.Bernoulli 27(4), 2159–2188

2021
[18]

Ho, and X

Guha, A., N. Ho, and X. Nguyen (2023). On excess mass behavior in Gaussian mixture models with Orlicz-Wasserstein distances. InInternational Conference on Machine Learning, pp. 11847–11870. PMLR

2023
[19]

Hairault, A., C. P . Robert, and J. Rousseau (2022). Evidence estimation in finite and infinite mixture models and applications.arXiv preprint arXiv:2205.05416

work page arXiv 2022
[20]

Heinrich, P . and J. Kahn (2018). Strong identifiability and optimal minimax rates for finite mixture estimation.Annals of Statistics 46(6A)

2018
[21]

Ho, N. and X. Nguyen (2016a). Convergence rates of parameter estimation for some weakly identifiable finite mixtures.The Annals of Statistics 44(6), 2726
[22]

Ho, N. and X. Nguyen (2016b). On strong identifiability and convergence rates of parameter estimation in finite mixtures.Electronic Journal of Statistics, 271–307
[23]

Nguyen, and Y

Ho, N., X. Nguyen, and Y. Ritov (2020). Robust estimation of mixing measures in finite mixture models.Bernoulli 26(2), 828–857

2020
[24]

Ishwaran, H. and L. F. James (2001). Gibbs sampling methods for stick-breaking priors. Journal of the American Statistical Association 96(453), 161–173

2001
[25]

Ishwaran, H. and L. F. James (2002). Approximate Dirichlet process computing in finite normal mixtures: smoothing and prior information.Journal of Computational and Graphical statistics 11(3), 508–532

2002
[26]

Ishwaran, H., L. F. James, and J. Sun (2001). Bayesian model selection in finite mixtures by marginal density decompositions.Journal of the American Statistical Association 96(456), 1316– 1332

2001
[27]

Ishwaran, H. and M. Zarepour (2002). Exact and approximate sum representations for the Dirichlet process.Canadian Journal of Statistics 30(2), 269–283

2002
[28]

Korwar, R. M. and M. Hollander (1973). Contributions to the theory of Dirichlet processes. The Annals of Probability 1(4), 705–711

1973
[29]

Lindsay, B. (1995). Mixture models: Theory, geometry, and applications. InNSFCBMS Re- gional Conference Series in Probability and Statistics, Volume 5. Institute of Mathematical Statistics

1995
[30]

Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates: I. Density estimates.The Annals of Statistics, 351–357. 20

1984
[31]

Martin, R. (2012). Convergence rate for predictive recursion estimation of finite mixtures. Statistics & Probability Letters 82(2), 378–384

2012
[32]

McLachlan, G. J. and K. E. Basford (1988). Mixture models. Inference and applications to clustering.Statistics: textbooks and monographs

1988
[33]

McLachlan, G. J. and D. Peel (2000).Finite mixture models. John Wiley & Sons

2000
[34]

Miller, J. W. and M. T. Harrison (2013). A simple example of Dirichlet process mixture in- consistency for the number of components.Adv. Neural Inf. Process. Syst. 26, 199–206

2013
[35]

Miller, J. W. and M. T. Harrison (2014). Inconsistency of Pitman-Yor process mixtures for the number of components.J. Mach. Learn. Res. 15, 3333–3370

2014
[36]

Miller, J. W. and M. T. Harrison (2018). Mixture models with a prior on the number of components.J. Am. Stat. Assoc. 113, 340–356

2018
[37]

Nguyen, X. (2013). Convergence of latent mixing measures in finite and infinite mixture models.Ann. Stat. 41, 370–400

2013
[38]

Nguyen, X. and Y. Wei (2026). Optimal transport based theory for latent structured models. arXiv preprint arXiv:2601.11465

work page arXiv 2026
[39]

Ohn, I. and L. Lin (2023). Optimal Bayesian estimation of Gaussian mixtures with growing number of components.Bernoulli 29(2), 1195–1218

2023
[40]

Pearson, K. (1894). Contributions to the mathematical theory of evolution.Proceedings of the Royal Society of London 54(326-330), 329–333
[41]

Pitman, J. and M. Yor (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator.The Annals of Probability 25(2), 855–900

1997
[42]

Rajkowski, L. (2019). Analysis of the Maximal a Posteriori Partition in the Gaussian Dirichlet Process Mixture Model.Bayesian Analysis 14(2), 477–494

2019
[43]

Rousseau, J. and K. Mengersen (2011). Asymptotic behaviour of the posterior distribution in overfitted mixture models.Journal of the Royal Statistical Society Series B: Statistical Methodol- ogy 73(5), 689–710

2011
[44]

Scricciolo, C. (2011). Posterior rates of convergence for Dirichlet mixtures of exponential power densities.Electronic Journal of Statistics 5, 270–308

2011
[45]

Sethuraman, J. (1994). A constructive definition of Dirichlet priors.Statistica sinica, 639–650

1994
[46]

Shen, W., S. T. Tokdar, and S. Ghosal (2013). Adaptive Bayesian multivariate density estima- tion with Dirichlet mixtures.Biometrika 100(3), 623–640

2013
[47]

van de Geer, S. (1996). Rates of convergence for the maximum likelihood estimator in mix- ture models.Journal of Nonparametric Statistics 6(4), 293–310

1996
[48]

van der Vaart, A. and J. A. Wellner (2023).Weak Convergence and Empirical Processes: With Applications to Statistics(Second ed.). Springer Nature

2023
[49]

van der Vaart, A. W. (2000).Asymptotic statistics, Volume 3. Cambridge university press

2000
[50]

Mukherjee, and X

Wei, Y., S. Mukherjee, and X. Nguyen (2023). MinimumΦ-distance estimators for finite mixing measures.arXiv preprint arXiv:2304.10052

work page arXiv 2023
[51]

Wei, Y. and X. Nguyen (2022). Convergence of de Finetti’s mixing measure in latent structure models for observed exchangeable sequences.The Annals of Statistics 50(4), 1859–1889. 21 A Proofs A.1 Proof of Lemma 1 Proof of Lemma 1.Consider point 1. FixN∈NandR∈(0, 1). Notice that by (5) we have that −log ∑j>N wj =− ∑N j=1 log(1−v j)∼Gamma(N,α), so that for e...

2022
[52]

P k and ˜Pk′ are independent, for every k,k ′ =0, . . . ,K
[53]

˜pk and ˜Qk are independent for every k=1, . . . ,K
[54]

,K and ( ˜p0, ˜p1,

˜Qk ∼DP (αP0(· ∩B k)/P0(Bk)) for k=0, . . . ,K and ( ˜p0, ˜p1, . . . ,˜pK) ∼Dirichlet(α 0,α 1, . . . ,αK) Proof.The equalities in (38), (39) and (40) follow immediately from the definitions in (35), (36) and (37). As regards point 1, notice that wj =v j j−1 ∏ i=1 (1−v i), ˜wj =v K+j K+j−1 ∏ i=K+1 (1−v i). for everyj≥1. Since{v j}j is a sequence of indepen...
[55]

,tK),t 0 =1− ∑K k=1 tk and∆ K is theK-dimensional simplex

GivenK 1 andK 2 as in the previous point, we need to upper bound ˆ ∆K t01 {RKt0≤rn} α0−1 ∏ k∈K1 tαk−1 k 1 {RKtk≤3rn} ! ∏ k∈K2 tαk−1 k 1 {|RKtk+p k−w∗ k |≤rn} ! dt, wheret = (t 1, . . . ,tK),t 0 =1− ∑K k=1 tk and∆ K is theK-dimensional simplex. Assume without loss of generality thatK∈ K 2 and, sinceα K <1, we have that tαK−1 K ≤ w∗ K −p K −r n RK αK−1 ≤2R ...

[1] [1]

Bystrova, J

Alamichel, L., D. Bystrova, J. Arbel, and G. Kon Kam King (2024). Bayesian mixture models (in) consistency for the number of clusters.Scandinavian Journal of Statistics 51(4), 1619–1660

2024

[2] [2]

Lijoi, G

Ascolani, F., A. Lijoi, G. Rebaudo, and G. Zanella (2023). Clustering consistency with Dirichlet process mixtures.Biometrika 110(2), 551–558

2023

[3] [3]

Bariletto, N., D. Le, A. Rinaldo, and N. Ho (2026). Convergence Rates for Latent Mix- ing Measures in Infinite Homoscedastic Location-Scale Mixture Models.arXiv preprint arXiv:2605.06845

work page internal anchor Pith review Pith/arXiv arXiv 2026

[4] [4]

Lijoi, L

Barrios, E., A. Lijoi, L. E. Nieto-Barajas, and I. Prünster (2013). Modeling with Normalized Random Measure Mixture Models.Statistical Science 28(3), 313–334

2013

[5] [5]

Chen, J. (1995). Optimal rate of convergence for finite mixture models.Annals of statistics 23(1), 221–233

1995

[6] [6]

Connor, R. J. and J. E. Mosimann (1969). Concepts of independence for proportions with a generalization of the Dirichlet distribution.Journal of the American Statistical Association 64(325), 194–206

1969

[7] [7]

Favaro, A

De Blasi, P ., S. Favaro, A. Lijoi, R. H. Mena, I. Prünster, and M. Ruggiero (2013). Are Gibbs- type priors the most natural generalization of the Dirichlet process?IEEE transactions on pattern analysis and machine intelligence 37(2), 212–229

2013

[8] [8]

Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems.The Annals of statistics, 209–230

1973

[9] [9]

Gassiat, E. and R. Van Handel (2014). The local geometry of finite mixtures.Transactions of the American Mathematical Society 366(2), 1047–1072

2014

[10] [10]

Genovese, C. R. and L. Wasserman (2000). Rates of convergence for the Gaussian mixture sieve.The Annals of Statistics 28(4), 1105–1127. 19

2000

[11] [11]

Ghosal, S., J. K. Ghosh, and A. W. van der Vaart (2000). Convergence rates of posterior distributions.Annals of Statistics, 500–531

2000

[12] [12]

Ghosal, S. and A. W. Van Der Vaart (2001). Entropies and rates of convergence for maximum likelihood and Bayes estimation for mixtures of normal densities.Annals of Statistics, 1233– 1263

2001

[13] [13]

Ghosal, S. and A. W. van der Vaart (2007). Posterior convergence rates of Dirichlet mixtures at smooth densities.Ann. Stat. 35, 697–723

2007

[14] [14]

Ghosal, S. and A. W. van der Vaart (2017).Fundamentals of Nonparametric Bayesian Inference. Cambridge University Press

2017

[15] [15]

Ghosh, J. K. and R. Ramamoorthi (2003).Bayesian nonparametrics. Springer

2003

[16] [16]

Gormley, I. C., T. B. Murphy, and A. E. Raftery (2023). Model-based clustering.Annual Review of Statistics and Its Application 10(1), 573–595

2023

[17] [17]

Ho, and X

Guha, A., N. Ho, and X. Nguyen (2021). On posterior contraction of parameters and inter- pretability in Bayesian mixture modeling.Bernoulli 27(4), 2159–2188

2021

[18] [18]

Ho, and X

Guha, A., N. Ho, and X. Nguyen (2023). On excess mass behavior in Gaussian mixture models with Orlicz-Wasserstein distances. InInternational Conference on Machine Learning, pp. 11847–11870. PMLR

2023

[19] [19]

Hairault, A., C. P . Robert, and J. Rousseau (2022). Evidence estimation in finite and infinite mixture models and applications.arXiv preprint arXiv:2205.05416

work page arXiv 2022

[20] [20]

Heinrich, P . and J. Kahn (2018). Strong identifiability and optimal minimax rates for finite mixture estimation.Annals of Statistics 46(6A)

2018

[21] [21]

Ho, N. and X. Nguyen (2016a). Convergence rates of parameter estimation for some weakly identifiable finite mixtures.The Annals of Statistics 44(6), 2726

[22] [22]

Ho, N. and X. Nguyen (2016b). On strong identifiability and convergence rates of parameter estimation in finite mixtures.Electronic Journal of Statistics, 271–307

[23] [23]

Nguyen, and Y

Ho, N., X. Nguyen, and Y. Ritov (2020). Robust estimation of mixing measures in finite mixture models.Bernoulli 26(2), 828–857

2020

[24] [24]

Ishwaran, H. and L. F. James (2001). Gibbs sampling methods for stick-breaking priors. Journal of the American Statistical Association 96(453), 161–173

2001

[25] [25]

Ishwaran, H. and L. F. James (2002). Approximate Dirichlet process computing in finite normal mixtures: smoothing and prior information.Journal of Computational and Graphical statistics 11(3), 508–532

2002

[26] [26]

Ishwaran, H., L. F. James, and J. Sun (2001). Bayesian model selection in finite mixtures by marginal density decompositions.Journal of the American Statistical Association 96(456), 1316– 1332

2001

[27] [27]

Ishwaran, H. and M. Zarepour (2002). Exact and approximate sum representations for the Dirichlet process.Canadian Journal of Statistics 30(2), 269–283

2002

[28] [28]

Korwar, R. M. and M. Hollander (1973). Contributions to the theory of Dirichlet processes. The Annals of Probability 1(4), 705–711

1973

[29] [29]

Lindsay, B. (1995). Mixture models: Theory, geometry, and applications. InNSFCBMS Re- gional Conference Series in Probability and Statistics, Volume 5. Institute of Mathematical Statistics

1995

[30] [30]

Lo, A. Y. (1984). On a class of Bayesian nonparametric estimates: I. Density estimates.The Annals of Statistics, 351–357. 20

1984

[31] [31]

Martin, R. (2012). Convergence rate for predictive recursion estimation of finite mixtures. Statistics & Probability Letters 82(2), 378–384

2012

[32] [32]

McLachlan, G. J. and K. E. Basford (1988). Mixture models. Inference and applications to clustering.Statistics: textbooks and monographs

1988

[33] [33]

McLachlan, G. J. and D. Peel (2000).Finite mixture models. John Wiley & Sons

2000

[34] [34]

Miller, J. W. and M. T. Harrison (2013). A simple example of Dirichlet process mixture in- consistency for the number of components.Adv. Neural Inf. Process. Syst. 26, 199–206

2013

[35] [35]

Miller, J. W. and M. T. Harrison (2014). Inconsistency of Pitman-Yor process mixtures for the number of components.J. Mach. Learn. Res. 15, 3333–3370

2014

[36] [36]

Miller, J. W. and M. T. Harrison (2018). Mixture models with a prior on the number of components.J. Am. Stat. Assoc. 113, 340–356

2018

[37] [37]

Nguyen, X. (2013). Convergence of latent mixing measures in finite and infinite mixture models.Ann. Stat. 41, 370–400

2013

[38] [38]

Nguyen, X. and Y. Wei (2026). Optimal transport based theory for latent structured models. arXiv preprint arXiv:2601.11465

work page arXiv 2026

[39] [39]

Ohn, I. and L. Lin (2023). Optimal Bayesian estimation of Gaussian mixtures with growing number of components.Bernoulli 29(2), 1195–1218

2023

[40] [40]

Pearson, K. (1894). Contributions to the mathematical theory of evolution.Proceedings of the Royal Society of London 54(326-330), 329–333

[41] [41]

Pitman, J. and M. Yor (1997). The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator.The Annals of Probability 25(2), 855–900

1997

[42] [42]

Rajkowski, L. (2019). Analysis of the Maximal a Posteriori Partition in the Gaussian Dirichlet Process Mixture Model.Bayesian Analysis 14(2), 477–494

2019

[43] [43]

Rousseau, J. and K. Mengersen (2011). Asymptotic behaviour of the posterior distribution in overfitted mixture models.Journal of the Royal Statistical Society Series B: Statistical Methodol- ogy 73(5), 689–710

2011

[44] [44]

Scricciolo, C. (2011). Posterior rates of convergence for Dirichlet mixtures of exponential power densities.Electronic Journal of Statistics 5, 270–308

2011

[45] [45]

Sethuraman, J. (1994). A constructive definition of Dirichlet priors.Statistica sinica, 639–650

1994

[46] [46]

Shen, W., S. T. Tokdar, and S. Ghosal (2013). Adaptive Bayesian multivariate density estima- tion with Dirichlet mixtures.Biometrika 100(3), 623–640

2013

[47] [47]

van de Geer, S. (1996). Rates of convergence for the maximum likelihood estimator in mix- ture models.Journal of Nonparametric Statistics 6(4), 293–310

1996

[48] [48]

van der Vaart, A. and J. A. Wellner (2023).Weak Convergence and Empirical Processes: With Applications to Statistics(Second ed.). Springer Nature

2023

[49] [49]

van der Vaart, A. W. (2000).Asymptotic statistics, Volume 3. Cambridge university press

2000

[50] [50]

Mukherjee, and X

Wei, Y., S. Mukherjee, and X. Nguyen (2023). MinimumΦ-distance estimators for finite mixing measures.arXiv preprint arXiv:2304.10052

work page arXiv 2023

[51] [51]

Wei, Y. and X. Nguyen (2022). Convergence of de Finetti’s mixing measure in latent structure models for observed exchangeable sequences.The Annals of Statistics 50(4), 1859–1889. 21 A Proofs A.1 Proof of Lemma 1 Proof of Lemma 1.Consider point 1. FixN∈NandR∈(0, 1). Notice that by (5) we have that −log ∑j>N wj =− ∑N j=1 log(1−v j)∼Gamma(N,α), so that for e...

2022

[52] [52]

P k and ˜Pk′ are independent, for every k,k ′ =0, . . . ,K

[53] [53]

˜pk and ˜Qk are independent for every k=1, . . . ,K

[54] [54]

,K and ( ˜p0, ˜p1,

˜Qk ∼DP (αP0(· ∩B k)/P0(Bk)) for k=0, . . . ,K and ( ˜p0, ˜p1, . . . ,˜pK) ∼Dirichlet(α 0,α 1, . . . ,αK) Proof.The equalities in (38), (39) and (40) follow immediately from the definitions in (35), (36) and (37). As regards point 1, notice that wj =v j j−1 ∏ i=1 (1−v i), ˜wj =v K+j K+j−1 ∏ i=K+1 (1−v i). for everyj≥1. Since{v j}j is a sequence of indepen...

[55] [55]

,tK),t 0 =1− ∑K k=1 tk and∆ K is theK-dimensional simplex

GivenK 1 andK 2 as in the previous point, we need to upper bound ˆ ∆K t01 {RKt0≤rn} α0−1 ∏ k∈K1 tαk−1 k 1 {RKtk≤3rn} ! ∏ k∈K2 tαk−1 k 1 {|RKtk+p k−w∗ k |≤rn} ! dt, wheret = (t 1, . . . ,tK),t 0 =1− ∑K k=1 tk and∆ K is theK-dimensional simplex. Assume without loss of generality thatK∈ K 2 and, sinceα K <1, we have that tαK−1 K ≤ w∗ K −p K −r n RK αK−1 ≤2R ...