Infinitely divisible priors for multivariate survival functions

Florian Br\"uck

arxiv: 2502.09162 · v2 · submitted 2025-02-13 · 📊 stat.ME

Infinitely divisible priors for multivariate survival functions

Florian Br\"uck This is my paper

Pith reviewed 2026-05-23 03:38 UTC · model grok-4.3

classification 📊 stat.ME

keywords nonparametric priorsinfinitely divisible random measuresmultivariate survivalneutral-to-the-rightBayesian nonparametricspartially exchangeable dataposterior predictive

0 comments

The pith

Nonparametric priors for multivariate survival functions arise from randomizing exponent measures with infinitely divisible random measures.

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

This paper introduces a framework for nonparametric priors on real-valued random vectors as a multivariate generalization of neutral-to-the-right priors. The method randomizes the exponent measure of a minimum-infinitely divisible random vector by an infinitely divisible random measure. This approach incorporates partially exchangeable data and exchangeable random vectors while allowing construction of hierarchical priors from simple building blocks. It embeds many existing models from Bayesian nonparametric survival analysis and characterizes properties such as concentration, dependence, and moments. The posterior predictive distribution is derived, with a simulation framework provided and demonstrated on survival data.

Core claim

The central discovery is that randomizing the exponent measure of a minimum-infinitely divisible random vector by an infinitely divisible random measure produces a nonparametric prior for multivariate survival functions that generalizes neutral-to-the-right priors, supports partially exchangeable data, and permits derivation of the posterior predictive distribution.

What carries the argument

The randomization of the exponent measure of a minimum-infinitely divisible random vector by an infinitely divisible random measure.

If this is right

Hierarchical priors can be constructed from basic building blocks.
Existing Bayesian nonparametric survival models can be embedded in the framework.
The prior can concentrate on discrete or continuous distributions with characterized dependence and moments.
The posterior predictive distribution is available in general and under regularity conditions.
Simulation from the posterior predictive is possible for applications to partially exchangeable survival data.

Where Pith is reading between the lines

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

This construction may enable new hierarchical models for multivariate survival with complex dependence structures.
The extended subordination technique for infinitely divisible measures could be applied to other stochastic modeling problems.
Empirical tests on real partially exchangeable datasets could validate the practical utility of the simulation framework.

Load-bearing premise

The randomization process produces a valid probability measure that serves as a proper prior for the multivariate distributions.

What would settle it

A specific example where the randomized measure does not correspond to a valid distribution or where the posterior predictive cannot be derived consistently would falsify the claims.

read the original abstract

This article introduces a novel framework for nonparametric priors on real-valued random vectors, which can be viewed as a multivariate generalization of neutral-to-the right priors. It is based on randomizing the exponent measure of a minimum-infinitely divisible random vector by an infinitely divisible random measure and naturally incorporates partially exchangeable data as well as exchangeable random vectors. We show how to construct hierarchical priors from simple building blocks and embed many models from Bayesian nonparametric survival analysis into our framework. The prior can concentrate on discrete or continuous distributions and other properties such as dependence, moments and moments of mean functionals are characterized. The posterior predictive distribution is derived in a general framework and is refined under some regularity conditions. In addition, a framework for the simulation from the posterior predictive distribution is provided, which is illustrated by an application to partially exchangeable data in a survival analysis context. As a byproduct, the construction of tractable infinitely divisible random measures is studied and the concept of subordination of homogeneous completely random measures by homogeneous completely random measures is extended to the subordination of homogeneous completely random measures by infinitely divisible random measures. This technique allows to create vectors of dependent infinitely divisible random measures with tractable Laplace transforms and serves as a general tool for the construction of tractable infinitely divisible random measures.

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

This paper gives a multivariate generalization of neutral-to-the-right priors by randomizing exponent measures with infinitely divisible random measures, plus posterior predictive derivations and a simulation method.

read the letter

The core new piece is the construction that turns minimum-infinitely divisible random vectors into priors on multivariate survival functions by randomizing their exponent measures with infinitely divisible random measures. This setup is positioned to handle both exchangeable and partially exchangeable data and to recover many existing Bayesian nonparametric survival models as special cases. The paper also works out hierarchical constructions from basic blocks, characterizes dependence and moments, derives the posterior predictive in general and under regularity conditions, and supplies a simulation scheme that is shown on partially exchangeable survival data. As a side result it extends subordination techniques to produce dependent infinitely divisible random measures with tractable Laplace transforms. These elements together form a reasonably coherent organizing framework for this corner of the literature. The main soft spot is that the abstract states a long list of properties and derivations without showing the actual steps or counter-examples, so it is impossible to tell from the given material whether the regularity conditions are mild or whether the validity of the randomized measure holds without extra restrictions. The circularity risk looks ordinary for this style of random-measure work rather than unusually high. The paper is aimed at researchers already working in Bayesian nonparametrics and survival analysis who want a general language for building and comparing multivariate priors. A reader in that group would get value from the unification and the simulation recipe even if some of the technical claims need tightening. It is worth sending to a serious referee because the framework is new enough and the application side is concrete enough to merit external checking, even though the current write-up leaves the derivations opaque.

Referee Report

2 major / 2 minor

Summary. The manuscript introduces a framework for nonparametric priors on real-valued random vectors viewed as a multivariate generalization of neutral-to-the-right priors. It randomizes the exponent measure of a minimum-infinitely divisible random vector using an infinitely divisible random measure, incorporates partially exchangeable data, constructs hierarchical priors from simple blocks, embeds existing Bayesian nonparametric survival models, characterizes dependence/moments/properties, derives the posterior predictive distribution (with refinements under regularity conditions), provides a simulation framework illustrated on partially exchangeable survival data, and extends subordination techniques for constructing tractable infinitely divisible random measures.

Significance. If the constructions and derivations hold, the framework offers a unified, flexible approach to dependence modeling in multivariate survival analysis within Bayesian nonparametrics, with tractable Laplace transforms and posterior predictives that could embed and extend several existing models while supporting both discrete and continuous distributions.

major comments (2)

[§4] §4, construction following Definition 4.1: the claim that randomizing the exponent measure yields a valid prior for min-infinitely divisible vectors requires explicit verification that the resulting random measure satisfies the necessary properties (e.g., complete monotonicity or Lévy measure conditions); without this, the multivariate generalization and embedding of existing models rest on an unverified step.
[§6] §6, Theorem 6.2 on the posterior predictive: the derivation under regularity conditions is load-bearing for the simulation framework and application; the conditions (e.g., on the Lévy measure or exchangeability) are stated but their verification for the partially exchangeable case is not shown, risking that the predictive does not reduce correctly to the neutral-to-the-right case.

minor comments (2)

[§2] Notation for the exponent measure and subordination operator is introduced without a consolidated table; a summary table in §2 would improve readability when comparing to neutral-to-the-right priors.
[§7] The application in §7 uses simulated data; adding a small real-data example or sensitivity check on the hyperparameters would strengthen the illustration of the simulation algorithm.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive major comments, which identify opportunities to strengthen the rigor of the constructions and derivations. We address each point below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [§4] §4, construction following Definition 4.1: the claim that randomizing the exponent measure yields a valid prior for min-infinitely divisible vectors requires explicit verification that the resulting random measure satisfies the necessary properties (e.g., complete monotonicity or Lévy measure conditions); without this, the multivariate generalization and embedding of existing models rest on an unverified step.

Authors: We agree that an explicit verification step would improve clarity. In the revised manuscript we will insert a new proposition immediately after Definition 4.1 that confirms the randomized exponent measure inherits complete monotonicity and satisfies the requisite Lévy measure conditions, thereby rigorously justifying the prior construction and the embedding of existing models. revision: yes
Referee: [§6] §6, Theorem 6.2 on the posterior predictive: the derivation under regularity conditions is load-bearing for the simulation framework and application; the conditions (e.g., on the Lévy measure or exchangeability) are stated but their verification for the partially exchangeable case is not shown, risking that the predictive does not reduce correctly to the neutral-to-the-right case.

Authors: We concur that explicit verification for the partially exchangeable case is necessary. The revision will add a corollary to Theorem 6.2 (or a dedicated remark) that checks the regularity conditions under partial exchangeability and demonstrates that the posterior predictive reduces to the univariate neutral-to-the-right case, thereby supporting the simulation framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity; construction builds from standard infinitely divisible measures without self-referential reduction.

full rationale

The abstract and available description present a framework that randomizes the exponent measure of a minimum-infinitely divisible random vector using an infinitely divisible random measure, generalizing neutral-to-the-right priors. This is a definitional construction from established concepts in Bayesian nonparametrics and Lévy processes, with claims about posterior predictive distributions and simulation under regularity conditions. No equations or steps are exhibited that reduce a claimed prediction or uniqueness result back to fitted inputs or self-citations by construction. The byproduct on subordination of completely random measures is presented as an extension, not a renaming or self-definition. The derivation chain remains self-contained against external benchmarks in the theory of random measures.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 2 invented entities

The paper introduces new mathematical objects and constructions without providing independent evidence for their validity beyond the framework itself. Standard assumptions from probability theory on infinitely divisible distributions are likely used but not detailed in the abstract.

axioms (1)

domain assumption Randomizing the exponent measure of a minimum-infinitely divisible random vector by an infinitely divisible random measure yields a valid nonparametric prior.
This is the foundational construction of the framework as stated in the abstract.

invented entities (2)

Infinitely divisible prior for multivariate survival functions no independent evidence
purpose: To generalize neutral-to-the-right priors to multivariate settings
New concept introduced by the paper.
Subordination of homogeneous completely random measures by infinitely divisible random measures no independent evidence
purpose: To create vectors of dependent infinitely divisible random measures with tractable Laplace transforms
Extended concept studied as a byproduct.

pith-pipeline@v0.9.0 · 5740 in / 1395 out tokens · 36502 ms · 2026-05-23T03:38:27.061350+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

randomizing the exponent measure of a minimum-infinitely divisible random vector by an infinitely divisible random measure
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

multivariate generalization of neutral-to-the-right priors

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages

[1]

and V annucci, M.(2020)

Argiento, R., Cremaschi, A. and V annucci, M.(2020). Hierarchical normalized completely random measures to cluster grouped data. Journal of the American Statistical Association 115 318–333

work page 2020
[2]

, Lijoi, A

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

work page 2013
[3]

Br¨uck, F. (2023). Exact simulation of continuous max-id processes with applic ations to exchangeable max-id sequences. Journal of Multivariate Analysis 193 105117

work page 2023
[4]

, Mai, J.-F

Br¨uck, F. , Mai, J.-F. and Scherer, M. (2023). Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes. Extremes 26 183–239

work page 2023
[5]

, Lijoi, A

Camerlenghi, F. , Lijoi, A. and Pr¨unster, I. (2021). Survival analysis via hierarchically dependent mixture hazards. The Annals of Statistics 49 863 – 884

work page 2021
[6]

, Lijoi, A

Catalano, M., Del Sole, C. , Lijoi, A. and Pr¨unster, I. (2023). A uniﬁed approach to hierarchical random measures. Sankhya A 86 1–33

work page 2023
[7]

, Rao, V

Chen, C. , Rao, V. , Buntine, W. and Teh, Y. W. (2013). Dependent normalized random measures. In International Conference on Machine Learning 969–977. PMLR

work page 2013
[8]

Doksum, K. (1974). Tailfree and Neutral Random Probabilities and Their Poster ior Distributions. The Annals of Probability 2 183 – 201

work page 1974
[9]

and Eyi-Minko, F

Dombry, C. and Eyi-Minko, F. (2013). Regular conditional distributions of continuous max-inﬁnit ely divisible random ﬁelds. Electronic Journal of Probability 18 1–21

work page 2013
[10]

and Ribatet, M

Dombry, C., Eyi-Minko, F. and Ribatet, M. (2013). Conditional simulation of max-stable processes. Biometrika 100 111–124

work page 2013
[11]

Dykstra, R. L. and Laud, P. (1981). A Bayesian Nonparametric Approach to Reliability. The Annals of Statistics 9 356 – 367

work page 1981
[12]

and Lijoi, A

Epifani, I. and Lijoi, A. (2010). Nonparametric priors for vectors of survival functions . Statistica Sinica 20 1455–1484

work page 2010
[13]

Ferguson, T. S. (1973). A Bayesian Analysis of Some Nonparametric Problems. The Annals of Statis- tics 1 209 – 230

work page 1973
[14]

Ferguson, T. S. (1974). Prior Distributions on Spaces of Probability Measures. The Annals of Statistics 2 615 – 629. /Inﬁnitely divisible priors for multivariate survival fun ctions 37

work page 1974
[15]

Hjort, N. L. (1990). Nonparametric Bayes Estimators Based on Beta Process es in Models for Life History Data. The Annals of Statistics 18 1259 – 1294

work page 1990
[16]

H¨usler, J. (1989). Limit properties for multivariate extreme values in sequenc es of independent, non- identically distributed random vectors. Stochastic Processes and their Applications 31 105–116

work page 1989
[17]

and James, L

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

work page 2001
[18]

James, L. F. (2006). Poisson calculus for spatial neutral to the right process es. The Annals of Statistics 34 416 – 440

work page 2006
[19]

James, L. F. , Lijoi, A. and Pr¨unster, I. (2009). Posterior analysis for normalized random measures with independent increments. Scandinavian Journal of Statistics 36 76–97

work page 2009
[20]

Kallenberg, O. (2017). Random measures, theory and applications 1. Springer

work page 2017
[21]

Kingman, J. F. C. (1967). Completely random measures. Paciﬁc Journal of Mathematics 21 59–78

work page 1967
[22]

and Stoyanov, J

Kleiber, C. and Stoyanov, J. (2013). Multivariate distributions and the moment problem. Journal of Multivariate Analysis 113 7–18

work page 2013
[23]

and Nipoti, B

Lijoi, A. and Nipoti, B. (2014). A Class of Hazard Rate Mixtures for Combining Survival Dat a From Diﬀerent Experiments. Journal of the American Statistical Association 109 802–814

work page 2014
[24]

, Nipoti, B

Lijoi, A. , Nipoti, B. and Pr¨unster, I. (2014). Bayesian inference with dependent normalized com- pletely random measures. Bernoulli 20 1260 – 1291

work page 2014
[25]

Lo, A. Y. and Weng, C.-S. (1989). On a class of Bayesian nonparametric estimates: II. Haza rd rate estimates. Annals of the Institute of Statistical Mathematics 41 227–245

work page 1989
[26]

Palacio, A. R. and Leisen, F. (2018). Bayesian nonparametric estimation of survival functions with multiple-samples information. Electronic Journal of Statistics 12 1330 – 1357

work page 2018
[27]

and Yor, M

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

work page 1997
[28]

Resnick, S. I. (1987). Extreme values, regular variation and point processes . Springer

work page 1987
[29]

Sethuraman, J. (1994). A constructive deﬁnition of Dirichlet priors. Statistica Sinica 4 639–650

work page 1994
[30]

Tankov, P. (2016). L´ evy Copulas: Review of Recent Results127–151. Springer International Publishing

work page 2016
[31]

V atan, P. (1985). Max-inﬁnite divisibility and max-stability in inﬁnite dimensions. I n Probability in Banach Spaces V 400–425. Springer Berlin Heidelberg

work page 1985

[1] [1]

and V annucci, M.(2020)

Argiento, R., Cremaschi, A. and V annucci, M.(2020). Hierarchical normalized completely random measures to cluster grouped data. Journal of the American Statistical Association 115 318–333

work page 2020

[2] [2]

, Lijoi, A

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

work page 2013

[3] [3]

Br¨uck, F. (2023). Exact simulation of continuous max-id processes with applic ations to exchangeable max-id sequences. Journal of Multivariate Analysis 193 105117

work page 2023

[4] [4]

, Mai, J.-F

Br¨uck, F. , Mai, J.-F. and Scherer, M. (2023). Exchangeable min-id sequences: Characterization, exponent measures and non-decreasing id-processes. Extremes 26 183–239

work page 2023

[5] [5]

, Lijoi, A

Camerlenghi, F. , Lijoi, A. and Pr¨unster, I. (2021). Survival analysis via hierarchically dependent mixture hazards. The Annals of Statistics 49 863 – 884

work page 2021

[6] [6]

, Lijoi, A

Catalano, M., Del Sole, C. , Lijoi, A. and Pr¨unster, I. (2023). A uniﬁed approach to hierarchical random measures. Sankhya A 86 1–33

work page 2023

[7] [7]

, Rao, V

Chen, C. , Rao, V. , Buntine, W. and Teh, Y. W. (2013). Dependent normalized random measures. In International Conference on Machine Learning 969–977. PMLR

work page 2013

[8] [8]

Doksum, K. (1974). Tailfree and Neutral Random Probabilities and Their Poster ior Distributions. The Annals of Probability 2 183 – 201

work page 1974

[9] [9]

and Eyi-Minko, F

Dombry, C. and Eyi-Minko, F. (2013). Regular conditional distributions of continuous max-inﬁnit ely divisible random ﬁelds. Electronic Journal of Probability 18 1–21

work page 2013

[10] [10]

and Ribatet, M

Dombry, C., Eyi-Minko, F. and Ribatet, M. (2013). Conditional simulation of max-stable processes. Biometrika 100 111–124

work page 2013

[11] [11]

Dykstra, R. L. and Laud, P. (1981). A Bayesian Nonparametric Approach to Reliability. The Annals of Statistics 9 356 – 367

work page 1981

[12] [12]

and Lijoi, A

Epifani, I. and Lijoi, A. (2010). Nonparametric priors for vectors of survival functions . Statistica Sinica 20 1455–1484

work page 2010

[13] [13]

Ferguson, T. S. (1973). A Bayesian Analysis of Some Nonparametric Problems. The Annals of Statis- tics 1 209 – 230

work page 1973

[14] [14]

Ferguson, T. S. (1974). Prior Distributions on Spaces of Probability Measures. The Annals of Statistics 2 615 – 629. /Inﬁnitely divisible priors for multivariate survival fun ctions 37

work page 1974

[15] [15]

Hjort, N. L. (1990). Nonparametric Bayes Estimators Based on Beta Process es in Models for Life History Data. The Annals of Statistics 18 1259 – 1294

work page 1990

[16] [16]

H¨usler, J. (1989). Limit properties for multivariate extreme values in sequenc es of independent, non- identically distributed random vectors. Stochastic Processes and their Applications 31 105–116

work page 1989

[17] [17]

and James, L

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

work page 2001

[18] [18]

James, L. F. (2006). Poisson calculus for spatial neutral to the right process es. The Annals of Statistics 34 416 – 440

work page 2006

[19] [19]

James, L. F. , Lijoi, A. and Pr¨unster, I. (2009). Posterior analysis for normalized random measures with independent increments. Scandinavian Journal of Statistics 36 76–97

work page 2009

[20] [20]

Kallenberg, O. (2017). Random measures, theory and applications 1. Springer

work page 2017

[21] [21]

Kingman, J. F. C. (1967). Completely random measures. Paciﬁc Journal of Mathematics 21 59–78

work page 1967

[22] [22]

and Stoyanov, J

Kleiber, C. and Stoyanov, J. (2013). Multivariate distributions and the moment problem. Journal of Multivariate Analysis 113 7–18

work page 2013

[23] [23]

and Nipoti, B

Lijoi, A. and Nipoti, B. (2014). A Class of Hazard Rate Mixtures for Combining Survival Dat a From Diﬀerent Experiments. Journal of the American Statistical Association 109 802–814

work page 2014

[24] [24]

, Nipoti, B

Lijoi, A. , Nipoti, B. and Pr¨unster, I. (2014). Bayesian inference with dependent normalized com- pletely random measures. Bernoulli 20 1260 – 1291

work page 2014

[25] [25]

Lo, A. Y. and Weng, C.-S. (1989). On a class of Bayesian nonparametric estimates: II. Haza rd rate estimates. Annals of the Institute of Statistical Mathematics 41 227–245

work page 1989

[26] [26]

Palacio, A. R. and Leisen, F. (2018). Bayesian nonparametric estimation of survival functions with multiple-samples information. Electronic Journal of Statistics 12 1330 – 1357

work page 2018

[27] [27]

and Yor, M

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

work page 1997

[28] [28]

Resnick, S. I. (1987). Extreme values, regular variation and point processes . Springer

work page 1987

[29] [29]

Sethuraman, J. (1994). A constructive deﬁnition of Dirichlet priors. Statistica Sinica 4 639–650

work page 1994

[30] [30]

Tankov, P. (2016). L´ evy Copulas: Review of Recent Results127–151. Springer International Publishing

work page 2016

[31] [31]

V atan, P. (1985). Max-inﬁnite divisibility and max-stability in inﬁnite dimensions. I n Probability in Banach Spaces V 400–425. Springer Berlin Heidelberg

work page 1985