Original: Approximating the clusters’ prior distribution in Bayesian nonparametric models
Revised: Approximating the clusters’ prior distribution in Bayesian nonparametric models

Original: Anonymous Authors

Original: Anonymous Institution

Original: Abstract
Revised: Abstract

Original:  Bayesian nonparametric models may induce clustering on the data. Knowledge of theprior distribution induced on the number of clusters isthus key for prior speciﬁcationand calibration.
Revised: In Bayesian nonparametrics, knowledge of the prior distribution induced on the number of clusters is key for prior speciﬁcation and calibration.

Original: However, evaluating this prior is infamously diﬃcult even for moderatesample size.
Revised: However, evaluating this prior is infamously diﬃcult even for moderate sample size.

Original: We evaluate several statistical approximations to the prior distribution onthe number of clusters for Gibbs-type processes, a class including the Pitman–Yor processand the normalized generalized gamma process.
Revised: We evaluate several statistical approximations to the prior distribution on the number of clusters for Gibbs-type processes, a class including the Pitman–Yor process and the normalized generalized gamma process.

Original: We introduce a new approximation basedon the predictive distribution of Gibbs-type process, which compares favourably with theexisting methods.
Revised: We introduce a new approximation based on the predictive distribution of Gibbs-type process, which compares favourably with the existing methods.

Original: We thoroughly discuss the limitations of these various approximationsby comparing them against an exact implementation of the prior distribution of the numberof clusters.
Revised: We thoroughly discuss the limitations of these various approximations by comparing them against an exact implementation of the prior distribution of the number of clusters.

Original: 1. Introduction
Revised: 1. Introduction

Original: Bayesian nonparametric (BNP) models may induce clustering on the data, for instancewhen they are used in mixture models.
Revised: Bayesian nonparametric (BNP) models may induce clustering on the data, for instance when they are used in mixture models.

Original: Knowledge of the prior distribution induced onthe number of clusters is thus key for application-driven prior speciﬁcation.
Revised: Knowledge of the prior distribution induced on the number of clusters is thus key for application-driven prior speciﬁcation.

Original: In this note,we focus on the large class of Gibbs-type processes (De Blasi et al., 2015), which containsas special cases the Dirichlet process, the Pitman–Yor process (PY) and the normalizedgeneralized gamma process (NGG), to cite a few.
Revised: In this note, we focus on the large class of Gibbs-type processes (De Blasi et al., 2015), which contains as special cases the Dirichlet process, the Pitman–Yor process (PY) and the normalized generalized gamma process (NGG), to cite a few.

Revised: We study the implicit prior on the number of clusters induced by these processes and provide a calibration method.

Original: Gibbs-type processes can be deﬁnedvia Poisson–Kingman partitions, see Pitman (2003).
Revised: Gibbs-type processes can be deﬁned via Poisson–Kingman partitions, see Pitman (2003).

Original: They are parameterized by a discountparameter σ < 1 and a nonnegative function h deﬁned on the positive half-line.
Revised: They are parameterized by a discount parameter σ < 1 and a nonnegative function h deﬁned on the positive half-line.

Original: We focus onσ ≥ 0 parameter, a restriction which entails inﬁnite-dimensionality of the processes understudy, thus ruling out Gibbs-type priors with ﬁnitely many species that prevail when σ < 0(see Pitman, 2006).
Revised: We focus on σ ≥ 0 parameter, a restriction which entails inﬁnite-dimensionality of the processes under study, thus ruling out Gibbs-type priors with ﬁnitely many species that prevail when σ <(see Pitman, 2006).

Original: Gibbs-type priors are almost surely discrete, taking the form of a weighted countablesum of Dirac masses.
Revised: Gibbs-type priors are almost surely discrete, taking the form of a weighted countable sum of Dirac masses.

Original: Constructive representations of the weight vector (convenient forsampling) include the stick-breaking representation (see Sethuraman, 1994;Pitman and Yor,1997;
Revised: Constructive representations of the weight vector (convenient for sampling) include the stick-breaking representation (see Sethuraman, 1994;

Original:  Favaro et al., 2016) and the Ferguson–Klass representation (Ferguson and Klass, 1972,only for some speciﬁc cases of Gibbs-type processes).
Revised: Pitman and Yor, 1997; Favaro et al., 2016) and the Ferguson–Klass representation (Ferguson and Klass, 1972, only for some speciﬁc cases of Gibbs-type processes).

Original: Because of discreteness, modelingobservations X n = ( X 1 , . . .
Revised: Because of discreteness, modeling observations X n = ( X 1 , . . .

Original: , X n ) with a Gibbs-type prior creates ties (clusters) in theobservations with positive probability.
Revised: , X n ) with a Gibbs-type prior creates ties (clusters) in the observations with positive probability.

Original: This clustering procedure inherits a prior distributionfrom the Gibbs-type prior.
Revised: This clustering procedure inherits a prior distribution from the Gibbs-type prior.

Original: Denote by K n ≤ n the number of clusters implied by the model in data X n .
Revised: Denote by K n ≤ n the number of clusters implied by the model in data X n .

Original: Weconsider the prior distribution for the number of clusters K n , supported on the set ofintegers { 1 , . . . , n } .
Revised: We consider the prior distribution for the number of clusters K n , supported on the set of integers { 1 , . . . , n } .

Original: The probability mass function ( p n,k ) 1 ≤ k ≤ n of this random variable has a fairly simple expression in terms of two triangular arrays of reals, denoted by V n,k and
Revised: The probability mass function ( p n,k ) 1 ≤ k ≤ n of this random variable has a fairly simple expression in terms of two triangular arrays of reals, denoted by V n,k and

Original: C n,k , for any positive integers n, k such that 1 ≤ k ≤ n (see De Blasi et al., 2015)
Revised: C n,k , for any positive integers n, k such that 1 ≤ k ≤ n (see De Blasi et al., 2015)

Original: [Equation]
Revised: [Equation]

Original: The ( V n,k ) triangular array encodes information about σ and the h function, while the ( C n,k ) triangular array, called generalized factorial coeﬃcients (Charalambides, 2005), only depends on σ .
Revised: The ( V n,k ) triangular array encodes information about σ and the h function, while the ( C n,k ) triangular array, called generalized factorial coeﬃcients (Charalambides, 2005), only depends on σ .

Original: More speciﬁcally,
Revised: More speciﬁcally,

Original: [Equation]
Revised: [Equation]

Original: where Γ is the Gamma function, f σ is the density of a positive σ -stable random variable, and( x ) n = x ( x + 1) . . .
Revised: where Γ is the Gamma function, f σ is the density of a positive σ -stable random variable, and ( x ) n = x ( x + 1) . . .

Original: ( x + n − 1) denotes the increasing factorial coeﬃcient, for any real number x andinteger n .
Revised: ( x + n − 1) denotes the increasing factorial coeﬃcient, for any real number x and integer n .

Original: See Appendix A for a more precise description of coeﬃcients V n,k , and C n,k .
Revised: See Appendix A for a more precise description of coeﬃcients V n,k , and C n,k .

Original: The specialcases of PY and NGG processes are provided in Equations (6) and (7), respectively.
Revised: The special cases of PY and NGG processes are provided in Equations (6) and (7), respectively.

Original: This note addresses the practical evaluation of the prior weights ( p n,k ) 1 ≤ k ≤ n , a notoriouslydiﬃcult task, even for moderate sample sizes of order 10 2 or 10 3 .
Revised: This note addresses the practical evaluation of the prior weights ( p n,k ) 1 ≤ k ≤ n , a notoriously diﬃcult task, even for moderate sample sizes of order 10 2 or 10 3 .

Original: The diﬃculty stems from theevaluation of the ( V n,k ) triangular array on the one hand, and of the generalized factorial coeﬃcients( C n,k ) on the other.
Revised: The diﬃculty stems from the evaluation of the ( V n,k ) triangular array on the one hand, and of the generalized factorial coeﬃcients ( C n,k ) on the other.

Original: Many of the summands involved in the computation of C n,k and V n,k overﬂowthe double precision exponent range, both in the positive and the negative domain, preventing therecourse to the usual log-transformation to address the risk of overﬂow.
Revised: Many of the summands involved in the computation of C n,k and V n,k overﬂow the double precision exponent range, both in the positive and the negative domain, preventing the recourse to the usual log-transformation to address the risk of overﬂow.

Original: Note that the arduousness of the ( V n,k ) evaluation actually depends on the speciﬁc Gibbs-typeclass under study.
Revised: Note that the arduousness of the ( V n,k ) evaluation actually depends on the speciﬁc Gibbs-type class under study.

Original: For instance, (2) greatly simpliﬁes in the case of the PY process while it remainsunwieldy for the NGG process (see (6) and (7) in Appendix).
Revised: For instance, (2) greatly simpliﬁes in the case of the PY process while it remains unwieldy for the NGG process (see (6) and (7) in Appendix).

Original: For this reason, illustrations in thisnote cover the arduous NGG case, while the more manageable PY case is treated in the Appendix.
Revised: For this reason, illustrations in this note cover the arduous NGG case, while the more manageable PY case is treated in the Appendix.

Original: Contributions 
Revised: Contributions Sec

Original: Sec 2 We propose a new approximation of the prior distribution on K n based on approximating thepredictive distribution of the process.
Revised: We propose a new approximation of the prior distribution on K n based on approximating the predictive distribution of the process.

Original: Three other existing approaches are also implemented:the ﬁnite-dimensional multinomial processes of Lijoi et al.
Revised: Three other existing approaches are also implemented: the ﬁnite-dimensional multinomial processes of Lijoi et al.

Original: (2020a,b); and the truncation oftwo sum representations of the processes, that are stick-breaking (Sethuraman, 1994; Pitmanand Yor, 1997; Favaro et al., 2016) and Ferguson–Klass (Ferguson and Klass, 1972).
Revised: (2020a,b); and the truncation of two sum representations of the processes, that are stick-breaking (Sethuraman, 1994; Pitman and Yor, 1997; Favaro et al., 2016) and Ferguson–Klass (Ferguson and Klass, 1972).

Original: Code ismade available in Julia1 and [-R. 
-]
Revised: Code is made available in Julia and [-R. 
-]R through the package GibbsTypePriors 1 .

Original: Sec 3.1 We explore the quality of the proposed approximations by comparing the induced prior probability mass function on K n against that obtained with an exact implementation based onarbitrary precision arithmetics (Kon Kam King et al.).
Revised: Sec 3.1 We explore the quality of the proposed approximations by comparing the induced prior probability mass function on K n against that obtained with an exact implementation based on arbitrary precision arithmetics 1 .

Original: We ﬁnd that our new approximationbased on the predictive distribution performs favourably when compared to the others, exceptfor a parameter range where the ﬁnite-dimensional multinomial process fares better.
Revised: We ﬁnd that our new approximation based on the predictive distribution performs favourably when compared to the others, except for a parameter range where the ﬁnite-dimensional multinomial process fares better.

Original: Sec 3.2 We demonstrate how knowledge of the prior on K n can inform the delicate question of specifying hyperparameters on BNP priors, illustrating how the ﬁrst two prior moments of K nrelate to each other as a function of the hyperparameters.
Revised: Sec 3.2 We demonstrate how knowledge of the prior on K n can inform the delicate question of specifying hyperparameters on BNP priors, illustrating how the ﬁrst two prior moments of K n relate to each other as a function of the hyperparameters.

Original: The functionalities proposed here are highly relevant for probabilistic programming languagessupporting BNP modelling, such as Turing (see Turing.jl, Ge etal., 2018), implemented in Julia andmaintaining modelling with discrete random probability measures (Trapp et al., 2019), and Edward(Tran et al. 
Revised: The functionalities proposed here are highly relevant for probabilistic programming languages supporting BNP modelling, such as Turing (see Turing.jl, Ge et al. , et al

Original: https://github.com/AABI-Gibbs-type-priors/AABI-Gibbs-type-priors 2.
Revised: Edward

Original: https://github.com/AABI-Gibbs-type-priors/AABI-RGibbs-type-priors
Revised: Tran et al.

Original: 2. Statistical approximations
Revised: 2. Statistical approximations

Original: In this section, we present four approaches, the ﬁrst of which is novel, leading to approximations ofthe prior distribution of K n , for diﬀerent families of Gibbs-type processes.
Revised: In this section, we present four approaches, the ﬁrst of which is novel, leading to approximations of the prior distribution of K n , for diﬀerent families of Gibbs-type processes.

Original: 2.1. Predictive distribution approximation
Revised: 2.1. Predictive distribution approximation

Original: The ﬁrst approximationwe consider relies on the second order approximation suggested by Arbelet al.
Revised: The ﬁrst approximation relies on a second order approximation suggested by Arbel et al.

Original: ; Arbel and Favaro (2020).
Revised: ; Arbel and Favaro (2020).

Original: This requires the function h to be continuously diﬀerentiable(with derivative denoted by h (cid:48) ) in order to deﬁne a second order term β n,k :=
Revised: This requires the function h to be continuously diﬀerentiable (with derivative denoted by h (cid:48) ) in order to deﬁne a second order term β n,k :=

Original: φ h ( n/k 1 /σ ), withφ h ( t ) : = − t h (cid:48) ( t ) / h ( t ), in the approximation.
Revised: φ h ( n/k 1 /σ ), with φ h ( t ) := − t h (cid:48) ( t ) / h ( t ).

Original: Theorem 1 in Arbel and Favaro (2020) provides aso-called second order approximation for the predictive distribution of the Gibbs-type process in theform of the ratio of successive V n,k parameters.
Revised: Theorem 1 in Arbel and Favaro (2020) provides a so-called second order approximation for the predictive distribution of the Gibbs-type process in the form of the ratio of successive V n,k parameters.

Original: This, in turn, suggests the following result on theratio of consecutive prior weights p n +1 ,k :
Revised: This, in turn, suggests the following result on the ratio of consecutive prior weights p n +1 ,k :

Original: Proposition 1 Ratios of consecutive weights satisfy the large n approximation for any 1 ≤ k ≤ n :
Revised: Proposition 1 Ratios of consecutive weights satisfy the large n approximation for any 1 ≤ k ≤ n :

Original: [Equation]
Revised: [Equation]

Original: Based on the large n approximation of Proposition 1, a simple recursive algorithm  can be employed inorder to compute an approximation to the prior weights p n +1 ,k without resorting to the cumbersomecomputation of the V n,k parameters.
Revised: Based on the large n approximation of Proposition 1, a simple recursive algorithm (see Appendix B.1.2) can be designed to compute an approximation to the prior weights p n +1 ,k without resorting to the cumbersome computation of the V n,k parameters.

Original: In the speciﬁc case of the NGG, β n,k takes a simple form (seedetails in Appendix B.1).
Revised: In the speciﬁc case of the NGG, β n,k takes a simple form (see details in Appendix B.1).

Original: 2.2. Finite-dimensional approximation
Revised: 2.2. Finite-dimensional approximation

Original: Finite-dimensional alternatives to BNP priors have been developed for cases where the assumptionthat the number of clusters grows with the sample size does not hold.
Revised: Finite-dimensional alternatives to BNP priors have been developed for cases where the assumption that the number of clusters grows with the sample size does not hold.

Original: Their ﬂexibility is suchthat they could also be considered as approximations to certain nonparametric priors.
Revised: Their ﬂexibility is such that they could also be considered as approximations to certain nonparametric priors.

Original: The recentworks by Lijoi et al.
Revised: The recent works by Lijoi et al.

Original: (2020a,b) develop ﬁnite-dimensional multinomial versions of the PY processand normalized random measures with independent increments (NRMI, Regazzini et al., 2003).
Revised: (2020a,b) develop ﬁnite-dimensional multinomial versions of the PY process and normalized random measures with independent increments (NRMI, Regazzini et al., 2003).

Original: The
Revised: The

Original: NGG is a special case of an NRMI, and the ﬁnite-dimensional NGG counterpart is termed the NGGmultinomial process.
Revised: NGG is a special case of an NRMI, and the ﬁnite-dimensional NGG counterpart is termed the NGG multinomial process.

Original: Being a ﬁnite-dimensional process, it requires an integer parameter H whichdeﬁnes the maximum number of clusters that can be obtained.
Revised: Being a ﬁnite-dimensional process, it requires an integer parameter H which deﬁnes the maximum number of clusters that can be obtained.

Original: Theorem 4 and Example 4 of Lijoiet al.
Revised: Theorem 4 and Example 4 of Lijoi et al.

Original: (2020a) provide the prior distribution of the number of clusters under the NGG multinomialprocess:
Revised: (2020a) provide the prior distribution of the number of clusters under the NGG multinomial process:

Original: [Equation]
Revised: [Equation]

Original: for any k ∈ { 1 , . . .
Revised: for any k ∈ { 1 , . . .

Original: , min( H, n ) } , where S (cid:96),k is the Stirling number of the second kind.
Revised: , min( H, n ) } , where S (cid:96),k is the Stirling number of the second kind.

Original: This distribution could alternatively be represented using the probabilities p n,k of the NGG process and inthis case, one could use the approximations for NGG to compute the NGG multinomial process.
Revised: This distribution could alternatively be represented using the probabilities p n,k of the NGG process and in this case, one could use the approximations for NGG to compute the NGG multinomial process.

Original: Moreover, it is also possible to compute the expectation through the weights of p n,k
Revised: Moreover, it is also possible to compute the expectation through the weights of p n,k

Original: (see details in
Revised: (see details in

Original: Appendix B.2).
Revised: Appendix B.2).

Original: 2.3. Stick-breaking truncation
Revised: 2.3. Stick-breaking truncation

Original: The stick-breaking representation is perhaps the most popular constructive representation of the
Revised: The stick-breaking representation is perhaps the most popular constructive representation of the

Original: Dirichlet process (Sethuraman, 1994).
Revised: Dirichlet process (Sethuraman, 1994).

Original: More speciﬁcally, the stick-breaking describes the distributionof the component weights by recursively breaking a stick of length one (i.e. the total probabilityof the process).
Revised: More speciﬁcally, the stick-breaking describes the distribution of the component weights by recursively breaking a stick of length one (i.e. the total probability of the process).

Original: This weight distribution is size-biased, meaning that the weights are stochasticallydecreasing (decreasing in expectation).
Revised: This weight distribution is size-biased, meaning that the weights are stochastically decreasing (decreasing in expectation).

Original: It turns out that any Gibbs-type process satisﬁes such a stick-breaking representation, see for instance Theorem 14.23 of Ghosal and Van der Vaart (2017) fordetails.
Revised: It turns out that any Gibbs-type process satisﬁes such a stick-breaking representation, see for instance Theorem 14.23 of Ghosal and Van der Vaart (2017) for details.

Original: In addition to the Dirichlet process case, Pitman and Yor (1997) present the PY case, while
Revised: In addition to the Dirichlet process case, Pitman and Yor (1997) present the PY case, while

Original: Lau and Cripps (2015); Favaro et al.
Revised: Lau and Cripps (2015); Favaro et al.

Original: (2016) present the NGG case.
Revised: (2016) present the NGG case.

Original: The stick-breaking representationprovides a simple way to approximate the process by truncating the representation at some ﬁnitelevel H .
Revised: The stick-breaking representation provides a simple way to approximate the process by truncating the representation at some ﬁnite level H .

Original: Many diﬀerent approaches have been proposed, with ﬁxed or random truncation levels,yielding varying guarantees on the truncation error, see for instance Ishwaran and James (2001);
Revised: Many diﬀerent approaches have been proposed, with ﬁxed or random truncation levels, yielding varying guarantees on the truncation error, see for instance Ishwaran and James (2001);

Original: Muliere and Tardella (1998); Arbel et al.
Revised: Muliere and Tardella (1998); Arbel et al.

Original: Here we adopt a simple ﬁxed truncation at level
Revised: Here we adopt a simple ﬁxed truncation at level

Original: H .
Revised: H .

Original: We use a stick-breaking representation of NGG process based on fractions of exponentiallytilted stable and gamma random variables (see Lemma 3.3 Lau and Cripps, 2015)(see details in
Revised: We use a stick-breaking representation of NGG process based on fractions of exponentially tilted stable and gamma random variables (see Lemma 3.3 Lau and Cripps, 2015)(see details in

Original: Appendix B.3).
Revised: Appendix B.3).

Original: 2.4. Ferguson–Klass truncation
Revised: 2.4. Ferguson–Klass truncation

Original: Lastly, we consider here a sub-class of Gibbs-type processes composed of those processes which areobtained by a normalisation step, called normalized random measures with independent increments(NRMI, Regazzini et al., 2003).
Revised: Lastly, we consider here a sub-class of Gibbs-type processes composed of those processes which are obtained by a normalisation step, called normalized random measures with independent increments (NRMI, Regazzini et al., 2003).

Original: An alternative series representation to the stick-breaking in thiscase has been developed by Ferguson and Klass (1972).
Revised: An alternative series representation to the stick-breaking in this case has been developed by Ferguson and Klass (1972).

Original: It is referred to as the Ferguson–Klassrepresentation.
Revised: It is referred to as the Ferguson–Klass representation.

Original: See Barrios et al.
Revised: See Barrios et al.

Original: for a review, and Arbel et al.
Revised: for a review, and Arbel et al.

Original: (2020) for an R packagedevoted to mixture modeling with NRMI.
Revised: (2020) for an R package devoted to mixture modeling with NRMI.

Original: Several truncation strategies have been proposed, see forinstance Argiento et al.
Revised: Several truncation strategies have been proposed, see for instance Argiento et al.

Original: ; Arbel and Pr¨unster (2017).
Revised: ; Arbel and Pr¨unster (2017).

Original: As opposed to the stick-breaking, the
Revised: As opposed to the stick-breaking, the

Original: Ferguson–Klass representation provides a vector of weights in a non-increasing order.
Revised: Ferguson–Klass representation provides a vector of weights in a non-increasing order.

Original: As a result, asimilar truncation strategy as that employed for the stick-breaking provides a lower truncation error(almost surely) for a given truncation level H .
Revised: As a result, a similar truncation strategy as that employed for the stick-breaking provides a lower truncation error (almost surely) for a given truncation level H .

Original: 3. Comparison of approximations and discussion 
Revised: 3. Comparison of approximations and discussion 3.1. Prior on K n

Original: 3.1. Prior on K n

Original: We compare all the presented schemes on an approximation of the prior on K n for a NGG process,for variousparameter values ( τ , σ , andn the number of observations).
Revised: We compare all the presented schemes for approximating the prior on K n for an NGG process parameterised by ( τ, σ ) (see deﬁnition in Appendix A), for various values of parameters τ , σ , and number of observations n .

Original: Our reference is an exactcomputation of the prior distribution for the NGG.
Revised: Our reference is an exact computation of the prior distribution for the NGG using arbitrary precision arithmetic in our package GibbsTypePriors , a strategy not as scalable as the predictive approximation.

Original: See Figure 1 for n = 100 and Figure 3 in Appendix C for n = 1000.
Revised: See Figure 1 for n = 100 and Figure 3 in Appendix C for n = 1000.

Original: For comparison, we speciﬁed the same truncation level H = 250 forboth the truncated methods and for the ﬁnite-dimensional process.
Revised: For comparison, we speciﬁed the same truncation level H = 250 for both the truncated methods and for the ﬁnite-dimensional process.

Original: See results for H = 1000 on Figure 4 in Appendix C.
Revised: See results for H = 1000 on Figure 4 in Appendix C.

Original: The predictive approximation is a good approximation both for smallvalues of τ and large values of σ , but the quality decreases with growing τ , for small values ofσ .
Revised: The predictive approximation is a good approximation both for small values of τ and large values of σ , but the quality decreases with growing τ , for small values of σ .

Original: This is coherent with the results for predictive approximation discussed in Arbel and Favaro(2020).
Revised: This is coherent with the results for predictive approximation discussed in Arbel and Favaro (2020).

Original: Interestingly, the NGG multinomial process performs well in all situations, outperformedonly by the predictive approximation for small values of τ and large values of σ .
Revised: Interestingly, the NGG multinomial process performs well in all situations, outperformed only by the predictive approximation for small values of τ and large values of σ .

Original: The stick-breakingtruncation performs well for low values of both parameters; but quickly becomes unreliable asthey increase.
Revised: The stick-breaking truncation performs well for low values of both parameters; but quickly becomes unreliable as they increase.

Original: This is expected as the truncation level necessary for a good approximation growsrapidly when the parameters increase.
Revised: This is expected as the truncation level necessary for a good approximation grows rapidly when the parameters increase.

Original: The Ferguson–Klass truncation performs better than thestick-breaking for the same truncation level, because it displays large weights ﬁrst while there isonly a stochastic order for stick-breaking, as discussed previously.
Revised: The Ferguson– Klass truncation performs better than the stick-breaking for the same truncation level, because it displays large weights ﬁrst while there is only a stochastic order for stick-breaking, as discussed previously.

Original: Results for n = 1000 (Figure 3in Appendix C) are similar to n = 100, and larger truncation level H
Revised: Results for n = 1000 (Figure 3 in Appendix C) are similar to n = 100, and larger truncation level H

Original: (Figure 4 in Appendix C)reveal the same trends.
Revised: (Figure 4 in Appendix C) reveal the same trends.

Original: Figure 2 shows another comparison for a larger range of parameter values.
Revised: Figure 2 shows another comparison for a larger range of parameter values.

Original: In line with Figure 1, we use the truncation level H =250 and the sample size n = 100.
Revised: In line with Figure 1, we use the truncation level H = and the sample size n = 100.

Original: Eachpoint on the graph represents the expectation and standard deviation ( E [ K n ] , std[ K n ]) of the priordistribution induced by a parameter pair ( τ, σ ), whose values are denoted by colour (see grid legend,panel (d)).
Revised: Each point on the graph represents the expectation and standard deviation ( E [ K n ] , std[ K n ]) of the prior distribution induced by a parameter pair ( τ, σ ), whose values are denoted by colour (see grid legend, panel (d)).

Original: All the approximations exhibit a similar pattern, despite huge variations on the range ofattainable pairs ( E [ K n ] , std[ K n ]) due to approximation failure for some parameters values, especiallyfor truncated approximations.
Revised: All the approximations exhibit a similar pattern, despite huge variations on the range of attainable pairs ( E [ K n ] , std[ K n ]) due to approximation failure for some parameters values, especially for truncated approximations.

Original: Increasing the truncation level H would mitigate the disparities among approximations, but at a higher computational cost.
Revised: The disparities among approximations could be mitigated by increasing the truncation level H , but for values of σ close to 1, excessively large values of H would be required for good precision (see eg Figure 2 in Arbel and Pr¨unster, 2017).

Original: For this large range of parametervalues, the predictive approximation seems to perform better than the other approximations exceptfor small values of σ and large values of τ (pink area).
Revised: For this large range of parameter values, the predictive approximation seems to perform better than the other approximations except for small values of σ and large values of τ (pink area).

Original: The NGG multinomial process performsreasonably well in this area of parameter space, but starts failing for larger values of σ .
Revised: The NGG multinomial process performs reasonably well in this area of parameter space, but starts failing for larger values of σ .

Original: Both stickbreaking and Ferguson–Klass approximations seem to fail for this low level of truncation, althoughthe Ferguson–Klass approach seems to fare better and could probably give good results with a highertruncation level.
Revised: Both stick-breaking and Ferguson–Klass approximations seem to fail for this low level of truncation, although the Ferguson–Klass approach seems to fare better and could probably give good results with a higher truncation level.

Original: All approximations present some degree of challenge for numerical stability, butwhile the existing solutions for stick-breaking and Ferguson–Klass methods could still be unstablefor some parameter values, those challenges seem solved by Kon Kam King et al.
Revised: All approximations present some degree of challenge for numerical stability, but while the existing solutions for stick-breaking and Ferguson–Klass methods could still be unstable for some parameter values, those challenges seem solved using arbitrary precision arithmetic (package GibbsTypePriors ) for NGG predictive and

Original: for NGG predictiveand NGG multinomial.
Revised: NGG multinomial.

Original: 3.2. Prior calibration
Revised: 3.2. Prior calibration

Original: A practical application of the approximations presented here is prior calibration, or in other wordschoosing prior hyperparameters to reﬂectpertinent expert information.
Revised: A practical application of the approximations presented here is prior calibration, i.e. choosing prior hyperparameters to reﬂect expert information.

Original: A natural approach is tochoose the moments of the prior distribution to reﬂect the amount of information known a priori,typically an expected number of clusters and an uncertainty for mixture models.
Revised: A natural approach is to use this information to specify the moments of the prior distribution, typically an expected number of clusters and an uncertainty.

Original: However, thestructure of Gibbs-type priors (1) and (2) renders this approach a challenging task, as moments donot have an invertible analytical expressionin general that would allow selecting hyperparameters,or even interpreting their qualitative impact on the distribution.
Revised: However, this is challenging for Gibbs-type priors (1) and (2), as moments do not have an invertible analytical expression that would allow selecting hyperparameters, or even interpreting their qualitative impact on the distribution.

Original: The approximations presentedin this note allow investigating the impact of the hyperparameters on various aspects of the priordistribution and ultimately choosing them to calibrate the prior.
Revised: The approximations presented in this note allow investigating the impact of the hyperparameters on various aspects of the prior distribution and ultimately choosing them to calibrate the prior.

Original: We consider calibrating an NGGprocess prior or a PY process, both involving two parameters which can be denoted τ ∈ R + andσ ∈ [0 , 1[.
Revised: We consider calibrating an NGG process prior or a PY process, both involving two parameters which can be denoted τ ∈ R + and σ ∈ [0 , 1).

Original: As a ﬁrst approximate guide, τ is often presented as inﬂuencing the location of thedistribution, while σ is often presented as having more bearing on the scale (e.g. Barrios et al.,2013).
Revised: As a ﬁrst approximate guide, τ is often presented as inﬂuencing the location of the distribution, while σ is often presented as having more bearing on the scale (e.g. Barrios et al., 2013).

Original: Figure 2 shows more precisely the impact of the parameters of the NGG process.
Revised: Figure 2 shows more precisely the impact of the parameters of the NGG process.

Original: Dashed(resp.
Revised: Dashed (resp.

Original: dotted) lines on the graph connect the points where σ (resp.
Revised: dotted) lines on the graph connect the points where σ (resp.

Original: τ ) is kept constant.
Revised: τ ) is kept constant.

Original: Followingdashed lines, we can see that for small ﬁxed σ (dark blue to pink), increasing τ strongly increases theexpected number of clusters.
Revised: Following dashed lines, we can see that for small ﬁxed σ (dark blue to pink), increasing τ strongly increases the expected number of clusters.

Original: For large σ however (green to yellow), increasing τ mostly decreases theprior standard deviation, while for intermediate σ , increasing τ both increases the expectation anddecreases the standard deviation.
Revised: For large σ however (green to yellow), increasing τ mostly decreases the prior standard deviation, while for intermediate σ , increasing τ both increases the expectation and decreases the standard deviation.

Original: Following dotted lines, we can see that for a ﬁxed τ , increasing σincreases the prior standard deviation up to a point where the standard deviations starts decreasing,in an almost parabolic relationship.
Revised: Following dotted lines, we can see that for a ﬁxed τ , increasing σ increases the prior standard deviation up to a point where the standard deviations starts decreasing, in an almost parabolic relationship.

Original: The relation appears closer to parabolic for large values ofτ , and the maximum of the parabola moves towards smaller expectations with larger values of τ .
Revised: The relation appears closer to parabolic for large values of τ , and the maximum of the parabola moves towards smaller expectations with larger values of τ .

Original: Another striking feature of Figure 2, particularly relevant for prior calibration, is that some regionsof ( E [ K n ] , std
Revised: Another striking feature of Figure 2, particularly relevant for prior calibration, is that some regions of ( E [ K n ] , std

Original: [ K n ]) are not attainable for any combination of parameters.
Revised: [ K n ]) are not attainable for any combination of parameters.

Original: This fact must be takeninto account when calibrating the prior distribution using its moments.
Revised: This fact must be taken into account when calibrating the prior distribution using its moments.

Original: Echoing the observationfrom the previous section, the ﬁgure also suggests that the predictive approximation could be usedfor prior calibration for large values σ , while NGG multinomial performs better for small values ofσ .
Revised: Echoing the observation from the previous section, the ﬁgure also suggests that the predictive approximation could be used for prior calibration for large values σ , while NGG multinomial performs better for small values of σ .

Original: 3.3. Discussion
Revised: 3.3. Discussion

Original: We have compared four approximations to the NGG process (and to PY in Appendix) showing thatthe predictive-based approximation fares best overall, though it fails and should be replaced by themultinomial approximation for small values of σ and large values of τ .
Revised: We have compared four approximations to the NGG process (and to PY in Appendix) showing that the predictive-based approximation fares best overall, though it fails and should be replaced by the multinomial approximation for small values of σ and large values of τ .

Original: To complement these results,note that all three approximations relying on a level H satisfy a weak convergence guarantee when
Revised: To complement these results, note that all three approximations relying on a level H satisfy a weak convergence guarantee when

Original: H → ∞ .
Revised: H → ∞ .

Original: There is no such equivalent for the predictive-based approximation, but the large n resultof Proposition 1 holds instead.
Revised: There is no such equivalent for the predictive-based approximation, but the large n result of Proposition 1 holds instead.

Original: Further research will address comparing computational eﬃciency ofthe methods.
Revised: Further research will address comparing computational eﬃciency of the methods.

Original: [Figure]
Revised: [Figure]

Original: [Figure]
Revised: [Figure]

Original: lllllll l l l lllll l l l l l lllll l l l l l llll l l l l l l llll l l l l ll llll l l llll l l l l llllll l l l l llllllllllllllll        100 Expectation
Revised: lllllll l l l lllll l l l l l lllll l l l l l llll l l l l l l llll l l l l ll llll l l llll l l l l llllll l l l l llllllllllllllll        100 Expectation

Original: S t d
Revised: S t d

Original: [Figure]
Revised: [Figure]

Original: predictive
Revised: predictive

Original: [Figure]
Revised: [Figure]

Original: lllllll l llllll l l l l lllll l l l ll llll l l l lll lllll lllll llllllllllllllllllllllllllllllllllllllll llllllllll       100 Expectation
Revised: lllllll l llllll l l l l lllll l l l ll llll l l l lll lllll lllll llllllllllllllllllllllllllllllllllllllll llllllllll       100 Expectation

Original: [Figure]
Revised: [Figure]

Original: d ll l l l l l l l l l l l l ll lll l l lll l lll l lll ll l lll ll ll ll llll llll ll ll        100 Expectation
Revised: d ll l l l l l l l l l l l l ll lll l l lll l lll l lll ll l lll ll ll ll llll llll ll ll        100 Expectation

Original: S t d l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0.
Revised: S t d l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0.

Original: [Figure]
Revised: [Figure]

Original: 1 Introduction
Revised: Introduction

Original: 12 Statistical approximations 32.
Revised: 2 Statistical approximations 2.

Original: Predictive distribution approximation . .
Revised: Predictive distribution approximation . .

Original: Finite-dimensional approximation . .
Revised: Finite-dimensional approximation . .

Original: Stick-breaking truncation . .
Revised: Stick-breaking truncation . .

Original: Ferguson–Klass truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Revised: Ferguson–Klass truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Original: 43 Comparison of approximations and discussion 43.
Revised: 3 Comparison of approximations and discussion 3.

Original: Prior on K n . .
Revised: Prior on K n . .

Original: Prior calibration . . .
Revised: Prior calibration . . .

Original: Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Revised: Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Original: A
Revised: A

Original:  of  PY and NGG
Revised: Description of speciﬁc classes of Gibbs-type processes: PY and NGG

Original: A.1 V n,k parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Revised: A.1 V n,k parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Original: A.2 C n,k parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Revised: A.2 C n,k parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Original: B Details on approximations
Revised: B Details on approximations

Original: B.1 Predictive distribution approximation . .. . . . . . . . . . . . . . . . . . . . . . . .
Revised: B.1 Predictive distribution approximation . .

Original: B.1.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Revised: B.1.1 Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Original: B.1.2 Algorithm for obtaining prior weights from Proposition 1 . . .
Revised: B.1.2 Algorithm for obtaining prior weights from Proposition 1 . . .

Original: B.2 Finite-dimensional approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Revised: B.2 Finite-dimensional approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Original: B.3 Stick-breaking truncation . .
Revised: B.3 Stick-breaking truncation . .

Original: B.4 Ferguson–Klass truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Revised: B.4 Ferguson–Klass truncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Original: C Comparison of approximations for NGG (extra plots)
Revised: C Comparison of approximations for NGG (extra plots)

Original: D Comparison of approximations for PY
Revised: D Comparison of approximations for PY

Original: D.1 Finite-dimensional approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Revised: D.1 Finite-dimensional approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Original: D.2 Stick-breaking truncation . . . .
Revised: D.2 Stick-breaking truncation . . . .

Original: D.3 Expectation and variance of the PY process . . . . . . . . . . . . . . . . . . . . . . .

Original: D.4 Comparison of approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Revised: D.3 Comparison of approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Original: Appendix A. Description of speciﬁc classes of Gibbs-type processes: PYand NGGA.1. V n,k parameters
Revised: Appendix A. Description of speciﬁc classes of Gibbs-type processes: PY and NGG A.1. V n,k parameters

Original: The functional parameter h deﬁning a Gibbs-type process can be equivalently written in terms ofan inﬁnite triangular array of positive parameters V n,k , for any n ≥ 1 and 1 ≤ k ≤ n , as follows: 
Revised: The functional parameter h deﬁning a Gibbs-type process can be equivalently written in terms of an inﬁnite triangular array of positive parameters V n,k , for any n ≥ 1 and 1 ≤ k ≤ n , as follows: [Equation]

Original: [Equation]
Revised: [Equation]

Original:  where Γ is the Gamma function and f σ is the density of a positive σ -stable random variable.
Revised: ) where Γ is the Gamma function and f σ is the density of a positive σ -stable random variable.

Original: The PY process, parameterized by σ ∈ (0 , 1) and α > − σ , is obtained by choosing
Revised: The PY process, parameterized by σ ∈ (0 , 1) and α > − σ , is obtained by choosing

Original: [Equation]
Revised: [Equation]

Original: which implies V n,k = [-(cid:81) k − 1 i =
 ( α + iσ-]
Revised: which implies V n,k = [-(cid:81) k − 1 i =
 ( α + iσ-][Equation]

Original: [Equation]
Revised: [Equation]

Original: (6) where the Pocchammer symbol ( α ) n is used to denote the rising factorial
Revised: where the Pochhammer symbol ( α ) n is used to denote the rising factorial

Original: α ( α + 1) · · · ( α + n − 1).
Revised: α ( α + 1) · · · ( α + n − 1).

Original: On the other hand, the NGG is parameterized by σ ∈ (0 , 1) and τ ≥ 0 and is deﬁned byh ( t ) =
Revised: On the other hand, the NGG is parameterized by σ ∈ (0 , 1) and τ ≥ 0 and is deﬁned by h ( t ) =

Original: e τ σ − τt , such that V n,k = 
Revised: e τ σ − τt , such that V n,k = σ

Original: [Equation]
Revised: [Equation]

Original: [Equation] where Γ( a, x ) = (cid:82) ∞
Revised: (7) where Γ( a, x ) = (cid:82) ∞

Original: x s a − 1 e − s d s is the incomplete gamma function.
Revised: x s a − 1 e − s d s is the incomplete gamma function.

Original: Note that the NGG have severalequivalent parametrizations.
Revised: Note that the NGG have several equivalent parametrizations.

Original: Another parametrization comes from the deﬁnition of the NGG throughnormalization of generalized gamma process.
Revised: Another parametrization comes from the deﬁnition of the NGG through normalization of generalized gamma process.

Original: Generalized gamma processes are discrete randommeasures P of the form
Revised: Generalized gamma processes are discrete random measures P of the form

Original: [Equation]
Revised: [Equation]

Original: where weights J i do not sum to 1 and θ i are location parameters, sampled iid from a measure P 0 , aprobability distribution over parameter space Θ (see Barrios et al., 2013).
Revised: where weights J i do not sum to 1 and θ i are location parameters, sampled iid from a measure P 0 , a probability distribution over parameter space Θ (see Barrios et al., 2013).

Original: ( J i , θ i ) are the points of
Revised: ( J i , θ i ) are the points of

Original: Poisson process with mean intensity:
Revised: Poisson process with mean intensity:

Original: [Equation]
Revised: [Equation]

Original: which depends on parameter κ ≥ 0 and σ ∈ [0 , 1), such that ( κ, σ ) (cid:54) =
Revised: which depends on parameter κ ≥ 0 and σ ∈ [0 , 1), such that ( κ, σ ) (cid:54) =

Original: (0 , 0), α > 0.
Revised: (0 , 0), α > 0.

Original: The NGG is obtained by a normalization step from P which consists in dividing it by its total mass.
Revised: The NGG is obtained by a normalization step from P which consists in dividing it by its total mass.

Original: It can be denoted NGG( α, κ, σ ; P 0 )
Revised: It can be denoted NGG( α, κ, σ ; P 0 )

Original: (see Barrios et al., 2013).
Revised: (see Barrios et al., 2013).

Original: Note the important change of parameter between the ( τ, σ ) and the ( α, κ, σ ) parameters:
Revised: Note the important change of parameter between the ( τ, σ ) and the ( α, κ, σ ) parameters:

Original: [Equation]
Revised: [Equation]

Original: A.2. C n,k parameters
Revised: A.2. C n,k parameters

Original: For any Gibbs-type process, the probability mass function ( p n,k ) 1 ≤ k ≤ n of the prior distribution onthe number of clusters K n can be described in terms of V n,k and of another triangular array ofreals called generalized factorial coeﬃcients (see Charalambides, 2005) and denoted by C n,k , for anypositive integers n, k such that 1 ≤ k ≤ n :
Revised: For any Gibbs-type process, the probability mass function ( p n,k ) 1 ≤ k ≤ n of the prior distribution on the number of clusters K n can be described in terms of V n,k and of another triangular array of reals called generalized factorial coeﬃcients (see Charalambides, 2005) and denoted by C n,k , for any positive integers n, k such that 1 ≤ k ≤ n :

Original: [Equation]
Revised: [Equation]

Original: An interesting property of the generalized factorial coeﬃcients is that they follow a recursiverelation:
Revised: An interesting property of the generalized factorial coeﬃcients is that they follow a recursive relation:

Original: [Equation]
Revised: [Equation]

Original: with the following corner cases: C 0 , 

Revised: with the following corner cases: C 0 ,

Original: = 1, C n, 

Revised: = 1, C n,

Original: = 0 and for all k > n , C n,k = 0.
Revised: = 0 and for all k > n , C n,k = 0.

Original: Appendix B. Details on approximations 
Revised: Appendix B. Details on approximations B.1. Predictive distribution approximation B.1.1. Proof of Proposition

Original: B.1. Predictive distribution approximation B.1.1. Proof of Proposition 1

Original: The second order approximation of Arbel and Favaro (2020) can be written as
Revised: The second order approximation of Arbel and Favaro (2020) can be written as

Original: [Equation]
Revised: [Equation]

Original: where β n,k
Revised: where β n,k

Original: = O (1).
Revised: = O (1).

Original: Indeed, inspection of the proof of Theorem 1 in Arbel and Favaro (2020) showsthat o (1 /n )
Revised: Indeed, inspection of the proof of Theorem 1 in Arbel and Favaro (2020) shows that o (1 /n )

Original: (resp.
Revised: (resp.

Original: o (1 /n 2 )) can be replaced by O (1 /n 2 ) (resp.
Revised: o (1 /n 2 )) can be replaced by O (1 /n 2 ) (resp.

Original: O (1 /n 3 )).
Revised: O (1 /n 3 )).

Original: Note that we adopt theform of Equation (1.13) of Arbel and Favaro (2020) which has the advantage of being a properprobability distribution (in contrast to approximations of Equation (1.10) and Equation (1.11) of
Revised: Note that we adopt the form of Equation (1.13) of Arbel and Favaro (2020) which has the advantage of being a proper probability distribution (in contrast to approximations of Equation (1.10) and Equation (1.11) of

Original: Arbel and Favaro (2020) which do not lead to a proper probability distribution).
Revised: Arbel and Favaro (2020) which do not lead to a proper probability distribution).

Original: A ﬁrst orderapproximation is obtained by simply setting β n,k = 0 in the above equations.
Revised: A ﬁrst order approximation is obtained by simply setting β n,k = 0 in the above equations.

Original: But it yields poorapproximation quality and shall not be used here.
Revised: But it yields poor approximation quality and shall not be used here.

Original: The prior distribution of the number of clustersfor n + 1 samples is deﬁned by weights p n +1 ,k = P
Revised: The prior distribution of the number of clusters for n + 1 samples is deﬁned by weights p n +1 ,k = P

Original: ( K n +1 = k ), which have the following form
Revised: ( K n +1 = k ), which have the following form

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original:  where (cid:80) n +1 i =1 p i = 1.
Revised: (14) where (cid:80) n +1 i =1 p i = 1.

Original: Therefore, the ratio of two consecutive weights is
Revised: Therefore, the ratio of two consecutive weights is

Original: [Equation]
Revised: [Equation]

Original: This involves a ratio of two consecutive V parameters which can be obtained as the result of the ratio of Equations (12) and (13):
Revised: This involves a ratio of two consecutive V parameters which can be obtained as the result of the ratio of Equations (12) and (13):

Original: [Equation]
Revised: [Equation]

Original: where step (a) is because β n,k
Revised: where step (a) is because β n,k

Original: = O (1) (Arbel and Favaro, 2020).
Revised: = O (1) (Arbel and Favaro, 2020).

Original: Hence, incorporating this into (15) yields
Revised: Hence, incorporating this into (15) yields

Original: [Equation]
Revised: [Equation]

Original: where we have used the monotony property of the C n,k sequence provided in Lemma 2 in order toupper bound the ratio C n +1 ,k +1 C n +1 ,k by one and to ensure that the O (cid:0) 1 n (cid:1) term can be placed out of thebracket.
Revised: where we have used the monotonicity of the C n,k sequence provided in Lemma 2 in order to upper bound the ratio C n +1 ,k +1 C n +1 ,k by one and to ensure that the O (cid:0) 1 n (cid:1) term can be placed out of the bracket.

Original: This concludes the proof.
Revised: This concludes the proof.

Original: Lemma 

Revised: Lemma

Original: For any integer n , the sequence k
Revised: For any integer n , the sequence k

Original: (cid:55)→ C n,k is non increasing for k ∈ { 1 , . . . , n } .
Revised: (cid:55)→ C n,k is non increasing for k ∈ { 1 , . . . , n } .

Original: Proof The proof is by induction, by noting that the diﬀerence C n +1 ,k − C n +1 ,k +1 can be separatedinto a sum of three non negative terms ( n − σk )
Revised: Proof The proof is by induction, by noting that the diﬀerence C n +1 ,k − C n +1 ,k +1 can be separated into a sum of three non negative terms ( n − σk )

Original: ( C n,k − C n,k +1 )
Revised: ( C n,k − C n,k +1 )

Original: + σ ( C n,k − 1 − C n,k ) + σ C n,k +1 .
Revised: + σ ( C n,k − 1 − C n,k ) + σ C n,k +1 .

Original: B.1.2. Algorithm for obtaining prior weights from Proposition1
Revised: B.1.2. Algorithm for obtaining prior weights from Proposition

Original: We describe here a simple algorithm that outputs prior weights p n +1 ,k , k ∈ { 1 , . . .
Revised: We describe here a simple algorithm that outputs prior weights p n +1 ,k , k ∈ { 1 , . . .

Original: , n + 1 } , from the predictive approximation ratio of Proposition 1.
Revised: , n + 1 } , from the predictive approximation ratio of Proposition 1.

Original: Note that we consider here a sample size of n +only for presentation convenience.
Revised: Note that we consider here a sample size of n +only for ease of presentation.

Original: Let x n,k := p n +1 ,k +1 p n +1 ,k as given in Equation 3, for k ∈ { 1 , . . . , n } .
Revised: Let x n := p n +1 ,k +1 p n +1 ,k as given in Equation 3, for k ∈ { 1 , . . . , n } .

Original: Starting from one end or the other (i.e. p n +1 , 1 or p n +1 ,n +1 ), we can recover all the other weights recursively by using the ratios x n,k s:
Revised: Starting from one end or the other (i.e. p n +1 , 1 or p n +1 ,n +1 ), we can recover all the other weights recursively by using the ratios x n s:

Original: [Equation]
Revised: [Equation]

Original: Combining this with (cid:80) n +1 i =1 p n +1 ,i = 1 yields p n +1 , 1 (cid:16) 1 + (cid:80) nj =1 (cid:81) ji =1 x i (cid:17)
Revised: Combining this with (cid:80) n +1 i =1 p n +1 ,i = 1 yields p n +1 , 1 (cid:16) 1 + (cid:80) nj =1 (cid:81) ji =1 x i (cid:17)

Original: = 1.
Revised: = 1.

Original: Hence all prior weights can be recursively obtained following 
Revised: Hence all prior weights can be recursively obtained following  

Original: [Equation]
Revised: [Equation]

Original: B.2. Finite-dimensional approximation
Revised: B.2. Finite-dimensional approximation

Original: Distribution of the prior number of clusters for the NGG multinomial process could be computedusing the distribution of NGG as follows (Theorem 4 in Lijoi et al., 2020a):
Revised: Distribution of the prior number of clusters for the NGG multinomial process could be computed using the distribution of NGG as follows (Theorem 4 in Lijoi et al., 2020a):

Original: [Equation]
Revised: [Equation]

Revised: [Equation]

Original: for any k ∈ { 1 , . . .
Revised: for any k ∈ { 1 , . . .

Original: , min( H, n ) } , where P ( K n, ∞ =
Revised: , min( H, n ) } , where P ( K n, ∞ =

Original: l + k ) are the probabilities in the NGG process.
Revised: l + k ) are the probabilities in the NGG process.

Original: These probabilities of NGG could be replaced by approximation for faster computation.
Revised: These probabilities of NGG could be replaced by approximation for faster computation.

Original: Moreover,the expected number of clusters induced by NGG multinomial process, could be computed using theapproximation for p n,k :
Revised: Moreover, the expected number of clusters induced by NGG multinomial process, could be computed using the approximation for p n,k :

Original: [Equation]
Revised: [Equation]

Original: We can use the computed p n,k to compute the expected number of clusters for NGG.
Revised: We can use the computed p n,k to compute the expected number of clusters for NGG.

Original: [Equation]
Revised: [Equation]

Original: B.3. Stick-breaking truncation
Revised: B.3. Stick-breaking truncation

Original: We used the algorithm suggested by Lau and Cripps (2015) for stick-breaking representation of the
Revised: We used the algorithm suggested by Lau and Cripps (2015) for stick-breaking representation of the

Original: NGG( α, κ, σ ; P 0 ) (see Appendix A for details on these parameters).
Revised: NGG( α, κ, σ ; P 0 ) (see Appendix A for details on these parameters).

Original: The stick-breaking truncationof the NGG can be written as:
Revised: The stick-breaking truncation of the NGG can be written as:

Original: [Equation]
Revised: [Equation]

Original: = [Equation] (cid:88)
Revised: (19) where p

Original: [Equation]
Revised: [Equation]

Original: , (19) wherep i

Original: = U
Revised: = U

Original: i
Revised: [Equation]

Original: i − 1 (cid:89)
Revised: [Equation]

Original: j = (1 − U j ) , i = 2 , . .
Revised: [Equation] = [Equation]

Original: , H − 1 , p 1 =
Revised: [Equation]

Original: U 1 , (20) and p H =
Revised: (20) and p H =

Original: (1 − (cid:80) H − 1 i =
Revised: (1 − (cid:80) H − 1 i =

Original: j p j )
Revised: j p j )

Original: =
Revised: =

Original: (cid:81)
Revised: (cid:81)

Original: Hj =1 (1 − U j ) and H deﬁnes truncation level.
Revised: Hj =1 (1 − U j ) and H deﬁnes truncation level.

Original: The sequence { U 1 , . . .
Revised: The sequence { U 1 , . . .

Original: , U H − 1 } has conditional density
Revised: , U H − 1 } has conditional density

Original: f U n | U n − 1 ,...,U 1 ( u n | u n− 1 , . . .
Revised: f U n [Equation] [Equation] ( u n | u n

Revised: [Equation]

Original: , u 1 ) equal to
Revised: , u 1 ) equal to

Original: [Equation]
Revised: [Equation]

Original: Here q i = (cid:81) ij =1 (1 − u j ) and q 0 = 1.
Revised: Here q i = (cid:81) ij =1 (1 − u j ) and q 0 = 1.

Original: For computing the probabilities p n,k , k = 1 , . . .
Revised: For computing the probabilities p n,k , k = 1 , . . .

Original: , H in (19), wesample U j , j = 1 , . . .
Revised: , H in (19), we sample U j , j = 1 , . . .

Original: , H − 1 using the following algorithm proposed in Lau and Cripps (2015) (see
Revised: , H − 1 using the following algorithm proposed in Lau and Cripps (2015) (see

Original: Lemma 3.3):
Revised: Lemma 3.3):

Original: • Sample augmented random variable ξ n conditional on { U n − 1 , . . .
Revised: • Sample augmented random variable ξ n conditional on { U n − 1 , . . .

Original: , U 1 } , that has density:
Revised: , U 1 } , that has density:

Original: [Equation]
Revised: [Equation]

Original:  • Sample X n , Z n conditional on ξ n : Z n ∼ gamma(1 − σ, κ/k n − 1 + ξ n ) and X n have exponentiallytilted stable distribution with parameters σ and λ = κ/k n − 1 + ξ n .
Revised: [Equation] • Sample X n , Z n conditional on ξ n : Z n ∼ gamma(1 − σ, κ/k n − 1 + ξ n ) and X n have exponentially tilted stable distribution with parameters σ and λ = κ/k n − 1 + ξ n .

Original: For sampling from the exponentially tilted stable distribution we used the R package copula (Hofertet al., 2020).
Revised: For sampling from the exponentially tilted stable distribution we used the R package copula (Hofert et al., 2020).

Original: To estimate the distribution of the prior number of clusters K n for the stick-breakingtruncation, we averaged over 2 × 10 2 Monte Carlo simulations then smoothed the result using a
Revised: To estimate the distribution of the prior number of clusters K n for the stick-breaking truncation, we averaged over 2 × 10 2 Monte Carlo simulations then smoothed the result using a

Original: Gamma density.
Revised: Gamma density.

Original: B.4. Ferguson–Klass truncation
Revised: B.4. Ferguson–Klass truncation

Original: We consider discrete random measure of the form (8), with jumps J i and random locations Z i (8).
Revised: We consider discrete random measure of the form (8), with jumps J i and random locations Z i (8).

Original: Ferguson–Klass representation consists in expressing random jumps
Revised: Ferguson–Klass representation consists in expressing random jumps

Original: J i in terms of underlying L´evyintensity.
Revised: J i in terms of underlying L´evy intensity.

Original: This representation produces the jumps in non increasing order J 1 ≥ J 2 , . ..
Revised: This representation produces the jumps in non increasing order J 1 ≥ J 2 , . .

Original:  , which canbe obtained by solving numerical equations ξ i = N ( J i ), where N ( v ) =
Revised: . , which can be obtained by solving numerical equations ξ i = N ( J i ), where N ( v ) =

Original: ν ([ v, ∞ )]) is a decreasingfunction and ξ 1 , ξ 2 − ξ 1 , . .
Revised: ν ([ v, ∞ )]) is a decreasing function and ξ 1 , ξ 2 − ξ 1 , . .

Original: are jump times of standard Poisson process of unit rate.
Revised: are jump times of standard Poisson process of unit rate.

Original: In case of thegeneralized gamma process N has the following form:
Revised: In case of the generalized gamma process N has the following form:

Original: [Equation]
Revised: [Equation]

Original: The correspondence between parametrisation in terms of κ and in terms of τ is τ = ακ σ /σ , c.f.
Revised: The correspondence between parametrisation in terms of κ and in terms of τ is τ = ακ σ /σ , c.f.

Original: Equation (9).
Revised: Equation (9).

Original: One can truncate the series in Equation (8) at some speciﬁed level H .
Revised: One can truncate the series in Equation (8) at some speciﬁed level H .

Original: We used thefollowing approach:
Revised: We used the following approach:

Original: • Sample ξ i ∼ PP for i = 1 , . . .
Revised: • Sample ξ i ∼ PP for i = 1 , . . .

Original: , H
Revised: , H

Original: • Compute J i =
Revised: • Compute J i =

Original: N − 1 ( ξ i ) for i = 1 , . .
Revised: N − 1 ( ξ i ) for i = 1 , . .

Original: , H .
Revised: , H .

Original: To solve equations ξ i = N ( J i ), we used a combination of quadrature methods to approximatethe integral (see, e.g., Burden and Faires, 1993) and a numerical procedure to solve the equation,or a root ﬁnding algorithm for numerical inversion and using the implementation of the uppergamma incomplete function from the R package expint .
Revised: To solve equations ξ i = N ( J i ), we used a combination of quadrature methods to approximate the integral (see, e.g., Burden and Faires, 1993) and a numerical procedure to solve the equation, or a root ﬁnding algorithm for numerical inversion and using the implementation of the upper gamma incomplete function from the R package expint .

Original: Both approaches are bound to be subjectto numerical stability over certain parameter ranges.
Revised: Both approaches are bound to be subject to numerical stability over certain parameter ranges.

Original: The prior distribution on K n is obtained byaveraging over 2 × 10 2 Monte Carlo simulations and the density is smoothed using a Gamma density.
Revised: The prior distribution on K n is obtained by averaging over 2 × 10 2 Monte Carlo simulations and the density is smoothed using a Gamma density.

Original: Appendix C. Comparison of approximations for NGG (extra plots)
Revised: Appendix C. Comparison of approximations for NGG (extra plots)

Original: Here we present in Figure 3 the prior distributions for K n for a large sample size of n = 1000.
Revised: Here we present in Figure 3 the prior distributions for K n for a large sample size of n = 1000.

Original: Experiments are also carried out with a large truncation level of H = 1000 (for truncated andﬁnite-dimensional approximations), see Figure 4.
Revised: Experiments are also carried out with a large truncation level of H = 1000 (for truncated and ﬁnite-dimensional approximations), see Figure 4.

Original: Comparing to similar Figure 1 and Figure 3,
Revised: Comparing to similar Figure 1 and Figure 3,

Original: we
Revised: we

Original: [Figure]
Revised: [Figure]

Original: t = 1, s = 0.25, n= [Figure] [Figure] t =10, s = 0.25, n =
Revised: n

Original: [Figure]
Revised: [Figure]

Original: k t = 1, s = 0.75, n=
Revised: n

Original: [Figure]
Revised: [Figure]

Original:  t=
Revised: k t

Original: [Figure]
Revised: [Equation]

Original:  s[Figure]
Revised: 10, s

Original: [Figure]
Revised: [Equation]

Original:  n =[Figure]
Revised: 0.75, n =

Original: [Figure]
Revised: [Figure]

Original:  [Figure]
Revised: can note that all the distributions that depend on H better approximate the true distribution. The quality of approximation deteriorates with growing sample size. [Figure] [Figure]

Original: [Figure]
Revised: [Figure]

Original: t = 1, s = 0.25, n =0.
Revised: t = 1, s = 0.25, n =0.

Original: t =10, s = 0.25, n =
Revised: t =10, s = 0.25, n =

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: k t = 1, s = 0.75, n =0.
Revised: k t = 1, s = 0.75, n =0.

Original: k t =10, s = 0.75, n =
Revised: k t =10, s = 0.75, n =

Original: [Figure] [Figure]
Revised: [Figure] [Figure]

Original: t = 1, s = 0.25, n =0.
Revised: t = 1, s = 0.25, n =0.

Original: t =10, s = 0.25, n =
Revised: t =10, s = 0.25, n =

Original: [Figure]
Revised: [Figure]

Original: k t = 1, s = 0.75, n =
Revised: k t = 1, s = 0.75, n =

Original: [Figure]
Revised: [Figure]

Original: k t =10, s = 0.75, n =
Revised: k t =10, s = 0.75, n =

Original: Appendix D. Comparison of approximations for PY 
Revised: Appendix D. Comparison of approximations for PY D.1. Finite-dimensional approximation

Original: D.1. Finite-dimensional approximation

Original: For the PY multinomial process the prior number of clusters is given in Theorem 3 of Lijoi et al.(2020b) and it equals
Revised: For the PY multinomial process the prior number of clusters is given in Theorem 3 of Lijoi et al. (2020b) and it equals

Original: [Equation]
Revised: [Equation]

Original: for any k ≤ min { H,n } , where S l,k is the Stirling number of second kind.
Revised: for any k ≤ min { H, n } , where S l,k is the Stirling number of second kind.

Original: D.2. Stick-breaking truncation
Revised: D.2. Stick-breaking truncation

Original: PY could be deﬁned using stick-breaking construction.
Revised: PY could be deﬁned using stick-breaking construction.

Original: Consider two independent families of randomvariables:
Revised: Consider two independent families of random variables:

Original: [Equation]

Original: [Equation]
Revised: [Equation]

Original: Deﬁne the random weights as:
Revised: Deﬁne the random weights as:

Original: [Equation]
Revised: [Equation]

Original: Deﬁne
Revised: Deﬁne

Original: P = ∞ (cid:88) k

Original: = [Equation]
Revised: [Equation]

Original: k

Original: [Equation]
Revised: [Equation]

Original: (26) then P ∼ PY( α, σ ; P 0 ), i.e P is the PY process with concentration parameter α , discount (ordiversity) parameter σ (with 0 ≤ σ ≤ 1) and base measure P 0 .
Revised: then P ∼ PY( α, σ ; P 0 ), i.e P is the PY process with concentration parameter α , discount (or diversity) parameter σ (with 0 ≤ σ ≤ 1) and base measure P 0 .

Original: D.3. Expectation and variance of the PY process
Revised: D.3. Comparison of approximations

Original: In the case of PY process, expectation and variance for the prior number of clusters can be computeddirectly, without computing the probabilities p n,k for the prior distribution of K n . We can deﬁne( α, σ ) based on the values of E [ K n ] and Var[ K n ], deﬁned below. By the P´olya urn representation of

Original: PY, we have

Original: [Equation]

Original: hence

Original: [Equation] Iterating yields [Equation] Using the law of total variance we have,
Revised: We compare the prior distribution for K n under the PY process and its approximations described above.

Original: [Equation] Using the fact that the variance of a two-point distribution with outcomes a and b and probabilities p and 1 − p is p (1 − p )( a − b ) 2 , we can write the ﬁrst term of the right-hand side of (28) as E [Equation]
Revised: Note that PY is not an NRMI (Regazzini et al., 2003), so as such it cannot be sampled according to Ferguson–Klass algorithm.

Original: Var

Original: [Equation] K n +1 | K 1 , . . . [Equation] n =
Revised: Figure 6 shows the relationship between the expected number of clusters and standard deviation for the PY process.

Original: [Equation] while (27) provides for second term of the right-hand side of (28) [Equation] Then combining (29) and(30), the following recursive formula holds
Revised: This ﬁgure is similar to Figure 2 for NGG and for comparison we use the same grid for parameters ( α, σ ) as the ( τ, σ ) grid of NGG, and the same values for n = 100, H = 250.

Original: [Equation]
Revised: [Figure]

Original: D.4. Comparison of approximations

Original: We compare the prior distribution for K n under the PY process and its approximations describedabove.

Original: Note that PY is not an NRMI (Regazzini et al., 2003), so as such it cannot be sampledaccording to Ferguson–Klass algorithm.

Original: [Figure]

Original: [Figure]

Original: ll

Original: [Figure]
Revised: [Figure]

Original: lll
Revised: lll

Original: [Figure]
Revised: [Figure]

Original: llll
Revised: lllll

Original: [Figure]
Revised: [Figure]

Original: [Table] [Figure] [Table] [Figure]
Revised: llllll

Original: [Table]

Original: [Figure]
Revised: [Figure]

Original: [Table] [Figure] [Table] [Figure]
Revised: l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l

Original: [Table]

Original: [Figure]
Revised: [Figure]

Original: [Table]

Original: [Figure]

Original: [Table]

Original: [Figure]

Original: [Table]

Original: [Figure]

Original: [Table]

Original: [Figure]

Original: [Table]

Original: [Figure]

Original: [Table] [Figure] 100 Expectation S t d ll l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l l 0.
Revised: l

Original: [Figure]
Revised: [Figure]

Original: a
Revised: l

Original: [Figure]
Revised: [Figure]

Original: Figure 6 shows the relationship between the expected number of clusters and standard deviationfor the PY process.

Original: This ﬁgure is similar to Figure 2 for NGG and for comparison we use the samegrid for parameters ( α, σ ) as the ( τ, σ ) grid of NGG, and the same values for n = 100, H = 250.