Compound random measures and their use in Bayesian nonparametrics

A new class of dependent random measures which we call {\it compound random measures} are proposed and the use of normalized versions of these random measures as priors in Bayesian nonparametric mixture models is considered. Their tractability allows the properties of both compound random measures and normalized compound random measures to be derived. In particular, we show how compound random measures can be constructed with gamma, $\sigma$-stable and generalized gamma process marginals. We also derive several forms of the Laplace exponent and characterize dependence through both the L\'evy copula and correlation function. A slice sampler and an augmented P\'olya urn scheme sampler are described for posterior inference when a normalized compound random measure is used as the mixing measure in a nonparametric mixture model and a data example is discussed.


Introduction
Bayesian nonparametric mixtures have become a standard tool for inference when a distribution of either observable or unobservable quantities is considered unknown.A more challenging problem, which arises in many applications, it to define a prior for a collection of related unknown distributions.For example, we may wish to estimate the distribution of survival times across different treatments or study-specific covariates, or the distribution of firm efficiency with different types of ownership (e.g.private, public, or not-for-profit).Suppose that x ∈ X denotes the value of covariates then, in a Bayesian nonparametric analysis, a prior needs to be defined across a collection of correlated distributions {p x |x ∈ X }.This problem was initially studied in a seminal paper on dependent Dirichlet processes [29] where generalisations of the Dirichlet process were proposed.Subsequent work used stick-breaking constructions of random measures as a basis for defining such a prior.This work is reviewed by [8].These priors can usually be represented as where w 1 (x), w 2 (x), . . .follow a stick-breaking process for all x ∈ X .A drawback with this approach is the stochastic ordering of the w i (x)'s for any x ∈ X which can lead to strange effects in the prior as x varies.
If X is countable, several other approaches to defining a prior on a collection of random probability measures have been proposed.The Hierarchical Dirichlet process (HDP) [37] assumes that px are a priori conditionally independent and identically distributed according to a Dirichlet process whose centring measure is itself given a Dirichlet process prior.This construction induces correlation between the elements of {p x |x ∈ X } in the same way as in parametric hierarchical models.This construction can be extended to more general hierarchical frameworks [see e.g.36, for a review].Alternatively, a prior can be defined using the idea of normalized random measures with independent increments which are defined by normalising a completely random measure.The prior is defined on a collection of correlated completely random measures {μ x |x ∈ X } which are then normalized for each of x, i.e. px = μx /μ x (Ω) where Ω is the support of μx .Several specific constructions have been proposed including various forms of superposition [16,26,27,28,3,1], the kernel-weighted completely random measures [13,15] and Lévy copula-based approaches [23,24,39] .In this paper, we develop an alternative method for constructing correlated completely random measures which is tractable, whose properties can be derived and for which slice sampling methods for posterior inference without truncation can be developed.The construction also provides a unifying framework for previously proposed constructions.Indeed, the σ-stable and gamma vector of dependent random measures, studied in the recent works of [23], [24] and [39], are special cases.Although these papers derive useful theoretical results, their application has been limited by the lack of a sampling methods for posterior inference.The algorithm proposed in this paper can also be used for posterior sampling for these nonparametric priors which is another contribution of the paper.
The paper is organized as follows.Section 2 introduces the concepts of completely random measures, normalized random measures and their multivariate extensions.Section 3 discusses the construction and some properties of a new class of multivariate Lévy process, Compound Random Measures, defined by a score distribution and a directing Lévy process.Section 4 provides a detailed description of Compound Random Measures with a gamma score distribution.Section 5 considers the use of normalized version of Compound Random Measures in nonparametric mixture models including the description of a Markov chain Monte Carlo scheme for inference.Section 6 provides an illustration of the use of these methods in an example and Section 7 concludes.

Preliminaries
Let (Ω, F, P) be a probability space and (X, X ) a measure space, with X Polish and X the Borel σ-algebra of subsets of X. Denote by M X the space of boundedly finite measures on (X, X ), i.e. this means that for any µ in M X and any bounded set A in X one has µ(A) < ∞.Moreover, M X stands for the corresponding Borel σ-algebra, see [6] for technical details.The concept of a completely random measure was introduced by [20].
Definition 1.Let μ be a measurable mapping from (Ω, F, P) into (M X ,M X ) and such that for any A 1 , . . ., A n in X , with A i ∩ A j = ∅ for any i = j, the random variables μ(A 1 ), . . ., μ(A n ) are mutually independent.Then μ is called a completely random measure (CRM).
A CRM can always be represented as a sum of two components: where the fixed jump points x 1 , . . ., x M are in X and the non-negative random jumps V 1 , . . ., V M are both mutually independent and independent from μc .The latter is a completely random measure such that where both the positive jump heights J i 's and the X-valued jump locations X i 's are random.The measure μc is characterized by the Lévy-Khintchine representation which states that where f : X → R is a measurable function such that f μc < ∞ almost surely and ν is a measure on R + × X such that for any B in X .The measure ν is usually called the Lévy intensity of μc .For our purposes, we will focus on the homogeneous case, i.e.Lévy intensities where the height and location contributions are separated.Formally, where ρ a measure on R + and α is a measure on X, which is usually called the centring measure.Some famous examples are the Gamma process, ν(ds, dx) = s −1 e −s ds α(dx), the σ-stable process, and the Beta process, A general class of processes that includes the gamma and σ-stable process is the Generalized Gamma process, Random measures are the basis for building Bayesian nonparametric priors.
Definition 2. Let μ be a measure in (M X ,M X ).A Normalized Random Measure (NRM) is defined as p = μ μ(X) .
The definition of a normalized random measure is very general and does not require that the underlying measure is completely random.The Pitman-Yor process (see [33]) is a well-known example of a Bayesian nonparametric priors which cannot be derived by normalizing a completely random measure.In this particular case, the unnormalized measure is obtained through a change of measure of a σ-stable process.However, many common Bayesian nonparametric priors can be defined as a normalization of a CRM and many other processes can be derived by normalising processes derived from CRMs.For instance, it can be shown that the Dirichlet Process, introduced by [12], is a normalized gamma process.Throughout the paper, we will assume that the underlying measure is a CRM and use the acronym NMRI (Normalized Random Measures with independent increments) to emphasize the independence of a CRM on disjoint intervals.
Although nonparametric priors based on normalization are extremely flexible, in many real applications data arise under different conditions and hence assuming a single prior can be too restrictive.For example, using covariates, data may be divided into different units.In this case, one would like to consider different distributions for different units instead of a single common distribution for all the units.In these situations, it is more reasonable to consider vectors of dependent random probability measures.
The definition in higher dimension is analogous (see [5]).Let U i (x) := ∞ x ν i (s) ds be the i-th marginal tail integral associated with ν i .If both the copula C and the marginal tail integrals are sufficiently smooth, then A wide range of dependence structures can be induced through Lévy copulas.For example the independence case, i.e.A×B ρ 2 (s 1 , s 2 ) ds 1 ds 2 = A ν 1 (s 1 ) ds 1 + B ν 2 (s 2 ) ds 2 for any A and B in B(R + ), corresponds to the Lévy copula where I A is the indicator function of the set A. On the other hand, the case of completely dependent CRMs corresponds to C (y 1 , y 2 ) = min{y 1 , y 2 } which yields a vector (μ 1 , μ2 ) such that for any x and y in X either μi ({x}) < μi ({y}) or μi ({x}) > μi ({y}), for i = 1, 2, almost surely.Intermediate cases, between these two extremes, can be detected, for example, by relying on the Lévy-Clayton copula defined by with the parameter θ regulating the degree of dependence.It can be seen that lim γ→0 C γ = C ⊥ and lim γ→∞ C γ = C .We close the section with the definition of vectors of normalized random measures with independent increments.Definition 4. Let (μ 1 , . . ., μd ) be a vector of CRMs on X and let pj = is called a vector of dependent normalized random measures with independent increments (VNMRI) on (X, X ).

Compound Random Measures
In this section, we will define a general class of vectors of NRMI that incorporates many recently proposed priors built using normalization, see for instance [23], [24], [39], [16] and [27] .
Following the notation in Eq. ( 2.3), we want to define a ρ d such for any j = 1, . . ., d.
In this setting we can define a compound random measure.
Definition 5. A Compound random measure (CoRM) is a vector of CRMs defined by a score distribution, a directing Lévy process and a centring measure α such that where g(y|z) is the probability mass function or probability density function of the score distribution with parameters z and ν is the Lévy intensity of the directing Lévy process which satisfies the condition where s is the Euclidean norm of the vector s = (s 1 , . . ., s d ).
The compound Poisson process with jump density g is a compound random measure with a score density g and whose directing Lévy process is a Poisson process.Therefore, compound random measures can be seen as a generalisation of compound Poisson processes.It is straightforward to show that μ1 , . . ., μd can be expressed as where m 1,i , . . ., m d,i i.i.d.
∼ g are scores and is a CRM with Lévy intensity ν (ds)α(dx).This makes the structure of the prior much more explicit.The random measures share the same jump locations (which have distribution α/α(X)) but the i-th jump has a height m j,i J i in the j-th measure and so the jump heights are re-scaled by the score (a larger score implies a larger jump height).Clearly, the shared factor J i leads to dependence between the jump heights in each measure.
In this paper, we will concentrate on the sub-class of CoRMs with a continuous score distribution which has independent dimensions and a single scale parameter so that where f is a univariate distribution.This implies that each marginal process has the same Lévy intensity of the form ν j (ds) = ν(ds) = f (s|z) ds ν (dz). (3.4) In Section 5.2, an algorithm is introduced to sample from the posterior of a hierarchical mixture models whose parameters are driven by a vector of normalized compound random measures.This sampler depends crucially on knowing the form of the Laplace Exponent and its derivatives.Some general results about the Laplace exponent and the dependence are available if we assume that the density z −1 f (s i /z) admits a moment generating function.
be the moment generating function of z −1 f (s j /z) and suppose that it exists.Then In the following theorem, the derivatives (up to a constant) of the Laplace exponent of a Compound random measure are provided. Then, Finally, it is possible to recover the underlying Lévy Copula of a compound random measure.Theorem 3.3.Let ρ d be the compound random measure defined in (3.2) and let F be the the distribution function of f .The underlying Lévy Copula of the compound random measure is where U −1 is the inverse of the tail integral function U (x) := ∞ x ν(s) ds.

CoRMs with independent gamma distributed scores
In this paper, we will focus on exponential or gamma score distributions.Throughout the paper we will write Ga(a, b) to be a gamma distribution (or density) with shape a and scale parameter b which has density x a−1 exp{−bx}.(4.1) We will consider CoRM processes with Ga(φ, 1) score distributions and so z −1 f (y/z) is the density of a gamma distribution with shape parameter equal to φ and scale parameter equal to 1/z.The Lévy intensities ν and ν and the score density f are linked by (3.4) and a CoRM can be defined by either deriving ν for a fixed choice of f and ν or by directly specifying f and ν .In this latter case, it is interesting to consider the properties of the induced ν.
Standard inversion methods can be used to derive the form of ν * for some particular choices of marginal process which are shown in Table 1.The results Table 1: The form of directing Lévy process in a CoRM which leads to particular marginal processes.
are surprising.A gamma marginal process arises when the directing Lévy process is a beta process and a stable marginal process arises when the directing Lévy process is also a stable process.Generalized gamma marginal processes lead to a directing Lévy process which is a generalization of the beta process (with a power of z which is less than 1) and re-scaled to the interval (0, 1/a).
Remark.Several authors have previously considered hierarchical models where µ 1 , . . ., µ d have followed i.i.d.CRM (or NRMII) processes whose centring measure are given a CRM (or NRMII) prior.This construction induces correlation between µ 1 , . . ., µ d and the hierarchical Dirichlet process is a popular example but we will concentrate on a hierarchical Gamma process [see e.g.32].In this case, µ j follows a Gamma process with centring measure α which also follows a Gamma process.This implies that we can write and we can write where J j,i ∼ Γ(s i , 1).This contrasts with the CoRM which implies that J j,i ∼ Γ(φ, 1/s i ) and α follows a Beta process.
It is interesting to derive the resulting multivariate Lévy intensities which can be compared with similar results in [23], [24] and [39].
Theorem 4.1.Consider a CoRM process with independent Ga(φ, 1) distributed scores.If the CoRM process has gamma process marginals then where |s| = s 1 + • • • + s d and W is the Whittaker function.If the CoRM process has σ-stable process marginals then 3 The result is proved in the appendix with the following corollary.
Corollary 4.1.Consider a CoRM process with independent exponentially distributed scores.If the CoRM has gamma process marginals we recover the multivariate Lévy intensity of [24], Otherwise, if σ-stable marginals are considered then we recover the multivariate vector introduced in [23] and [39], Alternatively, we can specify ν and derive ν.The forms for some particular processes are shown in Table 2 where U is the confluent hypergeometric function of the second kind and K is the modified Bessel function of the second kind.ν(s) Directing Lévy process The Lévy intensity of the marginal process in a CoRM with different directing Lévy processes.
Remark.There are several special cases if ν is the Lévy intensity of a beta process.Firstly, Therefore, these processes have a Lévy intensity similar to the Lévy intensity of the gamma process close to zero for any choice of φ and θ.The tails of the Lévy density are exponential.Therefore, the process has similar properties to the gamma process.
Remark.The generalized gamma process contains some special cases and the Lévy intensity of the marginal process for these process are shown in Table 3.With a generalized gamma directing Lévy process, It is straightforward to show ν(s) Directing Lévy process for small s.Therefore, the Lévy density close to zero is similar to the Lévy density of σ-stable process with parameter σ.For large s, we have Therefore, the tails will decays like exp{−s 1/2 }.
The next Theorems will provide an expression of the Laplace exponent when the scores are gamma distributed with φ ≥ 1 such that φ ∈ N. We want to stress the importance of the the Laplace transform in the Bayesian nonparametric setting.Indeed, it is the basis to prove theoretical results of the prior of interest.For instance, [23], [24] and [39] used the Laplace Transform to derive some distributional properties such as correlation, partition structure and mixed moments.Additionally, we will see that the Laplace transform plays a role in the novel sampler proposed in this paper.
Theorem 4.2.Consider a CoRM process with independent Ga(φ, 1) distributed scores.Suppose φ ≥ 1 such that φ ∈ N. Let λ ∈ (R + ) d be a vector such that it consists of l ≤ d distinct values denoted as λ = ( λ1 , . . ., λl ) with respective multiplicities n = (n 1 , . . ., n l ).Then where The proof of the previous Theorem is based on the result provided in Theorem 3.1 since the moment generating of a Gamma distribution exists and it is explicit.
Theorem 4.3.Consider a CoRM process with independent Ga(φ, 1) distributed scores.Suppose h=1 λi h z h be a function defined on the (j-1)-dimensional simplex where For the above integral we assume the usual convention that j i = 0 and j i = 1 whenever i > j.
In the following Corollary, the expression of the Laplace exponent is recovered for the special case of a CoRM with independent exponentially distributed scores.
Corollary 4.2.Consider a CoRM process with independent exponentially distributed scores.It follows that The proof of the corollary is omitted since it is a direct application of the results of the previous Theorems.Note that, if the vector has Gamma process marginals, i.e. ψ(λ i ) = log(1 + λ i ), then we recover the results in [24].If the vector has σ-stable process marginals, i.e. ψ(λ i ) = λ σ i , then we recover the result in [23] and [39].
Finally, we close the section with some results about the dependence structure of CoRM processes.First of all, it is possible to prove a result similar to Proposition 5 in [24].This result gives a close formula for the mixed moments of two dimensions of a CoRM process.The result is expressed in terms of an ordering on sets 0 ≺ s 1 ≺ • • • ≺ s j which is defined in [4].

Normalized Compound Random Measures
Vectors of correlated random probability measures can be defined by normalizing each dimension of a CoRM process.This will be called a Normalized Compound Random Measure (NCoRM) and is defined by the score distribution, the directing Lévy process and centring measure of the CoRM.The results derived in Table 1 can be used to define a NCoRM with a particular marginal process.For example, an NCoRM with Dirichlet process marginals arises by normalizing each dimension of a CoRM with gamma process marginals.It is also possible to compute the covariance of a two dimensions of an NCoRM process.Indeed, following [24], where g ρ is the function introduced in Equation (3.6).This result can be used to specify any parameters of the score distribution (or a prior for those parameters).Alternatively, if the scores are independent, the ratio of the same jump heights in the i-th and j-th dimension has the same distribution as the ratio of two independent random variables following the score distribution.For example, if the scores are independent and follow a gamma distribution with shape φ is chosen, this ratio follows an F -distribution with φ and φ degrees of freedom.

Links to other processes
Corollary 4.1 shows how the priors described in [23], [24] and [39] can be expressed in the CoRM framework.The CNMRI process [16][see also 27,3] can also be expressed in the NCoRM framework.The CNMRI prior express the random measure μg as where D is a (d × q)-dimensional selection matrix (with elements either equal to 0 or 1) and μ 1 , . . ., μ q are independent CRMs where μ k has Lévy intensity M k ν (ds)ᾱ(dx) for a probability measure ᾱ.A CNRMI process can be represented by a vector of CoRMs with score probability mass function directing Lévy intensity ν and centring measure ᾱ q k=1 M k .A CoRM process with independent scores can be used to construct a sub-class of CNRMI processes.A CoRM has a score distribution of the form f (s) = πδ s=1 +(1−π)δ s=0 , directing Lévy intensity ν (ds) and centring measure M ᾱ is identical to an unnormalized CNRMI process with q = 2 d , a D whose rows are the binary expansion of {0, 1, . . ., where f j (m) = π j δ m=1 + (1 − π j )δ m=0 .

Computational Method
We describe a method for a fitting nonparametric mixture model where the mixing measure is given a NCoRM prior.We assume that the data can be divided into d groups and y j,1 , . . ., y j,ng are the observations in the j-th group.The data are modelled as where k(y|θ) is a probability density function for y with parameter θ and p1 , . . ., pd are given a NCoRM prior.Using the notation of (3.3), we write Direct simulation from the posterior distribution is impossible since the state space is infinite dimensional.[11] describe an auxiliary variable method for the class of normalized random measure mixtures which involves intergrating out the unnormalized random measure.This makes their method difficult to extend to NCoRM mixtures and so our computational approach extends the slice sampling methods for normalized random measures with independent increments of [18].The posterior distribution can be expressed in a suitable form for MCMC by introducing latent variables v 1 , . . ., v d , s j,i and u j,i for i = 1, . . ., n j and j = 1, . . ., d, and integrating over certain jumps.Integrating over these latent variables leads to the correct marginal posterior distribution.A suitable form of posterior distribution for our MCMC method is where L = min j=1,...,d;i=1,...,ng {u j,i }.The jumps are divided into two disjoint groups The set A † has a finite number of elements which is denoted K and A has an infinite number of elements.The expectation can be expressed in terms of a univariate integral using a variation on Theorem 3.1 giving The full conditional distributions and a general discussion of methods for updating parameters are given below.Details of the implementation for specific processes are given in the appendix.

Updating v
The updating of v 1 , . . ., v d uses a variation on the interweaving approach of [38], which leads to better mixing than the standard full conditional distribution for v j .The parameter v j is updated in the following way.Firstly, we reparameterize to m † j,k = v j m † j,k and update v j from the full conditional density proportional to Secondly, we re-parameterized to m † j,k = m † j,k /v j and update v j from the full conditional density proportional to Both full conditional densities are sampled using a Metropolis-Hastings algorithm with random walk and an adaptive proposal distribution.
Updating J † and m † The density of the full conditional distribution of J † k is proportional to k .The elements of A † are also updated using a reversible jump Metropolis-Hastings method with a birth and a death move which are proposed with equal probability.The birth move involves proposing a new jump J † K+1 from a density proportional to ν J † K+1 for J † K+1 > L and m † 1,K+1 , . . ., m † d,K+1 i.i.d.
∼ f .The death move proposes to delete one of uniformly at random.The acceptance probability for the birth move is min and the acceptance probability if the k-th jump is proposed to be delete is min

Updating u
The full conditional distribution of u j,i is a uniform distribution on 0, J † s j,i for j = 1, . . ., d and i = 1, . . ., n j .Let κ be the min{u j,i } from the previous iteration and κ be the min{u j,i } from the current iteration.If κ > κ then the jumps for which J † j < κ are deleted.Otherwise, if κ < κ, a Poisson distributed number of jumps with mean Details on simulation for NCoRMs with Dirichlet process and normalized generalized gamma process marginals are provided in Appendix B.

Updating θ
The full conditional distribution of θ k is

Updating the parameters of the NCoRM prior
The full conditional distribution of the parameters of the NCoRM prior are proportional to The full conditional distribution of s j,i is a discrete distribution with a finite number of possible states proportional to m † j,s j,i I J † s j,i > u j,i k y j,i |θ s j,i , 1, . . ., n j , j = 1, . . ., d.

Illustrations
CALGB 8881 [25] and CALGB 9160 [2] clinical studies looked at the response of patients to different anticancer drug therapies.The response was white blood cell count (WBC) and patients had between four and 25 measurements taken over the course of the trial.The data was previously analysed by [30] who fit a nonlinear random effects model for the patient's response over time.
The model assumes that the mean response at time t with parameters θ = (z 1 , z 2 , z 3 , τ 1 , τ 2 , β 1 ) is given by where r = (τ 2 − t)/(τ 2 − τ 1 ) and g(θ, t) There were nine different combinations of the anticancer agent CTX, the drug GM-CSF and amifostine (AMOF) which are summarized in Table 4 Group Summaries of the data are available as part of the DPpackage in R where a non-linear regression model is fitted with f (θ j,i , t) as the mean for the i-th patient in the j-th group.We will consider the differences in the distribution of the estimated values θj,i 's across the nine studies.It is assumed that θj,i ∼ N(µ j,i , Σ j,i ), (µ j,j , Σ j,j ) ∼ pj where p1 , . . ., p9 are given a NCoRM process prior with independent Γ(φ, 1)distributed scores and Dirichlet process marginals.The centring measure α is N(µ| θ, 100Σ)IW(Σ|14, 4/9 × Σ) where θ and Σ are the sample mean and the sample covariance matrix of θ.This implies a prior mean of 1/9 × Σ.The parameter φ is given an exponential prior with mean 1.The results of the analysis are illustrated in Figure 1 which shows the posterior mean marginal density of each parameter.The results within each study are very similar with the main difference occurring between the two studies.All densities are very similar for the parameters z 1 , z 2 , z 3 and t 2 .There is a slight difference in the distribution for t 1 but much bigger differences for parameters β 0 and β 1 .The results for CALGB 8881 are unimodal whereas CALGB9160 includes additional modes at 0.5 for β 0 and −0.5 and 2 for β 1 .Figure 2 shows the posterior mean joint density of β 0 and β 1 which shows a bimodal distribution for CALGB9160 with one mode at roughly (−1.5, 0.5) (which is the mode for CALGB8881) and a second mode at roughly (−0.5, 0).This suggests that CALGB9160 may contains two groups who responded differently.The posterior median of φ was 1.03 with a 95% highest posterior density region of (0.46, 2.36).

Discussion
This paper describes compound random measures which are a new class of dependent nonparametric priors defined by a score distribution and a directing Lévy process.The tractability of these measures allows their properties to be derived and we concentrate on the class where scores are independent and gamma distributed.This allows the dependence between the measures in different dimensions to be modelled by the shape parameter of the gamma distribution.In this case, we show how compound random measures can be constructed with gamma, stable and generalized gamma process marginals.We also consider normalized version of compound random measures which can be used as priors for the mixing measure in a nonparametric mixture model.Inference can be made using a slice sampling algorithm.Normalized compound random measures includes many previously described priors which makes the links between these priors clearer.Posterior simulation has been impossible in models using some of these priors and application of the slice sampling algorithm makes inference feasible.
The compound random measure is defined using a completely random measure and a finite dimensional score distribution.For a given marginal process, the dependence between the distributions is controlled by the choice of finite dimensional score distribution.In this paper, we have concentrated on the case where the scores are independent and gamma distributed.The choice of a gamma distribution is convenient and allows properties to derived analytically but other choices such as the log normal distribution could be interesting in particular context.More generally, a multivariate score distribution could be used and a hierarchical structure for this distribution would naturally build on the work in this paper.Importantly, the modelling of dependence between random measures can be achieved by the modelling of dependence between random variables and so greatly reduces the difficulty of the problem.Future work will consider studying these classes of compound random measures.

A Proofs
Proof of Theorem 3.1 Stable marginals.The second part of the proof is straightforward and doesn't require additional algebra.
Proof of Theorem 4.2 From Equation (3.5) it follows λi ) φ under the hypothesis of independent Gamma distributed scores.The conclusion follows by noting that ∂ (n i −1)φ The last equality follows from a simple application of the Leibniz's formula, indeed where Since some of the k * 's could be zero then some terms could disappear in the expression above.For this reason, it's more convenient to write Υ φ l ( λ) as a sum over the set A φ,j instead of a sum over B φ,l .Thus, (1 + v i z) −φ , κ < z < κ.
A rejection sampler is used with rejection envelope proportional to z −1−σ (1 − z) σ+φ−1 , κ < z < κ.The acceptance probability is This rejection envelope is non-standard and can be sampled using a rejection sampler with the envelope

Table 4 :
CTX GM-CSF AMOF Study Number of patients 1 The levels of CTX (g m −2 ), GM-CSF (µg kg −1 ) and AMOF across the nine groups.CALGB 8881 is indicated as Study 1 and CALGB 9160 as Study 2.

1 Figure 1 :
Figure 1: The posterior mean marginal densities of each parameters in the CALGB example.The lines indicated a group in CALGB 8881 (solid line) and CALGB 9160 (dashed line).

Figure 2 :
Figure 2: The posterior mean joint densities of β 0 and β 1 in the CALGB example for the groups in CALGB 8881 and CALGB 9160.
x)dx be the tail integral of the marginal Lévy intensity and let U (x 1 , . . ., x d ) =

Table 3 :
The Lévy intensity of the marginal process in a CoRM with different directing Lévy processes.