Stirlings approximations for exchangeable Gibbs weights

Stirling’s appro ximations for exc hangeable Gibbs w eigh ts ∗ Annalisa Cerqu e tti † Dep artment of Metho ds and Mo dels for Ec onomics, T erritory and Financ e Sapienza University of R ome, Italy Septem b er 26, 2018 Abstract W e obtain some appr oximation results for the weigh ts appea ring in the exchange- able partition pro bability function iden tifying Gibbs partition models of parameter α ∈ (0 , 1), as in tro duce d in Gnedin and Pitman (2006). W e rely on approximation results for central and non-central gener alized Stirling n umbers a nd o n known re- sults for conditional a nd unconditional α diversit y . W e provide a n application to an approximate Bay esia n nonpara metric estimatio n of discovery probability in sp ecies sampling pro blems under no r malized inv ers e Ga ussian prio rs. 1 In tro du ction Exc hangeable Gibbs partitions (Gnedin and Pitman, 2006 ) are the largest class of infin ite exc hangeable partitions of the p ositiv e in tegers N and are c haracterized by a consistent sequence of exc hangeable partition probabilit y functions (EPPF) in the form p ( n 1 , . . . , n k ) = V n,k k Y j =1 (1 − α ) n j − 1 , for eac h n ≥ 1, ( n 1 , . . . , n k ) the sizes of the blo cks, α ∈ ( −∞ , 1) and V n,k co efficien ts satisfying the bac kwa rd recursiv e relation V n,k = ( n − k α ) V n +1 ,k + V n +1 ,k +1 , for V 1 , 1 = 1. The sp ecific form of the V n,k iden tifies the sp ecific α -Gibbs mo del, eac h aris- ing (cfr. Th. 12, Gnedin and Pitman, 20 06) as a mixture of extreme p artitions, namely: Fisher’s (1943) ( α, ξ ) p artitions, ξ = 1 , 2 , . . . , f or α < 0, Ew ens’ (1972) Poisson-Diric hlet ( θ ) partitions for α = 0, and Pitman’s (2003 ) Poi sson -K in gman P K ( ρ α | t ) conditional ∗ AMS (2000) subje ct classific ation . Primary: 60G 58. Secondary: 60G09. † Corresponding author, SAPIENZA Universit y of Rome, Via del Castro Laurenziano, 9, 00161 R ome, Italy . E-mail: annalisa .cerquetti@gmail .com 1 partitions for α ∈ (0 , 1), wh ere ρ α is the L ´ evy densit y of the Stable sub ordinator. F or α ∈ (0 , 1), θ > − α and mixing density γ θ , α ( t ) = Γ( θ + 1) Γ( θ /α + 1) t − θ f α ( t ) , where f α ( t ) is the density of the stable distribution, the P oisson-Kingman ( ρ α , γ α,θ ) mo del co rr esp onds to th e two -p ar ameter Poisson-Dirichlet partition mo del (Pitman, 1995, Pitman and Y or, 1997) with EPPF in the f orm p α,θ ( n 1 , . . . , n k ) = ( θ + α ) k − 1 ↑ α ( θ + 1) n − 1 k Y j =1 (1 − α ) n j − 1 , (1) whic h reduces to the Diric hlet ( θ ) partition mo del for α = 0. Recen tly exc hangeable Gibbs partitions and the corresp onding discrete random probability measures, ha ve found application as prior mod els in the Ba ye sian nonparametric treatmen t of sp ecies sampling problems, (cfr. Lijoi et al. 2007b, Lijoi et al. 2008). Here in terest typically lies in estimating the diversit y of a p opu lation of sp ecies with u n kno wn relativ e abu ndances, b y estimating b oth predictiv e sp e ci es richness (cfr. F a v aro et al. 2 009) and p osterior sp e ci es evenness (cfr. Cerquetti, 2012). By assuming the sequence of lab els o f differen t observ ed sp ecies to b e a r ealizat ion of an exc hangeable s equ ence ( X i ) i ≥ 1 with de Finetti measure b elonging to the Gibbs class, p osterior predictiv e results are obtained with re- sp ect to a future m -sample conditioning on th e multiplicit ies ( n 1 , . . . , n k ) of the first k sp ecies observ ed in a basic n -sample. With resp ect to the whole P K ( α, γ ) class, the ( α, θ ) partition mo del (1) stand s out for its mathematical tractabilit y , and a huge amoun t of results are a v ailable for this mo del b oth in random partitions theory lik e in Ba yesia n nonp arametric implemen ta- tions. But in general, without loss of generalit y , for mixing distributions in the form γ ( t ) = h ( t ) f α ( t ), for f α ( t ) the Stable dens ity , the Gibbs weig hts are obtained by mix- ing the conditional E P PF p α ( n 1 , . . . , n k | t ), (cfr Pitman, 2003 , eq. (66)) b y the sp ecific mixing distribution, which r esults in the follo wing in tegral V α,h n,k = Z ∞ 0 α k Γ( n − k α ) t − k α  Z 1 0 p n − 1 − k α f α ((1 − p ) t ) dp  h ( t ) dt. (2) It follo ws that calculating explicitly the P K ( ρ α , h × f α ) co efficien ts outside the ( α, θ ) class can be hard, mostly due to the lac k of explicit expression for th e α -stable densit y . T o giv e an example, in the Ba y esian nonparametric implementa tion of those mo d els, the most studied alternativ e to the ( α, θ ) mo d el is the nor malize d gener alize d Gamma class (Pitman, 2003; see Lijoi et al. 2005 , 2007a, Cerquetti, 20 07) whose mixing densit y arises b y the exp onen tialy tilting of th e stable density , hence γ ( t ) = h ( t ) × f α ( t ) = exp { ψ α ( λ ) − λt } f α ( t ) , 2 for ψ α ( λ ) = (2 λ ) α , λ > 0 the Laplace exponent of f α . By the r eparametrization λ = β 1 /α / 2, γ α,β ( s − 1 /α ) = exp ( β − 1 2  β s  1 /α ) f α ( s − 1 /α ) α − 1 s − 1 /α − 1 (3) and the corresp ondin g ( V n,k )’ (Cerquetti, 2007) are giv en b y V α,β n,k = e β 2 n α k Γ( n ) Z ∞ 0 λ n − 1 e − ( β 1 /α +2 λ ) α ( β 1 /α + 2 λ ) n − k α dλ, and can b e rewr itten (Lijoi et al. 2 007a) by the c hange of v ariable x = ( β 1 /α + 2 λ ) α , dλ = (2 α ) − 1 x 1 /α − 1 dx , as a linear com bination of incomplete Gamma functions V α,β n,k = e β α k − 1 Γ( n ) n − 1 X i =0  n − 1 i  ( − 1) i ( β ) i/α Γ( k − i α ; β ) . (4) The computational burd en implicit in w orking with those priors is then clear. Addition- ally , when d ealing with p osterior p redictiv e analysis in this setting, Bay esian n onpara- metric d istributions for qu an tities of interest in sp ecies samp ling problems are t ypically c haracterized by a ratio, that w e term p osterior Gibbs weights , of the kind V n + m,k + k ∗ V n,k , (5) for k ∗ the n umb er of differen t sp ecies observed in the additional m -sample, whose com- plexit y increases with the complexit y of the prior w eigh ts. It looks therefore interesti n g to in v estigate the possibility to obtain some sort of approxima tion r esult for b oth prior and p osterior Gibbs coefficient s. Here we prop ose a S tirling’s appro ximation by relying on appro ximation r esults for b oth g eneralized c entr al and non c entr al S tirling n um b ers. The p ap er is organize d as follo ws. In S ection 2. w e pro vide some pr eliminaries and obtain an appro ximation result for th e prior Gibbs we ights of the P K ( ρ α , h × f α ) cl ass. In Section 3 w e obtain the appro ximation r esu lt for the p osterior Gibbs weigh ts (5) by relying on a recen t result for conditional α div ersit y (Cerqu etti, 2001) and in Section 4. w e apply our findin gs to an appro ximate p osterior estimation of disco v ery probabilit y under n ormalized in v erse gaussian partitio n mo del (Pitman, 2003 , Lijoi et al. 2005 ) whic h corresp onds to the generalize d Gamma case for α = 1 / 2. 2 Stirling’s appro ximations for P K ( ρ α , h × f α ) w eigh ts F rom the fi rst order Stirlin g’s app ro ximation for ratio of Gamma fu nctions Γ( n + a ) Γ( n + b ) ∼ n a − b is an easy task to v erify that, for n → ∞ and k ≈ s n α , th e follo wing app ro ximation holds for the EPPF (1) of the P D ( α, θ ) mo d el p α,θ ( n 1 , . . . , n k ) ≈ ( α ) k − 1 Γ( k ) Γ( n ) Γ( θ + 1) Γ( θ /α + 1)  k n α  θ /α k Y j =1 (1 − α ) n j − 1 . (6) 3 Our fi rst aim is to generaliz e (6) to the P K ( ρ α , γ ) class. First recal l that gener al- ize d Stirling numb ers are com binatorial co efficien ts app earing in the expansion of ris- ing factorials ( x ) n = x ( x + 1) · · · ( x + n − 1) in terms of generalized rising factorials ( x ) k ↑ α = x ( x + α )( x + 2 α ) · · · ( x + ( k − 1) α ), hen ce ( x ) n = n X k =1 S − 1 , − α n,k ( x ) k ↑ α . An explicit expression for those num b er s is giv en by T oscano’s (1939) formula S − 1 , − α n,k = 1 α k k ! k X j =1 ( − 1) j  k j  ( − j α ) n , (7) additionally (cfr. e.g. Pitman, 2006) they corresp ond to p artial Bel l p olynomials of the kind B n,k (1 − α • ) = X { A 1 ,...,A k }∈P [ n ] k k Y j =1 (1 − α ) | A j |− 1 = n ! k ! X ( n 1 ,...,n k ) k Y j =1 (1 − α ) n j − 1 n j ! for α ∈ (0 , 1), wher e the first sum is ov er all p artitions ( A 1 , . . . , A k ) o f [ n ] in k b lo c ks, and the second sum in ov er all c omp ositions of n in to k parts. The follo wing resu lt relies on an approximat ion for generalized Stirling num b ers obtained in Pitman (1999) and on a local appro ximation result for th e distribution of a pr op er normalization of the n umb ers of blocks K n of an α ∈ (0 , 1) Gibbs partition. Prop osition 1. L et V α,h n,k b e the c o efficients in the EPPF of an exchange able Gibbs p artition a rising by a gener al P K ( ρ α , h × f α ) Poisson-Kingman mo del, then the fol low- ing Stirling’s appr oximation hold s for n → ∞ and k ≈ sn α V α,h n,k ≈ α k − 1 Γ( k ) Γ( n ) h "  k n α  − 1 /α # . Pr o of. In Pitman (199 9, cfr. eq. (96)) an asymptotic form ula f or the generalize d Stirlin g n umb ers S − 1 , − α n,k for n → ∞ and 0 < s < ∞ , with k ≈ s n α is deriv ed b y known lo cal limit appro ximations for the n umber of blocks in a partition generated b y a P D ( α, α ) mo del b y a stable d ensit y , n amely S − 1 , − α n,k ≈ α 1 − k Γ( n ) Γ( k ) g α ( s ) n − α , (8) where, g α ( s ) = α − 1 f α ( s − 1 /α ) s − 1 − 1 /α . No w, for K n the num b er of blo cks in a P K ( ρ α , h × f α ) partition mo del (Pitman, 2003) almost s u rely , for n → ∞ K n /n α → S a 4 for S α ∼ h ( s − 1 /α ) g α ( s ), wh ic h implies the follo wing local limit app ro ximation for k ≈ sn α holds for the distribution of K n P ( K n = k ) ≈ V α,h n,k S − 1 , − α n,k ≈ h ( s − 1 /α ) g α ( s ) n − α . Th u s the result follo ws by s ubstitution.  Example 2. The r esult in Pr op osition 1. agrees with the Stirling’s app ro xima- tion for the weig hts of the ( α, θ ) mo del obtained in (6). It is enough to notice that P D ( α, θ ) = P K ( ρ α , γ α,θ ) for γ θ , α = h α,θ × f α corresp ondin g to the θ p olynomial tilting of the stable densit y for h ( t ) = Γ( θ + 1) Γ( θ /α + 1) t − θ . As for the G eneralized Gamma m o del recalled in (3) and (4), the first ord er Stirling’s appro ximation for the Gib b s w eigh ts will corresp ond to V α,β n,k ≈ α k − 1 Γ( k ) Γ( n ) exp n β − n 2 ( β /k ) 1 /α o . 3 Stirling’s appro ximation for p osterior P K ( ρ α , h × f α ) w eigh ts As recalled in the In tr o duction, p osterior pr edictiv e d istr ibutions in Ba y esian nonpara- metric estimation in s p ecies sampling pr oblems u nder Gibbs pr iors are typically c harac- terized by a ratio of Gib b s wei ghts. As an example consid er the p osterior j oin t d istribu- tion of the rand om v ector K m , L m , S 1 , . . . , S K m for K m the n um b er of n ew sp ecies ge n - erated b y the additional m sample, L m the n umb er of new observ ations in new sp ecies, and S 1 , . . . , S K m the v ector of the sizes of the n ew sp ecies in exchange able r andom or der , namely P ( K m = k , L m = s, S K m = ( s 1 , . . . , s K m ) | n ) = = s ! s 1 ! · · · s k ∗ ! k ! V n + m,k + k ∗ V n,k  m s  ( n − k α ) m − s k ∗ Y i =1 (1 − α ) s i − 1 or the Ba yesia n nonparametric estimato r f or the num b er K m of new sp ecies in the additional sample (Lijoi et al. 2007 b ) E ( K m | K n = k ) = m X k ∗ =0 k ∗ V n + m,k + k ∗ V n,k S − 1 , − α, − ( n − k α ) m,k ∗ . Here S − 1 , − α, − ( n − k α ) m,k ∗ are non-central generalized S tirling num b ers, defined by the con- v olution relation (see Hsu an d Shiue, 1998, Eq. (16)) S − 1 , − α, − ( n − k α ) m,k ∗ = m X s = k ∗  m s  ( n − k α ) m − s S − 1 ,.α s,k ∗ , (9) 5 whose corresp ondin g T oscano’s formula (7) ma y b e deriv ed by (9) a s follo ws S − 1 , − α, − ( n − k α ) m,k ∗ = m X s = k ∗  m s  ( n − k α ) m − s 1 α ∗ 1 k ∗ ! k ∗ X j =1 ( − 1) j  k ∗ j  ( − j α ) s = = 1 α k ∗ 1 k ∗ ! k ∗ X j =1 ( − 1) j  k ∗ j  m X s = k ∗  m s  ( n − k α ) m − s ( − j α ) s = 1 α k ∗ 1 k ∗ ! k ∗ X j =1 ( − 1) j  k ∗ j  ( n − ( j + k ) α ) m . (10) In order to deriv e an appr o ximation resu lt for V n + m,k + k ∗ V n,k w e replicate the approac h ad op ted in the previous Section b y first looking for an ap- pro ximation for non c e ntr al generalized S tirling n umb ers b y exploiting their definition in terms of con vo lution of ce ntral generalized Stirling num b ers (9). Prop osition 3. The fol lowing appr oximation holds for non c entr al gener alize d Stir- ling numb ers S − 1 , − α, − ( n − k α ) m,k ∗ for α ∈ (0 , 1) and m lar ge S − 1 , − α, − ( n − k α ) m,k ∗ ≈ α 1 − k ∗ Γ( m ) Γ( k ∗ )Γ( n − k α ) m n − k α − α Z 1 0 (1 − p ) n − k α − 1 p − α − 1 g α ( z p − α ) dp (11) for g α ( s ) = f α ( s − 1 /α ) α − 1 s − 1 − 1 /α and z = k ∗ /m α . Pr o of. By the definition of n on-cen tral S tirling n umb ers S − 1 , − α, − ( n − k α ) m,k ∗ = m X s = k ∗  m s  ( n − k α ) m − s S − 1 , − α s,k ∗ and b y (8) S − 1 , − α, − ( n − k α ) m,k ∗ = m X s = k ∗  m s  Γ( s ) Γ( k ∗ ) s − α α 1 − k ∗ g α  k ∗ s α  ( n − k α ) m − s . This ma y b e rewritten as S − 1 , − α, − ( n − k α ) m,k ∗ = α 1 − k ∗ m X s = k ∗ m Γ( m ) Γ( s )Γ( m − s + 1) Γ( s ) Γ( k ∗ ) Γ( n − k α + m − s ) Γ( n − k α ) s − α − 1 g α  k ∗ m α m α s α  . F or z = k ∗ /m α , b y first order S tirlin g appro ximations f or ratio of Gamma functions S − 1 , − α, − ( n − k α ) m,k ∗ ≈ α 1 − k ∗ Γ( m ) m − α + n − k α − 1 Γ( n − k α )Γ( k ∗ ) m X s = k ∗  m − s m  n − k α − 1 g α ( z p − α )  s m  − α − 1 = 6 and b y the c han ge of v ariable p = s/m = α 1 − k ∗ m − α + n − k α Γ( m ) Γ( n − k α )Γ( k ∗ ) Z 1 0 (1 − p ) n − k α − 1 g α ( z p − α ) p − α − 1 dp.  No w w e are in a p osition to obtain the S tirling’s app ro ximation for the ratio V n + m,k + k ∗ /V n,k resorting to a general result for c onditional α diversity for exc hangeable Gibbs partitions driv en by the Stable sub ordinator, recen tly obtained in Cerquetti (201 1). Prop osition 4. The fol lowing Stirling’s app r oximation holds for the gener al p osterior Gibbs weights for a P K ( ρ α , h × f α ) p artition mo del for la r ge m and k ∗ ≈ sm α : V n + m,k + k ∗ V n,k ≈ h ( s − 1 /α ) s k α k ∗ Γ( k ∗ )Γ( n ) m − ( n − k α ) E n,k ,α ( h ( S − 1 /α ))Γ( m )Γ( k ) = α k + k ∗ − 1 h ( s − 1 /α ) s k Γ( k ∗ ) m − ( n − k α ) V n,k Γ( m ) . (12) Pr o of. Let Π b e a P K ( ρ α , γ ) partitio n of N d riv en by the stable sub ordin ator for some 0 < α < 1 and some mixin g probabilit y distribution th at w ithout loss of generalit y we assume in th e form γ ( t ) = h ( t ) f α ( t ). Fix n ≥ 1 and a p artition ( n 1 , . . . , n k ) o f n with k p ositiv e b o x-sizes, then b y a result in Cerqu etti (2011) for K m the n umb er of new blo c ks indu ced b y an additional m -sample K m m α | ( K n = k ) d − → S n,k α,h for S n,k α,h ha ving d en sit y f h,α n,k ( s ) = h ( s − 1 /α ) ˜ g α n,k ( s ) E α n,k [ h ( S − 1 /α )] , (13) for ˜ g α n,k ( s ) = Γ( n ) Γ( n − k α )Γ( k ) s k − 1 /α − 1 Z 1 0 p n − 1 − k α f α ((1 − p ) s − 1 /α ) dp the densit y of the p ro duct Y α,k × [ W ] α where Y α,k has densit y g α,k α ( y ) = Γ( k α + 1) Γ( k + 1) y k g α ( y ) for g α ( y ) = α − 1 y − 1 − 1 /α f α ( y − 1 /α ), indep enden tly of W ∼ β ( k α, n − k α ). Additionally E α n,k [ h ( S − 1 /α )] = V n,k ,h α 1 − k Γ( n ) Γ( k ) . Hence the follo wing local appro ximation for the p osterior distr ib ution of K m (Lijoi et al. 20 07) holds P α,h ( K m = k ∗ | K n = k ) = V n + m,k + k ∗ V n,k S − 1 , − α, − ( n − k α ) m,k ∗ ≈ h ( s − 1 /α ) ˜ g α n,k ( s ) E α n,k [ h ( S − 1 /α )] m − α . 7 Substituting (11 ) the result easily f ollo w s.  Example 5. [P oisson-Diric hlet ( α, θ ) mod el] Direct first order Stirling’s appr o xima- tion for the p osterior P D ( α, θ ) co efficien ts pro vides V n + m,k + k ∗ V n,k = ( θ + k α ) k ∗ ↑ α ( θ + n ) m ≈ α k ∗ (( k ∗ ) θ /α + k )Γ( k ∗ )Γ( θ + n ) Γ( m )Γ( θ /α + k ) m θ + n , while an application of (12) for h ( s − 1 /α ) = Γ( θ +1) Γ( θ/α +1) s θ /α yields V n + m,k + k ∗ V n,k ≈ α k ∗ + k − 1 h  k ∗ m α  − 1 /α i − θ  k ∗ m α  k Γ( k ∗ ) m − ( n − k α ) Γ( θ + 1) Γ( m )Γ( θ /α + 1) Γ( θ /α + 1)Γ( θ + n ) α k − 1 Γ( θ /α + k )Γ( θ + 1) and the t w o results agree . Example 6 . [Ge n eralized Gamma ( β , α ) mo del] Th e exac t form of the ratio in this case is giv en by V n + m,k + k + V n,k = α k ∗ P n + m − 1 i =0  n + m − 1 i  ( − 1) i ( β ) i/α Γ( k + k ∗ − i/α ; β ) ( n ) m P n − 1 i =0  n − 1 i  ( − 1) i ( β ) i/α Γ( k − i/α ; β ) while an application of (12) with h [( k ∗ /m α ) − 1 /α ] = exp ( β − m 2  β k ∗  1 /α ) pro vides V n + m,k + k ∗ V n,k ≈ α k ∗ exp  − m 2  β k ∗  1 /α   k ∗ m α  k Γ( k ∗ ) m − ( n − k α ) Γ( n ) Γ( m ) P n − 1 i =0  n − 1 i  ( − 1) i ( β ) i/α Γ( k − i α ; β ) . 4 Appro ximate estimation of disco v ery probabilit y u nder in v erse Gaussian partition mo del In Lijoi et al. (2007 b) a Ba ye sian nonparametric estimate, under squ ared loss fu nction of the probabilit y of observing a new sp ecies at the ( n + m + 1) th dra w condition- ally on a b asic sample with obser ved m ultiplicities ( n 1 , . . . , n k ), without observing the in termediate m obser v ations has b een obtained as ˆ D n,k m = m X k ∗ =0 V n + m +1 ,k + k ∗ +1 V n,k S − 1 , − α, − ( n − k α ) m,k ∗ . (14) The authors ev en deriv e explicit form ulas under Diric hlet, t wo- p arameter P oisson-Diric hlet and normalized In verse Gaussian priors, whic h corresp ond to the generalize d Ga mm a 8 prior mo del f or α = 1 / 2. Notice that the larger is the size o f the additional sample the more computationally hard wo uld be to calculate explicitly ( 14). By an applicatio n of (12), an appro ximate ev aluation of (14) for large m w ould corresp onds to ˆ D n,k m ≈ m X k ∗ =0 α k ∗ + k h ( s − 1 /α ) s k Γ( k ∗ + 1)( m + 1) − ( n − k α ) V n,k Γ( m + 1) S − 1 , − α, − ( n − k α ) m,k ∗ = = α k ( m + 1) − ( n − k α ) V n,k Γ( m + 1) m X k ∗ =0 h (( k ∗ /m α ) − 1 /α )( k ∗ /m α ) k α k ∗ Γ( k ∗ + 1) S − 1 , − α, − ( n − k α ) m,k ∗ , whic h sp ecializes under in v erse Gaussian mod el as follo ws ˆ D n,k m = ( m + 1) − ( n − k / 2) Γ( n ) Γ( m + 1) P n − 1 i =0  n − 1 i  ( − 1) i β 2 i Γ( k − 2 i ; β ) × × m X k ∗ =0 exp ( − m + 1 2  β k ∗  2 ) k ∗ √ m + 1 S − 1 , − 1 / 2 , − ( n − 2 k ) m,k ∗ , and b y th e T oscano’s formula for generalized Stirling num b ers r ecalled in (10) simplifi es to ˆ D n,k m = ( m + 1) − ( n − k / 2) Γ( n ) Γ( m + 1) P n − 1 i =0  n − 1 i  ( − 1) i β 2 i Γ( k − 2 i ; β ) × × m X k ∗ =0 exp ( − m + 1 2  β k ∗  2 ) k ∗ √ m + 1 k ∗ X j =1 ( − 1) j  k ∗ j  ( n − ( j + k ) / 2) m . T o h a v e an idea of the r eduction of the computational b urden obtained pro vided b y the appro ximation her e we rep ort the exact form ula obtained in Lijoi et al. (2007b) ˆ D n,k m = ( − β 2 ) m +1 ( n ) m +1 m X k ∗ =0 P n + m i =0  n + m i  ( − β 2 ) − i Γ( k + k ∗ + 1 + 2 i − 2( m + n ); β ) P n − 1 i =0  n − 1 i  ( − β 2 ) − i Γ( k + 2 + 2 i − 2 n ; β ) × × m X s = k  m s  2 s − k − 1 s − 1  2 k − 2 s Γ( s ) Γ( k ∗ ) n − j / 2 m − s . References Cerquetti, A. (2007) A note on Ba y esian nonparametric priors deriv ed from exp o- nen tially tilted Poisson-Kingman mo dels. Stat & P r ob L e tters , 77, 18, 1705– 1711. Cerquetti, A. (2011) Conditional α -div ersit y for exc h angeable Gibbs partitions d riv en b y the stable sub ordinator. Pr o c e e ding of the 7th Confer enc e on Statistic al Com- putation a nd Complex Systems, Padova, Italy, 2011 9 Cerquetti, A. (2012) A Ba y esian nonparametric estimator of Simp son’s index of even- ness under Gibbs priors. arXiv:1203.1 666 v1 [math.ST] Ewens, W. (1972) Th e sampling th eory of selectiv ely neutral alleles. The or. Pop. Biol. , 3, 87–112 . Fisher, R.A., Corb et, A.S. an d W illiams, C. B. (194 3) The relation b et we en the n umb er of sp ecies and the num b er of ind ividuals in a rand om sample of an animal p opulation. J. Animal E c ol. 12, 42–58. Gnedin, A. and Pitman, J. (20 06) Exchangeable Gibbs partit ions and Stirling tri- angles. J. M ath. Sci. , 138, 3, 5674– 5685. Hsu, L. C, & Shiue, P. J. (1998) A unified approac h to generalized Stirling num b er s . A dv. A ppl. Math. , 20, 366-384 . Lijoi, A. Mena, R.H. and Pr ¨ unster, I. (2005) Hierarc hical m ixture mo deling with normalized in verse Gaussian priors. J. Am. Stat. Asso c. , 100, 1 278-1291. Lijoi, A., Men a, R.H. and Pr ¨ unster, I. (2007a ) Controlli n g the reinforcemen t in Ba y esian non-parametric mixture mod els. J. R oy, Stat. So c. B , 69, 4, 715–7 40. Lijoi, A., Mena, R.H. and Pr ¨ unster, I. (2007b) Bay esian n onparametric estimation of the probabilit y of d iscov ering new sp ecies. Biometrika , 94, 769– 786. Lijoi, A., Pr ¨ unster, I. and W alker, S.G. (2008) Ba yesian nonparametric estimator deriv ed from conditional G ibb s structures. Ann. Appl. Pr ob ab. , 1 8, 15 19–1547. Pitman, J. (1995) Exchangeable and partially exc h an geable random partitions. Pr ob ab. Th. R el. Fields , 102 : 145-158. Pitman, J. (1996) Some dev elopments of the Blac kw ell-MacQueen urn sc heme. In T.S. F erguson, Shapley L.S., and MacQueen J.B., editors, Sta tistics, Pr ob ability and Game The ory , volume 30 of IMS L e ctur e Notes-Mono gr aph Series , page s 245–267 . Institute of Mathematical Statistics, Hayw ard , CA. Pitman, J. (199 9) Brownian moti on, bridge, exc u r sion and mea n d er c h aracterized b y sampling at indep end en t uniform times. Ele ctr on. J. Pr ob ab. , 4, 1- 33. Pitman, J . (2003) P oisson-Kingman partitions. In D.R. Goldstein, editor, Sci enc e and Statistics: A F e stschrift for T erry Sp e e d , v olume 40 o f Lecture Notes-Mo n ograph Series, pages 1–34. Institute of Mathematic al Statistics, Ha yward, California. Pitman, J. (2006) Combinatorial Sto chastic Pr o c esses . Eco le d ’Et´ e de Probabilit ´ e de Sain t-Flour XXXI I - 2 002. Lecture Notes in Mathematics N. 1875, S pringer. Pitman, J. an d Yor, M. (1997) Th e t w o-parameter P oisson-Diric h let distribution deriv ed from a stable su b ord inator. Ann. Pr ob ab. , 25, 85 5–900. 10 Toscano, L. (1939) Numeri di Stirling generalizz ati op er atori differenziali e p olinomi ip ergeometrici. Comm. Pontificia A c ademic a Scient. 3:721- 757. 11