Comment: Lancaster Probabilities and Gibbs Sampling

Statistic al Scienc e 2008, V ol. 23, No. 2, 187– 191 DOI: 10.1214 /08-STS252A Main article DO I: 10.1214/07-STS252 c  Institute of Mathematical Statisti cs , 2008 Comment: Lancaster Probabil i ties and Gibbs Sampling G´ era rd Letac 1. LANCASTER PROBABILITIES AS THE PROPER FRAMEW ORK It is a pleasure to congratulate the authors for this excellen t, original and p edagogical pap er. I read a p r eliminary dr aft at th e end of 2006 and I then men tioned to the authors that their w ork sh ould b e set within the framewo rk of Lancaster p robabilities, a r emoted corn er of the theory of probabilit y , now describ ed in th eir Section 6.1. The reader is referred to Lancaster ( 1958 , 1963 , 1975 ) and the syn thesis b y Koud ou ( 1995 , 1996 ) f or more details. Giv en probabilities µ ( dx ) and ν ( dy ) on sp aces X and Y , and giv en orthonormal b ases p = ( p n ( x )) and q = ( q n ( y )) of L 2 ( µ ) and L 2 ( ν ), a probabilit y σ on X × Y is said to b e of the Lancaster t yp e if either there exists a sequence ρ = ( ρ n ) in ℓ 2 suc h that σ ( dx, dy ) = " X n ρ n p n ( x ) q n ( y ) # µ ( dx ) ν ( dy ) or σ is a weak limit of suc h probabilities. Alter- nativ ely , one can say th at the sequence of signed measures [ P N n =0 ρ n p n ( x ) q n ( y )] µ ( dx ) ν ( dy ) con v erges w eakly to ward the pr obabilit y σ wh en N → ∞ (h ere ρ do es not need to b e in ℓ 2 ) . An accepta ble se- quence ρ = ( ρ n ) is called a Lancaster sequence for the quad r uple ( µ, ν, p, q ) . If p 0 = q 0 = 1 the margins of σ are ( µ, ν ) . W riting σ ( dx, dy ) = µ ( dx ) K ( x, dy ) = ν ( dy ) L ( y , dx ) the probabilit y kernel of the “ x -c h ain” considered in the pap er is k ( x, dx ′ ) = Z Y K ( x, dy ) L ( y , dx ′ ) G. L etac is Pr ofessor, L ab or atoir e d e St atistique et Pr ob abilit´ es, U niversit´ e Paul S ab atier, 31062 T oulouse, F r anc e e-mail: letac@cict.fr . This is an electro nic reprint of the original article published by the Institute of Mathematical Statistics in Statistic al Scienc e , 20 0 8, V ol. 23, No. 2 , 1 87–1 91 . This reprint differs from the origina l in pa gination a nd t yp ogr aphic detail. = " X n ρ 2 n p n ( x ) p n ( x ′ ) # µ ( dx ′ ) whic h clearly shows t hat p n is an eigenfunction for the eigen v alue ρ 2 n of the op erator T f ( x ) = R X f ( x ′ ) · k ( x, dx ′ ). I will n ot commen t here on the m ultiv ariate case X = R k and Y = R m . Ev erythin g wh ic h is kno wn ab out Lancaste r probabilities and wh ic h is sp ecific to this case is men tioned in S ection 7 of the pa- p er. T o m y knowledge, the Lancaster probabilities on th e torus ( R / Z ) 2 asso ciated to th e trigonometric orthonormal p olynomials ha ve nev er b een consid- ered. F or th e pr esen t time, the richest case is obvio usly the one w here X = Y = R and wh er e p = ( p n ) and q = ( q n ) are the orthonormal p olynomials obtained b y the Sc h midt orthonormalizatio n pro cess in L 2 ( µ ) and L 2 ( ν ) applied to the sequences ( x n ) and ( y n ), assuming furthermore that R e a | x | µ ( dx ) and R e a | y | ν ( dy ) are finite for some a > 0 . In the sequel, the term “Lancaster probabilities” will refer only to this real case. Th e follo wing should b e sp ecified clearly: Sa ying that conditions H1, H2 a nd H3 o f Section 3 are all fulfi lled is equiv alent to sa ying that P ( dx, dθ ) is a Lancaster pr ob- abilit y . An elegan t example can b e found in Buja ( 1990 , page 1049) with σ ( dx, dy ) = a + b B ( a, b ) x a − 1 y b − 1 1 A ( x, y ) dx dy where a, b > 0 and A = { ( x, y ); 0 < x, y ; x + y < 1 } . The margins are µ ( dx ) = β a,b +1 ( dx ) and ν ( dy ) = β b,a +1 ( dy ) and the Lancaster sequence is ρ n = ( − 1) n √ ab p ( a + n )( b + n ) . The present pap er on discussion is based on three observ ations. The fir st one is crucial: the t wo-c omp o- nen ts Gibbs sampler is v ery e asy t o p erform with a Lancaster pr obabilit y . This is the statemen t in Th e- orem 3.1. P arts a and b are w ell kno wn but part c is elegan t and s u rprizing. 1 2 G. LET A C 2. NA TURAL EXPONENTIAL F AM ILIES In order to explain the other t wo observ ations, let us introd uce some n otation: a (not necessarily b ound ed) p ositiv e measure µ on R is said to b e in M ( R ) if it is not concentrat ed on one p oin t and if its Laplace trans f orm L µ ( θ ) = e k µ ( θ ) = Z ∞ −∞ e θ x µ ( dx ) is such that the int erior Θ( µ ) of the int erv al D ( µ ) = { θ ∈ R ; L µ ( θ ) < ∞} is not empty . T o such a µ ∈ M ( R ) one asso ciates the one-dimensional natural exp onentia l f amily (NEF): F = F ( µ ) = { P ( µ, θ )( dx ) = e θ x − k µ ( θ ) µ ( dx ); θ ∈ Θ ( µ ) } . Since θ 7→ k µ ( θ ) is strictly co nv ex on Θ( µ ) the map θ 7→ m = k ′ ( θ ) = R ∞ −∞ xP ( µ, θ )( dx ) is injectiv e and its i mage M F = k ′ µ (Θ( µ )) is an op en in terv al called the d omain of the means of F . On e den otes by m 7→ θ = ψ µ ( m ) the in verse map from M F to Θ( µ ) . Fi- nally we sa y F or µ is steep if M F is the in terior of the con ve x sup p ort of µ. F or instance D ( µ ) = Θ( µ ) (in this case F is said to b e regular) implies that F is steep. The conv erse is not true. Diac onis and Ylvisak er ( 1979 ) sho w that if F is regular, if x 0 ∈ M F and if λ > 0 th en there exists a constan t C ( x 0 , λ ) such that π x 0 ,λ ( dθ ) = C ( x 0 , λ ) e λ ( θx 0 − k µ ( θ )) 1 Θ( µ ) ( θ ) dθ is a probability . W e call { π x 0 ,λ ; x 0 ∈ M F , λ > 0 } the Diaconis–Yl visak er family asso ciated to the NEF F . W e no w reparameterize it by th e mean. More sp ecif- ically , d en ote by ν x 0 ,λ ( dm ) = C ( x 0 , λ ) exp λ ( x 0 ψ µ ( m ) − k µ ( ψ µ ( m ))) · ψ ′ µ ( m ) 1 M F ( m ) dm the image of π x 0 ,λ ( dθ ) by θ 7→ m = k ′ µ ( θ ). Finally consider the distribution on R 2 defined by σ ( dx, dm ) = P ( µ, ψ µ ( m ))( dx ) ν x 0 ,λ ( dm ) . Note that th e marginal distribution µ 1 ( dx ) of σ ( dx, dm ) do es n ot b elong to F except in the normal case. (Pro ving this is an amusing exercise. It eve n holds when the reference measure dθ in the Diaconis–Yl visak er family is replaced b y an y other p ositiv e measure. 1 ) The second observ ation of the pap er, and a quite original one, is th at σ ( dx, dm ) is a Lancaster pr obabilit y if F is either binomial (Sec- tion 4.1), or P oisson (Section 4.2), or Gaussian (Sec- tion 4.3). An element of the Diaconis–Ylvi sak er f am- ily associated with the binomial case B ( θ , n ) is th e b eta distribution ν 1 ( dθ ) = β a,b ( dθ ) and the marginal distribution of X is the s o-calle d hyp ergeometric d is- tribution µ 1 ( dx ) = n X k =0  n k  ( a ) k ( b ) n − k ( a + b ) n δ k ( dx ) . (1) The construction of a L an caster pr ob ab ility with these margins ( µ 1 , ν 1 ) ha v e n ev er b een done b efore. Here th e Lancaster sequence is ρ j = n ! / ( a + b + n ) j ( n − j )! for 0 ≤ j ≤ n and ρ j = 0 if n < j. The Lan- caster pr obabilities obtained for F = P oisson and F = Gaussian are familiar and are men tioned in Koudou ( 1996 , Sect ion 3.3) and stud ied in Koudou ( 1995 ). My guess is that these 3 t yp es of NEF are the only ones w ith su c h a p rop erty: this is obviously false for the three other quadratic NEF (Nega tiv e b inomial, gamma, hyp erb olic), f or whic h ν x 0 ,λ ( dm ) has ve ry few momen ts. The reader can c hec k for example that the same is tr u e for the NEF ge nerated b y a s table la w of parameter α ∈ (0 , 1) concen trated on (0 , ∞ ) and d efined b y k µ ( θ ) = − c ( − θ ) α : recal l that α = 1 / 2 giv es the celebrated In v erse Gaussian distributions (the case α ∈ [1 , 2) has not to b e in v estigated sin ce it is n ot stee p). In order to explain the conte nt of the third ob- serv ation of the pap er, we in tro duce the Jorgensen set Λ( µ ) of µ ∈ M ( R ) . It is the set of λ ≥ 0 such that for λ > 0 there exists µ λ ∈ M ( R ) suc h that Θ( µ λ ) = Θ ( µ ) and suc h that L µ λ = ( L µ ) λ . W e im- p ose 0 ∈ Λ( µ ) . F or instance Λ( µ ) = [0 , ∞ ) if and only if F ( µ ) is made of infinitely divisible d istr ibutions. On th e other hand Λ( µ ) is the set of nonnegativ e in tegers if µ = δ 0 + δ 1 , namely if F is the Bernoulli family . In general Λ( µ ) can b e a quite complicate d additiv e semigroup: see Letac, Malouc he and Mau- rer ( 2002 ) for its description when µ is the con- v olution of a negativ e binomial distribution with a 1 The family G obtained in this wa y is also a conjugate family to F , whic h mea ns that the a p osteriori distribution π ( dθ | x ) is in G when the a priori distribution π is in G . F or this reason we do sp eak of the Diaconis–Ylvisaker family in- stead of the conjugate family of the paper, even if the later has the characteristi c prop erty men tioned in S ection 2.3.2. COMMENT 3 Bernoulli distrib ution. No w c onsider µ ∈ M ( R ) a nd λ and η in Λ( µ ) . Let ( X, Y ) ∼ P ( µ λ , θ ) ⊗ P ( µ η , θ ) . W rite S = X + Y ∼ P ( µ λ + η , θ ) (the distribution of Y kno wing S do es not dep end on θ ) and denote b y σ ( ds, dy ) the joint distribution of ( S, Y ) . The au- thors obs erv e that, wh en F h ap p ens to b e a quadratic NEF, σ is a Lancaste r probabilit y: this is the essence of Section 5. Ho we v er, this is a particular case of the follo wing classica l result menti oned in Eagleson ( 1964 ): sup p ose that λ, η , ξ are in Λ( µ ) and let ( X, Y , Z ) ∼ P ( µ λ , θ ) ⊗ P ( µ η , θ ) ⊗ P ( µ ξ , θ ) . Denote b y σ ( ds, dt ) the joint distrib u tion of ( S, T ) = ( X + Y , Y + Z ) . T h en σ is a Lancaster probab ility if F is a quadratic NEF. More sp ecifically if ( p ( λ ) n ) is the sequence of the orthonormal p olynomials f or P ( µ λ , θ ) and if 1 /c n ( λ ) is the p ositiv e square ro ot of the coefficient o f x n in p ( λ ) n the corresp onding Lan- caster sequence i s ρ n = c n ( η ) p c n ( λ + η ) c n ( η + ξ ) . (2) Th us Section 5 is based on the particular case λ = n 1 , η = n 2 , ξ = 0 of this result. 3. FINDING ALL LANCASTER F AMILIES WITH GIVEN MARGINS Giv en a pair of probabilities ( µ, ν ) on R s u c h that R e a | x | µ ( dx ) and R e a | y | ν ( dy ) are finite for some a > 0, consider the set L ( µ, ν ) of Lancaster probabilities σ with margins ( µ, ν ) and the set S ( µ, ν ) of cor- resp ond ing Lancaster sequ ences ρ = ( ρ n ) ∞ n =0 . They are isomorph ic compact conv ex sets which are com- pletely kno wn if w e kno w their extreme points. W e denote by I ( µ ) the smallest closed in terv al I suc h that µ ( I ) = 1 . W e consider sev eral cases: Case A. I ( µ ) is b oun ded, I ( ν ) is unboun ded. Case B. I ( µ ) = R and I ( ν ) is a h alf-line. Case C. I ( µ ) = I ( ν ) = R . Case D. I ( µ ) and I ( ν ) are half-lines. Case E. I ( µ ) and I ( ν ) are b ounded. Cases A and B are easy: the only Lancaster prob- abilit y is the pr o duct measure. Denote by a n > 0 and b y b n > 0 th e co efficien ts of x n in the orthonor- mal p olynomials p n and q n . Case C is quite inte rest- ing: from a r emark able result of T yan a nd Thomas ( 1975 ), extending an idea of Sarmano v and Brato ev a ( 1967 ), wh ich sa ys that if γ = lim inf ( a 2 n /b 2 n ) 1 / 2 n and if ρ ∈ S ( µ, ν ), there exists a probabilit y α ( dt ) on [ − γ , γ ] suc h that a n ρ n /b n = R γ − γ t n α ( dt ) . Simi- larly in t he case D, assuming without loss of gener- alit y that I ( µ ) and I ( ν ) are p ositiv e half-lines and if γ = lim inf ( a n /b n ) 1 /n then there exists a p robabil- it y α ( dt ) on [0 , γ ] such that a n ρ n /b n = R γ 0 t n α ( dt ) . The resu lts of Ty an and Thomas ( 1975 ) can also es- sen tially b e found again in T yan, Derin and Thomas ( 1976 ) and ha v e b een redisco vered b y C hristian Berg, quoted in Ismail ( 2005 , page 11 4) wh o do es not seem to b e aw are of this previous work. W e shall sp eak abou t case E later on. Not e that for µ = ν th e results by T ya n and Thomas are quite exciting since they mean that a Lancaster sequ ence m ust b e the moment sequence of a probabilit y either on [ − 1 , 1] (case C) or on [0 , 1] (case D). If w e are f or- tunate enough to p r o v e that ρ n = t n is a Lancaster sequence for all t ∈ [ − 1 , 1] (case C ) or all t ∈ [0 , 1] , b y the theorems of Ty an and Thomas, w e h av e a complete description of the Lancaster pr obabilities L ( µ, µ ) since they are parameteriz ed b y the proba- bilities α on [ − 1 , 1] or on [0 , 1] . Interestingly enough, this is known to happ en on ly for 4 t yp es of µ : Gaus- sian, Poisso n, negativ e binomial an d gamma. Th e corresp ondin g Lancaster probabilities (see Bar-Lev et al., 1994 ) are the only ones whic h b elong to a t wo- dimensional natur al exp onen tial f amily with v ari- ance fun ction of the f orm  a ( m 1 ) f ( m 1 , m 2 ) f ( m 1 , m 2 ) a ( m 2 )  . More sp ecifically one can conjecture the f ollo wing: • If I ( µ ) = R and if ( t n ) is in S ( µ, µ ) for all t ∈ [ − 1 , 1] then µ is Gaussian. • If I ( µ ) = [0 , ∞ ) and if ( t n ) is in S ( µ, µ ) for all t ∈ [0 , 1] then µ is either gamma, or Poi sson, or negativ e binomial. In the gamma case, it is interesting to consider the classical t wo- dimensional distribution o f Kibble ( 1941 ) and Moran ( 1967 ) with correlat ion r ∈ [0 , 1] and Jorgensen parameter q . It can b e defined by its Laplace transform Z ∞ 0 Z ∞ 0 e − sx − ty σ r ( dx, dy ) = (1 + s + t + (1 − r ) st ) − q . As observ ed by D’jac henko ( 1962 ), this probability is actually a Lancaster probabilit y w ith ρ n = r n , and th us an extremal one (the last three references are 4 G. LET A C tak en from J ohnson and Kotz, 1972 , pages 479– 482). This mea ns that i n general σ is a Lancaster proba- bilit y for the gamma margins µ = ν = γ q if and only if it is a mixing of Kibble and Moran distributions, whic h means that there exists a probabilit y distri- bution α ( dr ) on [0 , 1] s u c h that Z ∞ 0 Z ∞ 0 e − sx − ty σ ( dx, dy ) = Z 1 0 (1 + s + t + (1 − r ) st ) − q α ( dr ) . T ak e for instance α ( dr ) = β η,q − η ( dr ) to get bac k ( 2 ) for the g amma case a nd λ = ξ = q − η . F or the cases C and D and for ν not an affine transformation of µ , there is no kn o wn example where the set of the extreme p oin ts of L ( µ, ν ) can b e com- pletely describ ed. Koudou ( 1995 , 1996 ) h as shown that ρ n = t n is a Lancaster sequence: • for µ = P a and ν = P b ( P a means Poisson distri- bution with mea n a ) for 0 ≤ t ≤ ( a/b ) 1 / 2 if a ≤ b ; • for µ = N B a,λ and ν = N B a,λ [the negativ e bino- mial distribution N B a,λ is (1 − a ) λ P ∞ n =0 ( λ ) n n ! a n · δ n ( dx )] for 0 ≤ t ≤ ( a/b ) 1 / 2 if a ≤ b ; • for µ = N B a,λ and ν = γ λ for 0 ≤ t ≤ a 1 / 2 . In these thr ee cases, one can conjecture th at one has obtained all t he extreme p oin ts of S ( µ, ν ) . Consider a h yp erb olic distribution µ q as describ ed in Section 2.4 and simply defi n ed b y L µ q ( θ ) = (cos θ ) − q with q > 0 and Θ( µ q ) = ( − π 2 , π 2 ) . Lai and V ere-Jones ( 1975 ) ha v e pro ve d that ( t n ) i s nev er in S ( µ q , µ q ) (an other pro of is in Bar-Lev et al., 1994 ). F orm ula ( 2 ) applies h ere with c n ( q ) = ( q ) n n ! . In ( 2 ) w e tak e 0 ≤ η ≤ q and λ = ξ = q − η to sho w that the sequence ρ n = c n ( η ) c n ( q ) = 1 B ( η , q − η ) Z 1 0 t n t η − 1 (1 − t ) q − η − 1 dt is an elemen t of S ( µ q , µ q ) . This illustrates the results of T y an and Thomas with α ( dt ) = β η,q − η ( dt ) . On e can conjecture (as done by Lai and V ere-Jones for q = 1) that suc h a Lancaster sequence in dexed b y η ∈ [0 , q ] is an extreme p oint o f S ( µ q , µ q ), and that all extreme p oin ts are of this typ e. 4. THE CASE WHERE µ AND ν HA VE BOUNDED SUPPORT This is the case E ab o ve. F or th e v ariet y of re- sults already obtained in the literature, this is the ric hest case. F or fu ture researc h, it is the most c hal- lenging. If µ = ν supp ose that there exists x 0 suc h that | p n ( x ) | ≤ p n ( x 0 ) µ almost surely , and c onsider K ( x, y , z ) = ∞ X n =0 1 p n ( x 0 ) p n ( x ) p n ( y ) p n ( z ) . (3) Koudou ( 1995 ) has sho wn th at K ≥ 0 for almost all ( x, y , z ) in the µ sense implies that the extreme p oints of S ( µ, µ ) are d efi ned b y ρ n = p n ( x ) /p n ( x 0 ) when x describ es the supp ort of µ. This extend s a remark able pap er b y Eagleson ( 1969 ) devote d to the case wh ere µ is discrete with finite supp ort, where it is sho wn in particular that K ≥ 0 when µ is a binomial distribution. As men tioned in the pap er, the analysis by Koudou ( 1996 ) of Gasp er’s ( 1971 ) delicate results shows th at K ≥ 0 when µ = β a,b is a b eta d istribution su c h that a, b ≥ 1 / 2 [note that the case min ( a, b ) < 1 / 2 is op en]. The particular case a = b ≥ 1 / 2 deserves a sp ecial men tion. Using the transformation x 7→ 2 x − 1, we first mo v e the distributions fr om [0 , 1] to [ − 1 , 1] and w e introd uce ∆( x, y , z ) = 1 − x 2 − y 2 − z 2 + 2 xy z . F or − 1 < z < 1 w e consider the p lane domain U z = { ( x, y ); ∆ > 0 } . This d omain is limited by an ellipse E z tangen t to the sides of the unit square [ − 1 , 1] 2 . Denote µ a ( dx ) = 2 1 − 2 a B ( a,a ) (1 − x 2 ) a − 1 1 ( − 1 , 1) ( x ) dx . The n umb er x 0 in v olv ed in the definition of K in ( 3 ) is 1, and t he p olynomials p n are t he Jacobi polynomials with suitable paramet ers and normalize d such that they b ecome orthonormal with resp ect to µ α . With these notation, K is zero outside o f U z and is equal to K a ( x, y , z ) = C ( α )[(1 − x 2 )(1 − y 2 )(1 − z 2 )] 1 − a ∆ a − 3 / 2 in U z . The imp ortan t p oint is the follo wing. F or z ∈ ( − 1 , 1) consider the extremal Lancaster pr oba- bilities σ z ( dx, dy ) = K a ( x, y , z ) µ a ( dx ) µ a ( dy ) . These Lancaster pr obabilities σ z are the only ones (to - gether with the cen tered n on s ingular Gaussian dis- tributions with co v ariance of the t yp e h a b b a i ) to b e elliptically con toured. More sp ecifically , let E = R 2 ha v e the Euclidean stru cture suc h that U z is the unit disk. S a ying that σ z is el liptically con toured means that σ z is inv arian t by the orthogonal group O ( E ) of this Euclidean structur e. Th is characte rization is the consequence of an elegan t result of McGra w and COMMENT 5 W agner ( 1968 ). While most of the “results” ab out elliptically con toured distributions in R d are triv- ially reduced to co nsiderations ab out rotational in- v arian t distribu tions, this is not the case here. T h e reason is that the canonical basis of R 2 is struc- turally imp ortan t for L ancaster pr obabilities. In the other hand this canonica l basis is not orthonormal for the Euclidean structure asso ciated with a giv en elliptically con toured distribution and this mak es at- tractiv e the McGra w and W agner result. Koudou ( 1995 ) sh ows th at w e ha ve K ≥ 0 when µ ( dx ) = q + 1 2 π p p 2 − x 2 1 − x 2 1 ( − p,p ) ( x ) dx where q > 0 and p = 2 √ q / (1 + q ) . When q is an i n- teger this s tr ange pr obabilit y is the Planc herel mea- sure of the Gelfand pair asso ciated to th e homo- geneous tree wh ere ev ery v ertex has q + 1 neigh- b ors. T he corresp onding p olynomials are ca lled the Cartier–Dunau p olynomials in the literature (see Ar- naud, 1994 ). A general theory of the probabilities µ with b ound ed s u pp ort su c h that the function K of ( 3 ) is p ositiv e could b e a sub ject of researc h. As an example, I do not know whether K ≥ 0 or n ot w hen µ is t he hypergeometric distribution ( 1 ) co nsid er ed in the pap er, w here the orthonormal p olynomials are the Hahn polynomials. When µ and ν are t wo probabilities with b ounded supp ort suc h that ν is n ot an a ffine tr ansform of µ , the searc h of extreme p oin ts of the Lanca ster mea- sures do es not seem to h a v e b een done for any exam- ple. Supp ose that we ha v e found some ρ ∈ S ( µ, ν ) . A go o d w a y to create other elemen ts of S ( µ, ν ) is to pic k a ∈ S ( µ, µ ) and b ∈ S ( ν, ν ) . It is easy to see that ( a n ρ n b n ) ∞ n =0 is also in S ( µ, ν ) . Applying this remark to the int eresting pair ( µ 1 , ν 1 ) defi ned by ( 1 ) and to the new Lancaster s equence ρ j = n ! / ( a + b + n ) j ( n − j )! disco v ered by the authors w ould lead to a b etter understand ing of S ( µ 1 , ν 1 ) . 5. CONCLUSION W e r eferred to sev eral b r igh t pap ers by Eagle- son, Koudou, McGra w and W agner or T yan and Thomas, and to a gen uine masterpiece b y Gasp er . 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