Department of Agricultural & Resource Economics, UCB

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1 Department of Agriultural & Resoure Eonomis, UCB CUDARE Working Papers (University of California, Berkeley Year 2007 Paper 1048 Large Deviations Approah to Bayesian Nonparametri Consisteny: the Case of Polya Urn Sampling M. Grendar G. G. Judge R. K. Niven Institute of Measurement Sienes SAS, Bratislava, Slovakia University of California, Berkeley and Giannini Foundation The University of New South Wales at ADFA This paper is posted at the esholarship Repository, University of California. ub/1048 Copyright 2007 by the authors.

2 Large Deviations Approah to Bayesian Nonparametri Consisteny: the Case of Polya Urn Sampling Abstrat The Bayesian Sanov Theorem (BST identifies, under both orret and inorret speifiation of infinite dimensional model, the points of onentration of the posterior measure. Utilizing this insight in the ontext of Polya urn sampling, Bayesian nonparametri onsisteny is established. Polya BST is also used to provide an extension of Maximum Non-parametri Likelihood and Empirial Likelihood methods to the Polya ase.

3 Large Deviations Approah to Bayesian Nonparametri Consisteny: the Case of Pólya Urn Sampling M. Grendár, G. G. Judge, and R. K. Niven Sep 21, 2007 Abstrat The Bayesian Sanov Theorem (BST identifies, under both orret and inorret speifiation of infinite dimensional model, the points of onentration of the posterior measure. Utilizing this insight in the ontext of Pólya urn sampling, Bayesian nonparametri onsisteny is established. Pólya BST is also used to provide an extension of Maximum Non-parametri Likelihood and Empirial Likelihood methods to the Pólya ase. Keywords: Pólya L-divergene, Bayesian Maximum (A Posteriori Probability method, Maximum Non-parametri Likelihood method, Empirial Likelihood method AMS: 60F10, 60F15 1 Introdution In Bayesian nonparametri (or infinite dimensional statistis a stritly positive prior is put over a set Φ of probability distributions. In this ontext let r be the true data sampling distribution of a random sample X n X 1,X 2,...,X n. Provided that r Φ, as the sample size grows to infinity, the posterior distribution π( X n = x n over Φ is expeted to onentrate in a neighborhood of the true sampling distribution r. Whether and under what onditions this indeed happens is a subjet of Bayesian nonparametri onsisteny Department of Mathematis, FPV UMB, Tajovského 40, Banská Bystria, Slovakia; Institute of Mathematis and CS of the Slovak Aademy of Sienes (SAS, Institute of Measurement Siene of SAS. marian.grendar@savba.sk. To whom orrespondene should be addressed. Supported by VEGA 1/3016/06 grant. Graduate Shool, 207 Giannini Hall, University of California, Berkeley, CA, judge@are.berkeley.edu. The Shool of Aerospae, Civil and Mehanial Engineering, The University of New South Wales at ADFA, Canberra, ACT, 2600, Australia; and Niels Bohr Institute, Copenhagen University, 2100 Copenhagen Ø, Denmark; r.niven@adfa.edu.au. investigations. Surveys of the subjet inlude [7], [9], [25], [23], [24], [26]. More formally, as in [7], onsisteny of a sequene of posteriors with respet to a metri d an be defined as follows: The sequene {π( X n,n 1} is said to be d-onsistent at r, if there exists a Ω 0 R with r(ω 0 = 1 suh that for ω Ω 0, for every neighborhood U of r, π(u X n 1 as n goes to infinity. If a posterior is d-onsistent for any r Φ then it is said to be d-onsistent. There, two modes of onvergene are usually onsidered: onvergene in probability and almost sure onvergene and d is usually either Hellinger distane or a metri whih metriizes weak topology. Freedman s [5] lassi theorem on Bayesian nonparametri onsisteny for X taking on values from a finite set was in [1], [2], and independently in [6] proved by means of a Bayesian Sanov Theorem (BST. In [11] the onsisteny was via BST established for a ountable set of densities. BST (a.k.a. Sanov Theorem for Sampling Distributions is Bayesian ounterpart of Sanov Theorem for Empirial Measures [21], [3]. The latter is a basi result of Large Deviations (LD theory [4]. LD theory is a sub-field of probability theory where, informally, the typial onern is about the asymptoti behavior, on a logarithmi sale, of the probability of a given event. To promote and extend the Bayesian large deviations approah, we study Bayesian nonparametri onsisteny for a basi non-iid setting, where data are drawn aording to multiolor Pólya urn sheme. We demonstrate that data sampling distributions from the set Φ asymptotially a posteriori onentrate on the Pólya L-projetion(s of the true sampling distribution r on Φ. The statement holds also under misspeifition (i.e., when r / Φ. In [13] BST [11] was used to provide a probabilisti interpretation and justifiation of the Empirial Likelihood (EL method. Based on the Pólya BST pre- 1

4 sented in this paper we extend EL and Maximum Nonparametri Likelihood methods to the Pólya sampling ase. The paper is organized as follows. First, Pólya urn sampling is briefly desribed. Next, the urn is embedded into a Bayesian setting. The Pólya L-divergene, whih governs the exponential deay of posterior probability, is introdued next and then a Bayesian Sanov Theorem for Pólya Sampling is stated and proved. It diretly implies Bayesian nonparametri onsisteny. Next, the onsisteny result is used to provide an extension of Maximum Non-parametri Likelihood and Empirial Likelihood methods to the Pólya sampling ase. Finally, using the Bayesian large deviations approah, we provide a ouple of insights into Bayesian onsisteny. 2 Multiolor Pólya Urn Sampling Consider an urn ontaining α i > 0 balls of olors i, i = 1,2,...,m; and let m be finite. There is a total number N M i=1 α i of balls in the urn and when it is neessary to stress it, the urn will be alled an N-urn. We identify the set of possible olors with support X of a random variable X. A single ball is drawn from the urn, reorded and then returned together with Z balls of the same olor. Assuming n min(α 1,α 2,...,α m, the drawing is repeated n times. This sampling is known as the multiolor Pólya Eggenberger (PE urn sheme;.f. [8], [22], [15]. Prominent speial ases of the PE sheme are: iid sampling ( = 0, sampling without replaement ( = 1, and the ase of = 1. Given the PE sheme, the probability π(x n = x n q N ; that a sequene x n of n balls will be drawn from initial onfiguration q N of N balls is (.f. [22], [15]: π(x n = x n q N ; m i=1 α i(α i + (α i + (n i 1, N(N + (N + (n 1 (1 where vetor q N onsists of q N i α i N, and n i is the number of ourrenes of the i-th outome from X in the sample x n, i = 1,2,...,m. 3 Bayesian Embedding Let P(X be set of all probability mass funtions with the support X. Let Φ P(X and let Φ N denote intersetion of Φ with the set of all possible onfigurations of the N-urn. Let Φ N be the support of prior distribution π(q N of initial onfigurations q N of N-urn. Let r N be the true initial onfiguration of N-urn where r N is not neessarily in Φ N. From r N a sequene x n is drawn aording to the Pólya sampling sheme, that we haraterize by the parameter. Consequently, under this framework the bayesian arrives at the posterior probability distribution π(q N X n = x n ; of initial onfigurations of the N-urn. 4 Pólya L-divergene The Pólya L-divergene Lβ (q p of the probability mass funtion (pmf q P(X with respet to pmf p P(X is L β (q p m i=1 p i log(q i + βp i β m i=1 q i q i log. q i + βp i By the ontinuity argument Lβ 0 (q p m i=1 p i logq i 1. More onisely, Lβ (q p = L(q + βp p + 1 βi(q q + βp, where using standard onventions I( is the I-divergene I(a b a i log a i b i [3] and L( is the L-divergene L(b a a i logb i [10], [11]. Though base of the logarithm is immaterial, the natural logarithm will be used. The Pólya Lβ -projetion ˆq of p on A P(X is ˆq arginf q A Lβ (q p. The value of L β -divergene at an Lβ -projetion of p on A is denoted by L β (A p. Hereafter it is assumed finite. 5 Bayesian Sanov Theorem for Pólya Sampling Sanov Theorem for Empirial Measures [21] is wellknown; reader is direted to [4], [3]. Initiated by [18], the Sanov Theorem for Pólya Sampling was reently proven in [12]. The Bayesian ounterpart of Sanov Theorem was to the best of our knowledge first studied in [1] and [2] and independently in [6]. Our proof of Bayesian Sanov Theorem (BST for Pólya Sampling is based on [11], and also utilizes tools from [12]. 2

5 Asymptoti investigations of posterior onsisteny will be arried on under the following assumptions: 1 n and N go to infinity in suh a way that β(n n N β (0,1 as n, 2, r N onverges in the total variation metris to r P(X as n. Topologial qualifiers are meant in topology indued on the m-dimensional simplex by the usual topology on R m. Bayesian Sanov Theorem for Pólya Sampling. Let A Φ be an open set. Let β(n β (0,1, r N r as n. Then, for n, 1 n logπ(qn A x n ; = with probability one. {L β (A r L β (Φ r } Proof. The proof will be onstruted separately for > 0, < 0, = 0. For 0 Nq / (Z m N / Z, formula (1 an equivalently be expressed as [15]: π(x n q N ; = Γ( N Γ ( N + n m i=1 Γ( Nqi Γ( Nqi + n i, (2 where Γ( is the Gamma funtion. The following bounds [16] on the Gamma funtion Γ( are imposed: (b 1logb (a 1loga (b a < log Γ(b Γ(a < ( < b 1 ( logb a 1 loga (b a, (3 2 2 whih is valid for 0 < a < b. Let > 0 and note that the other restritions under whih (1 and (2 are equivalent are not ative, sine n min(α 1,α 2,...,α m. We use the bounds in (3 to get the upper bound U n of the probability π(q N A x n ;: U n = q N A π(qn m i=1 enl(qn i, 1 2n q N Φ π(qn m i=1 enl(qn i, 1 n and the lower bound L n similarly (to get L n just replae 1/2n with 1/n in U n. There, q l(q N N i,α ( i q N log( β(n α i + ( q N i β(n + νn i α + β(n q N log( i β(n + νn i, α { 1 n, 1 2n} and ν n is the empirial measure indued by the sample x n. Next, we use the simple bounds of [11] to develop the upper upper bound U n by Ū n and lower bound L n by L n, as follows: where Ū n = m i=1 enl( ˆqN i (A, 1 2n, 1 2n π( q N m i=1 enl( qn i (Φ, 1 n, 1 n, L n = ˆq N (S,α arg sup q N S m i=1 enl( ˆqN i (A, 1 n, 1 n π( q N m i=1 enl( qn i (Φ, 1 2n, 1 2n, m i=1 l(q N i,α, ( m q N (S,α arg sup l(q N i,α logπ(qn q N S i=1 n By the Strong Law of Large Numbers for Pólya Sampling (whih follows from [12], Thm. 2 and Borel Cantelli Lemma, ν n r, almost surely, as n. The Pólya L-divergene is ontinuous in q and A is open, by assumption. Thus, n 1 logū n onverges, } with probability one, to {Lβ (A r L β (Φ r, as n. This is the same as the point of almost sure onvergene of 1 n logl n and the Theorem for > 0 is thus proven. For 0 (1 Nq / (Z m (1 N / Z, the formula (1 an equivalently be expressed as (f. [12]: π(x n q N ; = Γ( 1 N n Γ ( 1 N m ( Γ 1 Nq i (. i=1 Γ 1 Nq i n i (4 Let < 0. Note that the other restritions under whih (1 and (4 are equivalent are not ative, sine n min(α 1,α 2,...,α m. The proof then an be onstruted along the same lines as for > 0. At the final {( stage one arrives at an expression ( of the } form m qi i=1 β r i log( qi β r i qi β log( qi β whih, after little algebra, an be seen to be the Pólya L-divergene. The ase of = 0 (i.e., iid urn has already been studied in [11]. The exponential deay rate funtion is L(A r L(Φ r, whih is the same as Lβ 0 (A r Lβ 0 (Φ r, implied by the ontinuity of the Pólya L- divergene Lβ (.. 3

6 6 Bayesian Nonparametri Consisteny for Pólya Sampling Bayesian nonparametri onsisteny for Pólya sampling is just a orollary of the above Pólya BST. To see this let for ε > 0, Aε C (Φ {q : Lβ (q r (Φ r > ε,q Φ}. L β Corollary. Let there be a finite number of Pólya L-projetions of r on Φ. As n, π(q Aε C (Φ x n ; 0, with probability one. Standard Bayesian onsisteny (i.e., under orret speifiation; r Φ follows as a speial ase of the Corollary. Informally, the posterior probability onentrates on the Pólya L-projetions of r on Φ. Observe that the Pólya L-projetion of r on Φ is an asymptoti instane of the sampling distribution with the supremal (over Φ value of the posterior probability; hene it is asymptoti form of the Maximum A-posteriori Probable sampling distribution. 7 MNPL and EL in Pólya ase Consider the problem of seleting an initial Pólya urn omposition q N from a set Φ of suh ompositions, when there is a sample x n (or empirial pmf ν n whih the sample indues drawn from the true urn r N, aording to PE sampling sheme with parameter. Pólya BST ditates that we selet in the asymptoti ase (n, ν n r, β(n β Lβ -projetion or r on Φ. Most ommonly, Φ is formed by moment onstraints that define a linear family of distributions L (u {q : X q i (u j (x i a j = 0, j = 0,1,2,...,J}, where u j is a real-valued funtion on X, u 0 = 1 R m, a R J+1, a 0 = 1. The Pólya L-projetion whih Pólya BST selets in this ase has the form βr i ˆq i (β,,λ =, e β J j=0 λ j(u j (x i a j 1 where λ R J+1 are Lagrange multipliers. If the hoie is to be made among all possible q N for a fixed, suffiiently large N, then by Pólya BST, we should selet the Pólya L-projetion ˆq(β,,λ of r on P(X, whih in this ase is just r, regardless of and β. For n, N not suffiiently large, there are two possibilities. It is possible either to selet the initial onfiguration with highest value of the posterior probability 1 π(q N ν n ; or the Pólya L-projetion of ν n on Φ N. As n, the two methods selet the same onfiguration(s. Observe that when = 0, the latter method selets onfiguration(s with the highest value of non-parametri likelihood m i=1 νn i logq i. Pólya BST thus extends Maximum Non-parametri Likelihood (MNPL method into the Pólya sampling: Pólya MNPL selets the urn onfiguration(s with the highest value of negative of Lβ (q νn. In the ase of iid sampling it was observed (f. [13] that the Bayesian Sanov Theorem [11] provides a probabilisti Bayesian interpretation and justifiation of Empirial Likelihood method [19], [17] in the parameter estimation ontext [20]. EL, viewed as estimation method, double-maximizes the non-parametri likelihood riterial funtion subjet to parametrized onstraints [20], [19], [17]. The above disussion thus diretly shows how to extend EL into the Pólya sampling framework: the negative of Pólya L-divergene has to be double-maximized subjet to parametri onstraints. 8 Summary The main advantage of Bayesian Sanov Theorem (BST approah to Bayesian nonparametri onsisteny over the traditional one lays not that muh on the tehnial side as on the oneptual one. BST identifies the rate funtion governing exponential deay of the posterior measure, and this in turn identifies points of onentration of the posterior as those distributions whih minimize the rate funtion. In the ase of i.i.d. sampling the posterior onentration points identified by BST are those distributions ˆq whih in the feasible set Φ maximize r logq. If Φ is the set of all distributions (with the same support, then ˆq is unique and equal r. Traditional approahes to Bayesian nonparametri onsisteny do not see the onentration point (i.e., r, under orret speifiation as a solution of the optimization problem. BST also shows that the points of asymptoti onentration of posterior probability are asymptoti instanes of a posteriori most probable (MAP sampling distributions. This fat implies that the mean posterior sampling distribution (i.e., the preditive distribution 1 Hene the name of the method assoiated with this seletion sheme: Bayesian Maximum Probability method; f. [14] 4

7 is, in general, not the point of onvergene under misspeifiation. In this paper we have used under the Pólya sampling sheme the Bayesian Sanov Theorem (BST to identify sampling distributions, on whih the posterior probability asymptotially onentrates. This way, Bayesian nonparametri onsisteny for Pólya sampling was established both under orret speifiation of model as well as under misspeifiation. In [13] it was pointed out that the non-parametri likelihood riterion, as well as methods that are based on its maximization (i.e., Maximum Non-parametri Likelihood (MNPL and Empirial Likelihood (EL methods are limited to independent sampling. The point was made on the grounds of a Bayesian large deviations interpretation of the methods. On the same ground the Pólya extension of BST implies that under Pólya sampling it is the Pólya non-parametri likelihood funtion (i.e., negative of the Pólya L-divergene that has to be maximized. Referenes [1] Ben-Tal, A., Brown, D. E. and R. L. Smith. (1987. Posterior onvergene under inomplete information. Teh. rep U. of Mihigan, Ann Arbor. [2] Ben-Tal, A., Brown, D. E. and R. L. Smith. (1988. Relative Entropy and the onvergene of the posterior and empirial distributions under inomplete and onfliting information. Teh. rep U. of Mihigan. [3] Csiszár, I. and P. Shields. (2004. Information theory and statistis: a tutorial. Foundations and Trends in Communiations and Information Theory, vol. 1, pp [4] Dembo, A. and O. Zeitouni. (1998. Large Deviations Tehniques and Appliations. New York:Springer-Verlag. [5] Freedman, D. A. (1963. On the asymptoti behavior of Bayes estimates in the disrete ase. Ann. Math. Statist., 34: [6] Ganesh, A. and N. O Connell. (1999. An inverse of Sanov s Theorem. Stat. & Prob. Letters, 42: [7] Ghosal, A., Ghosh, J. K. and R. V. Ramamoorthi (1999. Consisteny issues in Bayesian Nonanparametris. Asymptotis, Nonparametris and Time Series: A Tribute to Madan Lal Puri, , Dekker. [8] Eggenberger, F. and G. Pólya. (1923. Über die Statistik verketter Vorgänge. Z. Angew. Math. Meh., vol. 1, pp [9] Ghosh, J. and R. V. Ramamoorthi. (1999. Bayesian Nonparametris. New York: Springer Verlag. [10] Grendár, M. (2005. Conditioning by rare soures. Ata Univ. Belii, Math. 12: Online at [11] Grendár, M. (2006 L-divergene onsisteny for a disrete prior. Jour. Stat. Res. 40/1: Correted at arxiv. [12] Grendár, M and R. K. Niven. (2006. The Pólya urn: Limit Theorems, Pólya Divergene, Maximum Entropy and Maximum Probability. Available at arxiv:ond-mat/ ; submitted. [13] Grendár, M. and G. Judge. (2007. A Bayesian Large Deviations Probabilisti Interpretation and Justifiation of Empirial Likelihood. Preprint, ARE UC, Berkeley; under revision. Available at ub/1035. There is a typo in LST, inherited from [11]. The sets M e and N should be replaed. Also, Nε C should be modified aordingly. [14] Grendár, M. (2007. Maximum Probability and Maximum Entropy, Bayesian Maximum Probability and Maximum Non-parametri Likelihood. (Probabilisti Regularization of Inverse Problems, unpublished manusript. [15] Johnson, N. L., Kotz, S. and N. Balakrishnan. (1997. Disrete Multivariate Distributions. New York: Wiley. [16] Kečkić, J. D. and P. M. Vasić. (1971. Some inequalities for the gamma funtion. Publ. Inst. Math. (Beograd(N.S., vol. 11, pp [17] Mittelhammer, R., Judge, G. and D. Miller (2000. Eonometri Foundations. Cambridge:CUP. 5

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