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A Sufficient Statistics Construction of Exponential Family Lévy Measure

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A Sufficient Statistics Construction of Exponential Family Lévy Measure A Sufficient Statistics Construction of Exponential Family L´evy Measure Densities for Nonparametric Conjugate Models Robert Finn Brian Kulis Ohio State University CSE Department Ohio State University CSE Department Abstract conjugacy assumes its role in these two arenas cur- rently exist in unsettled contrast with another. To be sure, in the parametric setting, conjugate families such Conjugate pairs of distributions over infinite as the normal gamma, multinomial Dirichlet, and the dimensional spaces are prominent in machine Bernoulli beta, are familiar and often employed in the learning, particularly due to the widespread task of inference. Furthermore, such families afford adoption of Bayesian nonparametric method- the researcher convenient formulae for the updating ologies for a host of models and applications. of posterior parameters as a function of the prior pa- Much of the existing literature in the learning rameters and observed data. Thus, the acceptance and community focuses on processes possessing application of conjugate families is received with a uni- some form of computationally tractable con- fied and uncontroversial disposition. Unfortunately, at jugacy as is the case for the beta process and the present, no such unification exists for the treat- the gamma process (and, via normalization, ment of conjugate families in the case of nonparamet- the Dirichlet process). For these processes, ric Bayesian models, a class of models which greatly conjugacy is proved via statistical machinery enrich the researcher's toolkit and as such deserve the tailored to the particular model. We seek to effort required to achieve a greater understanding of address the problem of obtaining a general how to unify the notion of conjugacy in an infinite construction of prior distributions over infi- dimensional setting. nite dimensional spaces possessing distribu- tional properties amenable to conjugacy. Our Bayesian nonparametric (BNP) modeling is a promi- result is achieved by generalizing Hjort's con- nent and widely used technique in the machine learn- struction of the beta process via appropriate ing community, providing a broad class of statistical utilization of sufficient statistics for exponen- models which are more flexible than classical nonpara- tial families. metric models and more robust than both classical and Bayesian parametric models. BNP models such as the Chinese restaurant process, the Indian buffet process, 1 Introduction and Dirichlet process mixture models have obtained great success in problem domains such as clustering, dictionary learning, and density estimation, respec- Since Raiffa and Schlaifer [1] first formalized the no- tively [2{5]. This success is in large part due to the tion of conjugate prior families in 1961, they have re- adaptive nature of BNP models. As such, this model- peatedly earned their distinguished role as the key to ing framework permits the data to determine the level the operability of Bayesian modeling. Indeed, in both of model complexity rather than entail the specifica- the parametric and nonparametric cases, the ability to tion of the complexity level by the researcher. In or- perform statistical inference in a computationally effi- der for this adaptive framework to yield a computa- cient manner hinges on the existence of such conjugate tionally tractable model, one is commonly required to prior families. construct a pair of conjugate distributions defined over Although conjugacy is key in both parametric and non- an infinite dimensional space. parametric modeling venues, the manners in which Success in constructing such conjugate pairs in an in- th finite dimensional setting has been achieved in specific Appearing in Proceedings of the 18 International Con- cases producing, for example, the Dirichlet, gamma, ference on Artificial Intelligence and Statistics (AISTATS) 2015, San Diego, CA, USA. JMLR: W&CP volume 38. and beta processes. In each of these cases, the con- Copyright 2015 by the authors. struction of a suitable likelihood/prior pair and sub- 250 A Sufficient Statistics Construction of Exponential Family L´evy Measure Densities sequent illustration of the desired form for the poste- tributional properties are determined, in a sense to rior yielding conjugacy, is model specific. For exam- be made clear in the sequel, by an exponential fam- ple, in the case of the Dirichlet process, conjugacy of ily. This provides the first of the two constructions the multinomial and Dirichlet processes arises directly required to produce a conjugate pair over an infinite from the conjugacy of their marginals. In contrast, dimensional space. conjugacy in the case of the beta and Bernoulli pro- cesses is defined and obtained in a seemingly less direct manner, i.e. via the form of the densities for the pro- 2 Statistical Preliminaries cesses' respective L´evymeasures. This modification of the definition of conjugacy from the finite dimensional, We first recall the definition of a completely random or parametric, setting seems to yield a tractable def- measure, and then provide a brief account of their con- inition relying on a specification of the distributional nection to stochastic processes, while also supplying a properties of the infinitesimal increments of the pro- representation theorem in terms of Poisson processes. cesses under consideration. In the second part of this section we will gather req- uisite facts regarding exponential families and their While this modified definition provides a framework sufficient statistics. for conjugacy in a number of cases, e.g. of the gamma process as well, the actual proof techniques employed 2.1 Completely random measures in establishing this form of conjugacy are conceptu- ally orthogonal to one another in each case and require A random measure, Φ, is a function whose domain is a lengthy, involved, and dense arguments. Obtaining a measure space (Ω; F; µ) and whose range is a space of unified treatment of conjugacy in the infinite dimen- measures over a state space (S; Σ), where we take Σ to sional setting via updating formulae for the parameters be a σ-algebra of subsets of S. In other words, for each governing the density of the processes' L´evymeasure, ! 2 Ω we have Φ(!; ·) is a measure on (S; Σ), and for analogous to the updating of the parameters of density each fixed A 2 Σ, the function Φ(·;A):Ω −! R+ is of the random variable in the finite dimensional case, an F-measurable function. In this way we may view a is a goal which we begin to work towards in this paper. random measure Φ as a collection of random variables We describe the steps achieved in this paper below. over Ω indexed by the elements of Σ. The construction of families which are conjugate in A random measure Φ is said to be a com- the modified sense requisite for a nonparametric set- pletely random measure if for any finite collection ting, entails both the construction of a prior with dis- A1;A2;:::;An of elements of Σ which are pair- tributional qualities appropriate to the modeling task wise disjoint, the corresponding random variables at hand and the ability to derive a conjugate posterior Φ(·;A1); Φ(·;A2);:::; Φ(·;An) are independent. To from this amenable form of the prior. With respect to motivate a bit of intuition in this situation, take for the latter of the two constructions, i.e. the existence example S = [0; +1) and Σ to be the collection of of a conjugate posterior, a number of distribution spe- Borel measurable subsets of S. Then the condition of cific techniques have been successful, e.g. in the cases complete randomness implies for any t1 < t2 < : : : < of the gamma and beta processes. A limited number of tn we have Φ(·; (t1; t2]); Φ(·; (t2; t3]);:::; Φ(·; (tn−1; tn]) more general construction techniques either implicitly are independent. This is reminiscent of the situa- lurk in the theory of conditional measures over infinite tion where X(t) is a nondecreasing stochastic process dimensional spaces, or have explicitly been formulated with independent increments in the sense that for any to prove the existence of a conjugate prior, albeit with t1 < t2 < : : : < tn we have X(t2) − X(t1);X(t3) − varying success. For example, recently Orbanz [6] has X(t2);:::;X(tn) − X(tn−1) are independent, and Φ obtained a mathematical framework for proving the is the unique Borel measure for which Φ((a; b]) = existence of a conjugate posterior over an infinite di- X(b+) − X(a+) for all a < b. As particular exam- mensional space given one has in hand an appropriate ples of this definition, both the beta and the gamma prior over that space. Prominent cases where such processes satisfy the conditions of a completely ran- suitable priors exist are related to the class of expo- dom measure. nential families. In other words, one may apply the Orbanz construction to produce a conjugate posterior Kingman showed [7, 8] that any completely random given one has shown the existence of a prior whose measure Φ has a decomposition into independent com- distributional qualities are derived, i.e. appropriately pletely random measures of the form Φ = Φf +Φd+Φo, related, to an exponential family. In this paper we ad- where Φf corresponds to the fixed atoms of Φ, Φd is dress the issue of obtaining a general construction for the deterministic component of Φ, and Φo is a purely priors over an infinite dimensional space whose dis- atomic measure. In general, it is the measure Φo that is of interest. One can show that for any A 2 Σ the 251 Robert Finn, Brian Kulis random variable Φo(·;A) is infinitely divisible in the sense that for any n there exists a decomposition of A into pairwise disjoint sets A1;A2;:::;An 2 Σ such that 1 E[exp(−Φo(Ai))] = fE[exp(−Φo(A))]g n i = 1; : : : ; n: This property implies that the transform E[e−tΦo(A)] has the form Z −tΦ (A) −ts E[e o ] = exp − (1 − e )ν(dx; ds) : A×(0;1] The measure ν is referred to as the L´evymeasure [9] of the random variable Φo(A) and is of great importance as it determines the random variable Φo(·;A).
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