nips nips2008 nips2008-105 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Manfred Opper, Ulrich Paquet, Ole Winther
Abstract: A series of corrections is developed for the fixed points of Expectation Propagation (EP), which is one of the most popular methods for approximate probabilistic inference. These corrections can lead to improvements of the inference approximation or serve as a sanity check, indicating when EP yields unrealiable results.
Reference: text
sentIndex sentText sentNum sentScore
1 dk Abstract A series of corrections is developed for the fixed points of Expectation Propagation (EP), which is one of the most popular methods for approximate probabilistic inference. [sent-6, score-0.605]
2 These corrections can lead to improvements of the inference approximation or serve as a sanity check, indicating when EP yields unrealiable results. [sent-7, score-0.736]
3 1 Introduction The expectation propagation (EP) message passing algorithm is often considered as the method of choice for approximate Bayesian inference when both good accuracy and computational efficiency are required [5]. [sent-8, score-0.247]
4 However, while such empirical studies hold great value, they can not guarantee the same performance on other data sets or when completely different types of Bayesian models are considered. [sent-10, score-0.022]
5 In this paper methods are developed to assess the quality of the EP approximation. [sent-11, score-0.044]
6 We compute explicit expressions for the remainder terms of the approximation. [sent-12, score-0.067]
7 This leads to various corrections for partition functions and posterior distributions. [sent-13, score-0.612]
8 Under the hypothesis that the EP approximation works well, we identify quantities which can be assumed to be small and can be used in a series expansion of the corrections with increasing complexity. [sent-14, score-0.769]
9 The computation of low order corrections in this expansion is often feasible, typically require only moderate computational efforts, and can lead to an improvement to the EP approximation or to the indication that the approximation cannot be trusted. [sent-15, score-0.84]
10 2 Expectation Propagation in a Nutshell Since it is the goal of this paper to compute corrections to the EP approximation, we will not discuss details of EP algorithms but rather characterise the fixed points which are reached when such algorithms converge. [sent-16, score-0.526]
11 EP is applied to probabilistic models with an unobserved latent variable x having an intractable distribution p(x). [sent-17, score-0.098]
12 In applications p(x) is usually the Bayesian posterior distribution conditioned on a set of observations. [sent-18, score-0.054]
13 Since the dependency on the latter variables is not important for the subsequent theory, we will skip them in our notation. [sent-19, score-0.071]
14 1 It is assumed that p(x) factorizes into a product of terms fn such that p(x) = 1 Z where the normalising partition function Z = an approximation to p(x) in the form fn (x) , (1) n dx q(x) = n fn (x) is also intractable. [sent-20, score-1.29]
15 We then assume gn (x) (2) n where the terms gn (x) belong to a tractable, e. [sent-21, score-0.535]
16 To compute the optimal parameters of the gn term approximation a set of auxiliary tilted distributions is defined via 1 q(x)fn (x) qn (x) = . [sent-24, score-0.775]
17 (3) Zn gn (x) Here a single approximating term gn is replaced by an original term fn . [sent-25, score-0.825]
18 Assuming that this replacement leaves qn still tractable, the parameters in gn are determined by the condition that q(x) and all qn (x) should be made as similar as possible. [sent-26, score-1.039]
19 This is usually achieved by requiring that these distributions share a set of generalised moments (which usually coincide with the sufficient statistics of the exponential family). [sent-27, score-0.267]
20 Note, that we will not assume that this expectation consistency [8] for the moments is derived by minimising a Kullback–Leibler divergence, as was done in the original derivations of EP [5]. [sent-28, score-0.195]
21 Such an assumption would limit the applicability of the approximate inference and exclude e. [sent-29, score-0.114]
22 the approximation of models with binary, Ising variables by a Gaussian model as in one of the applications in the last section. [sent-31, score-0.091]
23 The corresponding approximation to the normalising partition function in (1) was given in [8] and [7] and reads in our present notation1 ZEP = Zn . [sent-32, score-0.338]
24 (4) n 3 Corrections to EP An expression for the remainder terms which are neglected by the EP approximation can be obtained by solving for fn in (3), and taking the product to get fn (x) = n Hence Z = dx R= n n Zn qn (x)gn (x) q(x) qn (x) q(x) = ZEP q(x) n . [sent-33, score-1.562]
25 (5) fn (x) = ZEP R, with dx q(x) n qn (x) q(x) and p(x) = 1 q(x) R n qn (x) q(x) . [sent-34, score-1.094]
26 (6) This shows that corrections to EP are small when all distributions qn are indeed close to q, justifying the optimality criterion of EP. [sent-35, score-0.95]
27 Exact probabilistic inference with the corrections described here again leads to intractable computations. [sent-37, score-0.603]
28 However, we can derive exact perturbation expansions involving a series of corrections with increasing computational complexity. [sent-38, score-0.756]
29 Assuming that EP already yields a good approximation, the computation of a small number of these terms maybe sufficient to obtain the most dominant corrections. [sent-39, score-0.089]
30 On the other hand, when the leading corrections come out large or do not sufficiently decrease with order, this may indicate that the EP approximation is inaccurate. [sent-40, score-0.612]
31 Two such perturbation expansions are be presented in this section. [sent-41, score-0.158]
32 1 The definition of partition functions Zn is slightly different from previous works. [sent-42, score-0.09]
33 1 Expansion I: Clusters n (x) The most basic expansion is based on the variables εn (x) = qq(x) − 1 which we can assume to be typically small, when the EP approximation is good. [sent-44, score-0.194]
34 Expanding the products in (6) we obtain the correction to the partition function R= dx q(x) (1 + εn (x)) (7) n =1+ εn1 (x)εn2 (x) q + εn1 (x)εn2 (x)εn3 (x) q + . [sent-45, score-0.235]
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