A Variational Bayes Approach to Decoding in a Phase-Uncertain Digital Receiver

ISSC 2011, T rinit y College Dublin, June 23–24 A V ariational Ba y es Approac h to Deco ding in a Phase-Uncertain Digital Receiv er Arijit Das † and An thony Quinn ∗ Dep artment of Ele ctr onic and Ele ctric al Engine ering T rinity Col le ge Dublin E-mail: † dasa@tcd.ie ∗ aquinn@tcd.ie Abstr act — This pap er presents a Ba y esian approach to symbol and phase inference in a phase-unsync hronized digital receiv er. It primarily extends [10] to the multi-sym b ol case, using the v ariational Bay es (VB) appro ximation to deal with the combinatorial complexit y of the phase inference in this case. The work provides a fully Ba yesian ex- tension of the EM-based framew ork underlying current turb o-sync hronization metho ds, since it induces a von Mises prior on the time-inv ariant phase parmeter. As a result, w e ac hieve tractable iterativ e algorithms with impro ved robustness in lo w SNR regimes, compared to the current EM-based approac hes. As a corollary to our analysis we also disco ver the importance of prior regularization in elegan tly tackling the significan t prob- lem of phase am biguity . Keywor ds — V ariational Ba yes approximation, phase synchronization, phase ambiguit y resolution, von Mises distribution, soft deco ding I Introduction Comm unication Systems face uncertain ties of the transmitter and receiv er oscillators, the noisy dis- p ersiv e nature of the c hannel, among others for transferring messages from the transmitter to the receiv er [11]. Incorrect alignment of the transmit- ter and receiver oscillators sho ws itself as a rota- tion of the receiv ed data on the complex plane and renders deco ding impossible. The process of align- ing these oscillators is known as synchronization and is an important asp ect in the pro cess of de- co ding [2]. Curren t state of the art in dealing with this problem is a joint iterativ e estimation of the phase and the sym b ol sequence, in an algorithm kno wn as turbo-synchronization [3]. This algorithm is based on the framework of an EM-algorithm where probabilit y distribution on the sym b ol decisions are used recursively to improv e the estimate of phase, and vice-v ersa. This use of probabilit y dis- tributions ov er the p ossible symbols while making a decision, defined as a ”soft” decision, was the no v elt y in this approac h [3]. The algorithm could also be defined under other frameworks, suc h as one based on the sum-pro duct algorithms [4], [7]. The EM-algorithm ho wev er, considers certaint y- equiv alence measures on the parameter v alues whic h lead to the loss information ab out their sta- tistical uncertain ty . This w ork aims to general- ize this framew ork in to a fully Ba yesian treatment, whic h is flexible enough to ”learn” the v aried chan- nel conditions along with the uncertainties asso- ciated with the modelled parameters. This w as ho w ev er partially achiev ed in [9], wherein the au- thors used a uniform prior o ver the phase. Goo d Ba y esian learning mo dels take in to consideration all the statistical information, the data has to offer and therefore advice better decisions under adv erse conditions. This pap er presents the adv antage of using a generalization of the uniform distribution on the unit circle, the V on-Mises distribution in resolv- ing phase ambiguit y . Literature consists of either using a huge num b er of pilot symbols or mo dified lik eliho od estimators [4] which are generally ineffi- cien t with resp ect to p o wer and time, to deal with the problem. In the course of our analysis, we find a natural elegant solution to the problem, which do es not add any computational burden to the syn- c hronization routine and exalts the imp ortance of prior regularization in Ba y esian Analysis. As constellation alphabets belong to finite fields, the receiver data mo dels are inevitably mixture distributions whic h are not in the exponential fam- ily . Finite dimensional sufficient statistics [5] and conjugate prior distributions [1] are a unique char- acteristic to the exp onen tial family . They are im- p ortan t in devising recursive learning algorithms. Unfortunately as this is not the case, it leads to mo dels with substan tially exp ensiv e inference pro- cedures. A deterministic approximation metho d, kno wn as V ariational Ba yesian (VB) Metho d [14], is applied in this pap er to deal with the problem. The rest of the pap er is structured as follows. Section I I defines the system mo del encompassing unkno wn phase and symbol sequence. Section I II describ es the single sym b ol transmission inference. Section IV describ es the com binatorial complexit y , the V ariational Ba yes Method and how it is ap- plied to the Multi-Sym b ol transmission inference. In section V, n umerical results for the p erformance of the describ ed receiver and its c omparisons with the current state of the art are provided. Section VI discusses the results and its implications. Sec- tion VI I presen ts the concluding remarks and fu- ture course of actions. I I System Model Structured redundancy is usually added in the transmitted data [2], in a process known as co d- ing, to protect from the noisy nature A W GN c han- nel. Equal partitions of the binary message stream formed from uniformly sampled and quantized analog signals, are bijectively mapp ed to a com- plex symbol a = [ a 1 , ...a m , ...a M ] 0 from a constel- lation alphab et A of size M. These symbols are then mo dulated onto a carrier signal which can b e sen t through a ph ysical c hannel. On the receiver side, the rev erse op eration called demo dulation at- tempts to deco de or identify the transmitted sym- b ol sequence, which are now corrupted by noise [3]. W e observe the data v ector x i ∈ C n dur- ing ev ery sym b ol perio d with i ∈ 1 , ...K with x i = [ x 0 1 , x 0 2 , ...x 0 i ] 0 and assume the prior proba- bilities corresp onding to the sym b ol vector a to b e α = [ α 1 , ...α M ] 0 . φ represents the phase dif- ference of the received signal to the transmitted, with noise r in an A W GN c hannel [10]. Ease of inferential analysis dictates the augmentation of the mo del using a laten t v ariable. Let it b e l i = [ l 1 ,i , ...l M ,i ] 0 ∈ {  M (1) , ... M ( M ) } , where  M ( m ) = [ δ ( m − 1) , ...δ ( m − M )] 0 and δ is the Kro- nec k er Delta. Hence l i is a v ector p ointer to the comp onen t active at the i th sym b ol p erio d. The data generation mo del for such a sym b ol transfer in a single sym b ol p erio d can b e defined as: f ( x i | φ, r , α ) = X l i f ( l i | α ) f ( x i | a, φ, r , l i ) (1) f ( x i | φ, r , l i ) ≡ C N x  l 0 i ag e j φ , r  (2) f ( l i | α ) ≡ M u l i (1 , α ) = l 0 i α (3) where C N : Complex Normal and M u : Multi- nomial distributions, φ ∈ [ − π , π ], r ∈ (0 , ∞ ), g i = [ s 1 e j ω , ..., s n e j ω n ] 0 with s i ∈ { 1 , 2 ...n } as the pulse sequence and ω the carrier digital frequency whic h are assumed to b e known. In this analysis w e assume kno wn SNR, based on whic h we can calculate the noise v ariance, r . a) Prior Distribution to Phase: V on-Mises Dis- tribution As we will see later, the VB approximation on the join t distributions lead to marginal distributions of the phase from the exp onential family , hence the use of conjugate priors remains imp ortan t. V on- Mises distribution [12], [8] is a useful distribution o v er the unit circle defined from [ − π , π ] on the real line, which serves as a conjugate prior to the phase for our conditional data mo del. It has one com- plex parameter, which serves as b oth lo cation and concen tration parameter for the distribution and has b een used in v aried applications on directional data. f ( φ | κ 0 ) ≡ M ( κ 0 ) = 1 2 π I 0 ( | κ 0 | ) e ( < { κ 0 e − j φ } ) (4) where κ 0 ∈ C . In the signal pro cessing literature this distribution is also kno wn as the Tikhonov distribution [12] and has b een previously used to mo del the phase error from the estimated phase us- ing a phase-lo c k ed lo op (PLL). In this w ork V on- Mises/Tikhono v distribution is considered to b e the prior distribution of the phase in a Bay esian framew ork which was explicitly introduced in [10]. As the V on-Mises/Tikhonov distribution is con- fined to the unit circle, means and v ariances out- side [ − π , π ] are undefined. Hence circular mo- men ts are used whic h are defined as: E [ φ ] = ∠ κ 0 (5) V ar cir [ φ ] = 1 − E [cos ( φ − E [ φ ])] = 1 − I 1 ( | κ 0 | ) I 0 ( | κ 0 | ) (6) Limiting b ehaviour of this distribution under v ari- ous conditions of its h yp erparameter are as follo ws lim | κ 0 |→∞ f ( φ | κ 0 ) = N φ  ∠ κ 0 , 1 | κ 0 |  (7) lim | κ 0 |→ 0 f ( φ | κ 0 ) = U φ ([ − π , π ]) (8) where N is the normal distribution and U is the uniform distribution. I I I Single Symbol Transmission No w we present the inference pro cedure for the single sym b ol transmission, where exact inference w as shown to b e p ossible in [10]. This analy- sis forms the basis for our generalization into the m ulti-sym b ol cases, wherein the V ariational Ba y es Metho d is applied. The data generation mo del in this case is tak en to b e f ( x, l, φ | g , κ 0 , α, β 0 , a ) = f ( x | a, φ, r , l ) f ( l | α ) f ( φ | κ 0 ) (9) where f ( φ | κ 0 ) ≡ M ( κ 0 ) (10) f ( l | α ) ≡ M u l t (1 , α ) = l 0 α (11) f ( x | a, φ, r , l ) ≡ C N x  l 0 ag e j φ , r  (12) The marginal p osterior distributions for phase Fig. 1: Graphical Mo del - Single-sym bol model in the case of uncertain phase and the symbol can b e easily calculated in this case and are giv en in [10] to b e f ( φ | x ) = X l p l M ( κ l ) (13) f ( l | x ) = M u l (1 , p l ) (14) where κ l = κ 0 + 2 r ( l 0 ag ) H x (15) = κ 0 + 2 r ( l 0 a ) H n X k =1 s k x k e − j ω k (16) p l ∝ ( l 0 α ) e ( − 1 r ( l 0 ag ) H ( l 0 ag ) ) I 0 ( | κ l | ) (17) The mean v alue of φ for each comp onent is given b y ∠ κ l . Also κ l can b e characterized as a Distrete Time F ourier T ransform on the incoming data, whic h is intuitiv e since κ l is a sufficient statistic to the phase φ . Here ho wev er w e get more infor- mation than from a simple DTFT, namely about the confidence o v er the exp ected phase. Another imp ortant observ ation ab out κ l can b e seen with the follo wing algebraic expansion: | κ l | = [ κ l κ l ] 1 / 2 (18) = [ 4 r 2 ( l 0 ag ) H x ( l 0 ag ) H x + κ 0 κ 0 + 2 r κ 0 ( l 0 ag ) H x + 2 r κ 0 ( l 0 ag ) H x ] 1 / 2 (19) F rom the abov e equation we can see that when κ 0 = 0 (Uniform Prior), all except the first term v anishes. That implies then that when the prior is a uniform distribution, the deco ding distribution is inv ariant to the symbols on the same radius in a QAM constellation, whic h is the origin of phase am biguit y . How ever a non-zero κ 0 mak es κ l de- p end on the v alue of the sym b ols and hence results in a unique MAP from the deco ding distribution. IV Mul ti-Symbol Transmission Real w orld comm unication systems inv olv e mul- tiple sym b ol p erio ds, which are observed sequen- tially and a deco ding distribution for the symbols is the de sired output. The joint data mo del under the indep endent symbol sequence assumption (rea- sonable in turb o processing where pseudorandom in ter-lea v ers are t ypically emplo y ed [9]) is then f ( x K | φ, r , α ) = K Y i =1 X l i f ( l i | α ) f ( x i | a, φ, r , l i ) (20) As a consequence of this product o ver a sum- mation, M K term mixture mo del app ears after K time perio ds, a com binatorial explosion which leads to in tractable exact inference. a) V ariational Bayes Metho d A deterministic appro ximation, kno wn as the V ari- ational Bay es Metho d is employ ed to deal with this computational intractabilit y . The Metho d aims at minimizing the Kullback-Leibler Div ergence be- t w een the candidate approximation and the origi- nal joint distribution. This divergence can be in- terpreted as an information difference [15] b etw een the tw o distributions. The in tuition then is to get a functional form of the p osterior marginals, so as to minimize the information lost due to the forced in- dep endence [13]. These metho ds hav e found many applications in machine learning, artificial intelli- gence and more recen tly in signal pro cessing [9]. The essential result whic h forms the basis of this metho d is as follows: Theorem IV.1 V ariational Bayes Metho d [14]: L et f ( θ | x ) b e the p osterior distribution of multi- variate p ar ameter, θ . The latter is p artitione d into q sub-ve ctors of p ar ameters or q-no des: θ = [ θ 0 1 , θ 0 2 , ...θ 0 q ] 0 (21) L et ˘ f ( θ | x ) b e an appr oximate distribution r e- stricte d to the set of c onditional ly indep endent dis- tributions for θ 1 , θ 2 ,... θ q : ˘ f ( θ | x ) = q Y i =1 ˘ f ( θ i | x ) (22) Then, the minimum of Kul lb ack-L eibler diver genc e KL  ˘ f ( θ | x ) || f ( θ | x )  is r e ache d for ˜ f ( θ i | x ) ∝ e  E ˜ f ( θ /i | x ) [ln( f ( θ, x ))]  (23) wher e i = 1 , ..., q and θ /i in θ , and ˜ f ( θ /i | x ) = Q q j =1 ,j 6 = i ˜ f ( θ j | x ) . Now ˜ f ( θ | x ) is the VB- apr oximation, and ˜ f ( θ i | x ) ar e the VB-mar ginals. b) Offline VB Appr oximation Consider the situation, where w e observe a batch of sym b ol transfer p erio ds together and are required to infer the marginal distributions for all the trans- ferred symbols and the phase parameter. Prior in- dep endence of the symbols is assumed. Ho w ev er it should b e noted, that ap osteriori the symbols are indep endent conditional on φ . Expanding the join t probability , ov er all unknown parameters, we ha v e f ( x K , l K , φ | κ 0 , α, r ) = f ( φ | κ 0 ) K Y i =1 f ( x i | a, φ, r , l i ) f ( l i | α ) (24) where f ( φ | κ 0 ) ≡ M ( κ 0 ) (25) f ( l i | α ) ≡ M u l i (1 , α ) = l 0 i α (26) f ( x i | a, φ, r , l i ) ≡ C N x i  l 0 i ag i e j φ , r  (27) F rom the abov e joint distribution, we find the Fig. 2: Graphical Mo del - Multi-sym bol model in the case of uncertain phase p osterior distributions for the lab el field and the phase. Integrating out the lab el field entails a M K summation, making it in tractable [13]. W e there- fore resort to using Theorem I I.1 and derive the appro ximate p osterior distributions of the param- eters as ˜ f ( φ | x K ) ∝ e  E ˜ f ( l K | x K ) [ln f ( x K , l K ,φ | κ 0 ,α,r )]  = M φ ( κ l K ) (28) ˜ f ( l i | x K ) ∝ e  E ˜ f ( φ | x K ) ˜ f ( l K \ i | x K ) [ln f ( x K , l K ,φ | κ 0 ,α,r )]  = M u l i (1 , p l i ) (29) with the shaping parameters as κ l K = κ 0 + 2 r K X i =1 ( ˆ l 0 i ag i ) H x i (30) p l i ∝ ( l 0 i α ) e ( − 1 r ( l 0 i ag i ) H ( l 0 i ag i )+ 2 r δ i ) (31) and the asso ciated VB moment as δ i = E ˜ f ( φ | x K )  <  ( l 0 i ag i ) H x i e − j φ  = < n ( l 0 i ag i ) H x i e − j φ κ l K o I 1 ( | κ l K | ) I 0 ( | κ l K | ) (32) κ l K again can be c haracterized as a DTFT. Imp or- tan t subsets of this metho d discussed in the litera- ture include [9] where the authors hav e assumed an uniform prior to the phase whic h implies κ 0 = 0. F urther if w e mak e the ratio I 1 ( | κ l K | ) I 0 ( | κ l K | ) = 1 w e get the EM-algorithm based turb o- sync hronization routine as discussed in [3]. c) Online VB Appr oximation In this case w e observ e eac h sym b ol transfer p e- rio d sequentially and are required to infer the marginal distributions for each transferred sym- b ol and phase parameter, after eac h sym b ol p e- rio d. Again prior independence of the symbols is assumed. T o deal with the increasing com bina- torial complexit y after each time step w e conflate the p osterior distribution of φ which is a mixture of M V on-Mises comp onents using V ariational Bay es Metho d in to a single V on-Mises p osterior. The as- so ciated sufficient statistic can then be used as the prior for the next sym b ol perio d. Expanding the join t probability , ov er unkno wn parameters after eac h time step, w e ha v e f ( x t , l t , φ | x t − 1 ) = f ( x t | a, φ, r , l t ) f ( l t | α ) ˜ f ( φ | x t − 1 ) (33) where f ( φ | x 0 ) = f ( φ | κ 0 ) ≡ M ( κ 0 ) (34) f ( l t | α ) ≡ M u l t (1 , α ) = l 0 t α (35) f ( x t | a, φ, r , l t ) ≡ C N x t  l 0 t ag t e j φ , r  (36) On equation (13) from Section I I I of the p osterior distribution of the phase, we apply the VB Method and derive the approximate conflated p osterior dis- tribution as ˜ f ( φ | x t − 1 ) ∝ e  E ˜ f ( l t − 1 | x t − 1 ) [ln f ( x t − 1 ,l t − 1 ,φ | x t − 2 ,κ 0 ,α,r )]  = M φ ( κ l t − 1 ) (37) (38) with the shaping parameter as ˆ κ l t − 1 = ˆ κ l t − 2 + 2 r ( ˆ l 0 t − 1 ag t − 1 ) H x t − 1 (39) p l t − 1 ∝ ( l 0 t − 1 α ) e ( − 1 r ( l 0 t − 1 ag ) H ( l 0 t − 1 ag ) ) I 0 ( | κ l t − 1 | ) (40) ˆ l t − 1 = E [ l t − 1 ] (41) Using this shaping parameter as initialization for the following sym b ol p erio d w e get from the single sym b ol transmission case: f ( φ | x t ) = X l p l t M ( κ l t ) (42) f ( l t | x t ) = M u l t (1 , p l t ) (43) where κ l t = ˆ κ l t − 1 + 2 r ( l 0 t ag t ) H x t (44) p l i ∝ ( l 0 i α ) e ( − 1 r ( l 0 i ag ) H ( l 0 i ag ) ) I 0 ( | κ l t | ) (45) κ l t again can b e characterized as a DTFT as ear- lier. V Simula tion Fig. 3: - Performance of the VB-based deco ding algorithms in the multi-sym b ol case The ab ov e figure describ es the prop ortion of suc- cessful iden tifications of the sym b ol transferred for v arious v alues of SNR, using the metho ds devel- op ed in this pap er. In eac h batc h 20 sym b ols are transmitted through an A WGN c hannel. In the lo w-SNR regime, from -15dB to 5dB we see that the Bay esian algorithms are significantly b etter. Indep enden t deco ding algorithm uses the exact de- co ding algorithm ov er each subsequent sym b ol p e- rio d ignoring the previous p erio ds. This metho d turns out to b e the fastest and most accurate for this sim ulation. The Online VB and the Offline VB approximations provide almost as go o d results, though they would b e more robust in a more gen- eral setting. W e can also clearly see the p erfor- mance of [9] and [3] is significantly low er with re- sp ect to the deco ding success. It should b e noted, to deal with the problem of phase ambiguit y for these tw o metho ds, 5 pilot symbols are used in fron t of the original 20. VI Discussion W e get tw o main results from the analysis p er- formed in this pap er. Primarily we are able to extend the EM-Algorithm based synchronization in to a fully Bay esian synchronization pro cedure. This can form the basis for future generalizations and analysis of ev en more complicated scenarios. Secondly , as a corollary to our analysis we are able to resolv e the problem of phase am biguity . As shown in the Section I I I, setting κ 0 = 0 ren- ders κ l indep enden t of the exp ected phase E [ φ ], whic h thereb y results in the apparent ambiguit y in the decoding distribution. The notion of non- zero κ 0 seems quite inconsequential, but a concen- trated prior ho wev er informs the presence of a uni- mo dal phase distribution and regularizes the ob- serv ation mo del which is intrinsically rotationally in v arian t for QAM. It m ust be noted how ever, that as the num b er of incoming observ ations increases the sampling distribution of the MAP symbol b e- comes relatively inv ariant to the v alue of κ 0 , as long as κ 0 > 0. Computationally it presen ts no ov erhead, as it in v olv es only the addition of a constan t. In con- trast past approaches lik e Differen tial Co ding [2], angle differential QAM sc heme [6] and the use of pilot symbols [3] [4], among others whic h hav e b een prop osed to deal with this rotational in v ariance are visibly more complex and lac k the simplicit y of this observ ation. In [3] the authors use the EM-Algorithm to deal with phase sync hronization. This essentially im- plies using the sifting prop erty of the Kroneck er Delta function and thus using the exp ectations of the parameters in the asso ciated functions rather than the expectations of the functions of the pa- rameters itself. Significan tly low er accuracy of the deco ding distribution in the low-SNR regime is ob- serv ed whic h conv erges in p erformance with in- creasing SNR. VI I Conclusions In this pap er we hav e b een able to extend the EM framew ork for tackling the phase-sync hronization problem, to a fully Bay esian VB based algorithm wherein as a virtue of its mo delling ability the problem of phase ambiguit y does not arise. This mak es the concept of using ”soft” information truly complete as no statistical information is lost at any point in the iterations. F urther the ba- sic framework dev elop ed motiv ates further relax- ations in to more general channels, suc h as those with m ulti-path fading, like Rician F ading [2]. In future work we would also be relaxing the condi- tion on kno wn noise parameter and deal with it in a Ba y esian framew ork. VI I I Ackno wledgments This research was supported b y Science F ounda- tion Ireland, gran t 08/RFP/MTH1710 References [1] Persi Diaconis and Donald Yl visaker . Con- jugate Priors for Exp onential F amilies . The Annals of Statistics , 7 (2):pp. 269–281, 1979. [2] Andrea Goldsmith . Wir eless Communications . Cam bridge Universit y Press, New Y ork, NY, USA, 2005. [3] C. Herzet, N. Noels, V. Lottici, H. Wymeersch, M. Luise, M. Moeneclaey, and L. V andendorpe . Co de-Aided T urbo Sync hronization . Pr o c e edings of the IEEE , 95 (6):1255 –1271, june 2007. [4] C. Herzet, K. Woradit, H. Wymeersch, and L. V andendorpe . Co de-Aided Maxim um- Lik eliho o d Ambiguit y Resolution Through F ree-Energy Minimization . Signal Pr o c essing, IEEE T r ansactions on , 58 (12):6238 –6250, Dec 2010. [5] Christian Hipp . Sufficient Statistics and Ex- p onen tial F amilies . The Annals of Statistics , 2 (6):pp. 1283–1292, 1974. [6] Jeng-Kuang Hw ang, Yu-Lun Chiu, and Chia- Shu Liao . Angle differential-QAM scheme for resolving phase ambiguit y in contin uous transmission system . Int. J. Commun. Syst. , 21 :631–641, June 2008. [7] Michael I. Jordan . An introduction to v ari- ational metho ds for graphical mo dels . In Machine L e arning , pages 183–233. MIT Press, 1999. [8] C. G. Kha tri and K. V. Mardia . The V on Mises-Fisher Matrix Distribution in Orien- tation Statistics . Journal of the R oyal Statisti- c al So ciety. Series B (Metho dolo gic al) , 39 (1):pp. 95–106, 1977. [9] Mauri Nissila and Subbara y an P asup a thy . Adaptiv e iterative detectors for phase- uncertain c hannels via v ariational b ound- ing . T r ans. Comm. , 57 :716–725, March 2009. [10] Anthony Quinn, Jean-Pierre Barbot, and P ascal Larzabal . The Bay esian Inference of Phase . In Pr o c e e dings of IEEE International Confer enc e on A c oustics, Spe e ch and Signal Pr o- c essing, 2011 , ICASSP’11, 2011. [11] C.E. Shannon . Comm unication in the pres- ence of noise . Pr o c e e dings of the IEEE , 72 (9):1192 – 1201, sept 1984. [12] Yuriy S. Shmaliy . V on Mises/Tikhonov- based distributions for systems with differ- en tial phase measurement . Signal Pr o c essing , 85 (4):693 – 703, 2005. [13] V. Smidl and A. Quinn . The V ariational Ba yes Approximation In Ba yesian Filter- ing . In IEEE International Confer enc e on A c ous- tics, Sp e ech and Signal Pr o c essing, 2006 , 3 , page I I I, Ma y 2006. [14] V a cla v Smidl and Anthony Quinn . The V ariational Bayes Metho d in Signal Pr o c essing . Springer, Dublin, Ireland, 2006. [15] Mar tin J. W ainwright and Michael I. Jor- dan . Graphical Mo dels, Exp onential F ami- lies, and V ariational Inference . F oundations and T r ends in Machine L e arning , 1 :1–305, 2008.