Proceedings of the 41st Hawaii International Conference on System Sciences - 2008

Toward a Comprehensive Model in Internet Auction Fraud Detection Bin Zhang [email protected]

Yi Zhou [email protected] Carnegie Mellon University Pittsburgh, PA 15213 USA

Abstract

Fraud detection has become a common concern of the online auction websites. Fraudsters often manipulate reputation systems and commit nondelivery fraud. To deal with fraud in group behavior we consider network level features, such as users’ beliefs of other users. In this paper we use the loopy belief propagation algorithm and apply it to network level fraud detection, classifying fraudsters, accomplices, as well as honest users. Our method shows good classification accuracy using real data.

1. Introduction Development in internet technology and electronic payment has made online activities such as shopping and auction more prevalent and convenient. The increasing number of people migrating to the internet to buy or sell items spurs the growth of online auctions. However, such development has also exacerbated fraud such as account hijacking and non-delivery of goods. Fraudsters often control multiple online accounts categorized as fraud and accomplice to fraud. Fraudsters use accomplices to boost their feedback ratings and fraudulent accounts to complete transactions and receive payment without delivering product. Fraud detection has become a common problem faced by the online auction companies. However, the sheer volume of users and transactions users make the problem hard to solve. This partly explains why there is no mature fraud detection system used in online auction companies. On the other hand, such characteristics as large volume data and the existence of social network structure underlining the data make this problem interesting and relevant to machine learning research. In this study we integrate machine learning techniques to identify whether an individual is a fraudster or an honest user in online auctions.

Christos Faloutsos [email protected]

The most prevalent auction fraud is the nondelivery fraud [3] meaning the seller fails to deliver products once they receive the payments. In most of auction sites, such as eBay, there are reputation systems existing to assess users’ reputations based on their historical transaction records. However, these systems are very simple and can be easily foiled. A common sense observation is that the fraudsters generally work together to boost their collective reputations. In this manner, fraudsters conduct fraudulent auctions, but at the same time gain their reputation from other well-behaved transactions. To deal with fraud in group behavior we must consider network level features. The most recent and perhaps the only existing work using the network level features for fraud detection is Chau et al.’s. Their 2-level fraud spotting (2LFS) algorithm makes use of both user level and network level features and combines them together to classify users as ‘fraud’, ‘accomplice’ and ‘honest’ [4]. For the remainder of this paper, we refer a ‘user’ as individual who has registered an account at an online auction site. Our work will be based on 2LFS because we believe that fraudsters work in groups and that network level features are important. We investigate the problem from both the feature and algorithm perspectives.

2. Problem definition E-commerce has increased dramatically since 1999. According to [5], [7], [16], the sales of B2C ecommerce including online auctions was 8,951 billion in 2004. Specifically, the sales of online auctions rose from $3.3 billion in 1999 to $8.5 billion in 2001. It is one of the fastest growing and most profitable segments of e-commerce. However increases in auction sales are accompanied by increases in fraud. In 2005, the IC3 (Internet Crime Complaint Center) received 231,493 online complaints, an 11.6% increase

1530-1605/08 $25.00 © 2008 IEEE

1

Proceedings of the 41st Hawaii International Conference on System Sciences - 2008

compared with 207,449 complaints in 2004. The total loss due to fraud was $183.12 million dollars. This is up from $68 million in total reported losses in 2004. Internet auction fraud was the most reported offense, comprising 62.7% of referred complaints. Nondelivered merchandise and/or payment accounted for 15.7% of complaints. Data from [11] shows these two kinds of fraud are the top 2 categories of complaints referred to law enforcement. As the largest online auction site with 200 million users and 4.4 million daily transactions [8], eBay is not a simple online auction website, but a virtual business transaction community. According to economics principles, transaction must take place based on mutual trust between buyers and sellers [12]. EBay thus adopted a community trust model to guarantee the trust and safety of transactions [2]. This model incorporates seven elements including; individual identities, common symbol system, reciprocal influence, shared narrative, emotional connection, uncommunity and status. Individual identities are the most critical element of online communities [1] and are implemented with feedback rating and icons.

2.1. Feedback rating Ebay feedback ratings allow its users to build reputations making successful transactions possible. After transactions have been completed and buyers receive the merchandise, both the buyer and seller rate each other based on a 3 point scale (1, 0, or -1) and leave comments (such as ‘fast delivery’, ‘five-star seller’, ‘highly recommend’), as shown in Figure 1. Users have running totals of feedback points attached to their screen names [14]. The number that appears next to a screen name is percentage of positive feedback. Higher numbers indicate higher user reliability, as a seller, buyer or both.. The user may click on the number next to the screen name to see the details of total number of positive, negative, and neutral comments related to a user’s feedback [2].

dissatisfaction. People could be paid for providing feedback, but more refined schemes such as payment on the basis of concurrence with future evaluations of others [15]. Second, it is especially difficult to elicit negative feedback. For example, on eBay it is quite common to negotiate before resorting to negative feedback. Therefore, bad transactions not resolved by such processes are reported in terms of user feedback. Even then, “fear of retaliatory negative feedback or simply a desire to avoid further unpleasant interactions may keep a dissatisfied buyer quiet” [14]. Third, there is no guarantee feedback is honest. For example, sellers may threaten to cancel buyers’ transactions unless positive feedback is posted. On the other hand, a seller might use accomplices to rate himself positively, thus artificially inflating his/her reputation [14]. We must understand the limitations of the rating system to identify potential fraud. Often a series of positive comment within a short time followed by several negative comments in a short time frame as well imply fraud; the positive feedback of this type of fraud often comes from 1-cent auctions. Such a process is used by fraudsters to quickly increase feedback ratings making them seem like a trusted seller. Fraudsters create new accounts and only sell 1-cent value merchandise and deliver as promised, giving them positive feedback on more than 200 auctions within weeks. Then fraudsters start selling merchandise with very high values, like plasma TVs; only these high price items payment is received but product is never delivered. User we need to be very cautious to this kind of users. If we see that they have just created an account within the past 3 months, and they have a feedback of 200+, we need to take into account that the comments might be skewed.

2.2. Weakness of feedback and fraud The eBay rating has recently caused great concern among users in its community. Resnick et al. summarized the rating system problem in three ways. First of all, people often don’t bother to provide feedback especially for successful transactions. Such a problem arises because once a buyer’s merchandise is delivered there is little incentive for him/her to spend time filling out user feedback. Many people participate in feedback systems as a testament to their community spirit. Users may also express their gratitude or

Figure 1. eBay feedback system There are not so much works related to fraud detection. As what has been pointed out in Chau et al’s [4], available fraud detection systems are mainly reputation systems which are very simple and easy to cheat. And as we have described before, the fraud auction is not an individual behavior but a group

2

Proceedings of the 41st Hawaii International Conference on System Sciences - 2008

behavior. That feature leads us to consider network level features for fraud detections. We have used Markov random fields to model the transaction network on eBay and implemented the belief propagation algorithm to classify the users as fraudsters or honest.

3. Literature review 3.1. 2LFS algorithm 2LFS [4] is the first work making use of network level features. It sets up the problem on the network level as a Markov random field (MRF). In their formulation, each user was a node in the graph and the presence of an edge between two nodes indicated that transactions were made between the two users. This was equivalent to consider that the online users were in a large social network and they established connections with each other through transactions. Users were divided into three categories: ‘fraud’, ‘accomplice’ and ‘honest’. ‘Fraud’ and ‘accomplice’ were people committed frauds. The difference were that ‘accomplice’ always behaved legally and they gained reputation/trustworth from ‘honest’ in order to help boost reputation of ‘fraud’. For each node in the graph, its value was observed on the user level and but the observation was noisy because fraudsters tent to disguise themselves. Then the question was that given the user level analysis of users’ states how to classify users in the context of the whole social network. This was a typical inference problem for Markov random field and Chau et al. used belief propagation (BP) to do the inference.

3.2. Markov propagation

random

field

and

belief

Markov random field (MRF) is a model about the joint probability distribution of a set of random variables and has been widely in machine learning [18]. The concept of MRF was brought up by Dobruschin [6] first in 1968 as a pure mathematical problem. Geman and Geman simplified and applied Dobruschin’s idea to computer vision [19]. Geman and Geman [10] uses S to present a set of points S = {s1 ,s 2 , s 3 ,…, s N } , the neighbors of S is

Figure 2. Pairwise Markov Random Field G = {G s , s ∈ S } , so we can use {S , G} to represent any graph, and X is an MRF with regard to G if: P ( X = ω ) > 0 for all ω ∈ Ω ; P ( X s = xs | X r = x r , r ≠ s ) = P ( X s = x s | X r = xr , r ∈ Gs ) for every s ∈ S where Ω = {ω = ( x s1 , x s2 ,… , x sN : x si ∈ Λ,1 ≤ i ≤ N )} Λ = {0, 1, 2, 3,…, L − 1} L = number of states

Yedidia generalized Geman’s idea to pairwise inference problems [19]. The inference problems on Markov random field arise from many specific instances, including computing the marginals and modes of probability distributions. And the typical inference problem for Markov random field is to calculate the conditional distribution of a set of nodes in the graph which are often called hidden nodes, given values to another set of nodes which are called observed nodes. This inference problem is called as exact inference. As the Figure 2 shown, xi, i = [1, 5] in this case, are hidden nodes that we cannot get their values, yi are observed nodes that we can measure. We assume there is statistical relation between xi and yi. Such relationship is also called evidence by Yedidia et al. [19], and represented by φ i ( x i , yi ) . Also for hidden nodes that are neighbors, we have statistical relation ψ ij ( x i , x j ) ; it is also called as compatibility function. Since compatibility function is only decided by nodes i and j, we call this kind of MRF as pairwise MRF. We then represent the joint probability of a hidden node xi and observed node yi as:

3

Proceedings of the 41st Hawaii International Conference on System Sciences - 2008

(

)

p {xi }i =1 , {y i }i =1 = N

N

1 ∏ψ ij ( xi , x j )∏i φi ( xi , yi ) Z i, j

where {xi }iN=1 is the collection of all hidden nodes, and

{yi }iN=1 is the collection of all observed nodes.

In [17], Wainwright and Jordan gave a comprehensive review for graphical models, including both the Bayesian networks and Markov random fields and proposed the junction tree algorithm which includes the inference algorithms under its umbrella, like the sum-product algorithm, Viterbi and forwardbackward algorithms for Hidden Markov models and etc. The junction tree algorithm is exactly the belief propagation. They also stated that the junction tree algorithm has a computational complexity that is exponential in the treewidth of the graph. So only when the graph is suitably sparse, for example a tree, the junction tree algorithm is a viable computational framework for exact inference. Wainwright and Jordan proposed the so called variational methods to relax the optimization problem and to provide approximated, but good enough solutions. The loopy belief propagation is right the variational method of the belief propagation. ‘Loopy’ means that there are cycles in message passing in belief propagation and the algorithm simply ignores it. Although the convergence of the loopy belief propagation is not guaranteed, it has been widely used in practice and for some networks has shown to be accurate and to converge quickly. On the other hand, for some networks the loopy belief propagation may give poor results or fail to converge. Although the convergence of Loopy belief propagation is not guaranteed, it still has been widely used in practice. Because first, Loopy BP converges in most cases; second, when Loopy BP converges, it provides very good approximation to the correct marginals [13]. Freeman and Pasztor [9] illustrated the ideal impressively that the Markov random field could be used as a general statistical model for interpreting images and scenes. They also showed the belief propagation had pretty good performance and fast convergence. This work is another example to illustrate the successful use of the Markov random field and the belief propagation on a big and loopy network. The belief propagation used by Chau et al. [4] in the 2-LFS algorithm is indeed loopy belief propagation. We will implement the algorithm and study the convergence of the algorithm in our project as well. In [19], Yedidia et al. thoroughly studied the belief propagation and the principles behind the belief propagation. They connected the belief propagation

algorithm to the Bethe approximation of statistical physics and showed that the belief propagation can only converge to a stationary point of an approximate free energy, known as the Bethe free energy in statistical physics. By this connection, they characterized the fixed points of belief propagation and made connections with the variational methods. They also derived a generalized belief propagation algorithm, which is shown to be more complex, but more accurate than the ordinary belief propagation. We have implemented the ordinary belief propagation and will try to implement the generalized belief propagation to see if it will give a more accurate solution than that found using ordinary belief propagation at the expense of computational complexity.

4. Method 4.1. Data description We have 5 million transaction records from 1 million dated between 1999 and 2006 crawled from eBay, each record includes information such as the seller ID, buyer ID, item number, feedback score and transaction time. The prices of the items can be found on the eBay website. We have crawled the prices from eBay for about 6,000 items. Although the price information is only useful for classifying the users at the user level as described in [4], it may not be the focus of our project, and we will use the price information in our future study.

4.2. Loopy belief propagation for fraud detection In this paper, we want to implement the loopy belief propagation and apply it to the network-level fraud detection. Figure 3 illustrates a part of the network arising from the interactions between the users on eBay. The white nodes represent registered users and their states. These states, fraud, accomplice or honest, are assumed uncertain in the Markov random field. The gray nodes stand for observations whose states are obtained by the available information in hand but might be wrong. The present of an edge between two white nodes means there are transactions occurring between two corresponding users. Thus, we want to infer the states of the white nodes (users) in the graph given the values of the gray nodes. White nodes are users’ category, such as fraud, accomplice, and honest user, and grey nodes are observations status, including fraud and honest user.

4

Proceedings of the 41st Hawaii International Conference on System Sciences - 2008

First of all, we need to make our decision on the states of the observations with the transaction data in our hand. We can decide the value of observation by looking at the feedback score that user has got. If he or she got a negative score, we set the observation of the user as fraud; otherwise, we set the observation as honest. Then in our Markov random field, the hidden node (white) has three states, i.e. fraud, accomplice and honest, and the observed node (gray) has two states, i.e. fraud and honest. To initialize our MRF, we set the pairwise potential functions as below:

In our experiment, we set εp to 0.05 and εo to 0.2 as proposed by Chau et al. [4]. We try the loopy belief propagation algorithm on a data set of 89,825 transactions among 55,867 users. By observing the users’ feedback scores, we found 117 users with negative scores in the group of users. We consider them as fraudsters. Then, when substantiating the observations in the Markov random field, we set the states of the observations of these users to ‘fraud’ and those of other observations to ‘honest’. Thus, we get a Markov random field like that in Figure 3. On the other hand, we know that there are some users have been identified as fraudsters by eBay. In such case, we can further substantiate the state for the hidden (white) nodes, thus we will get a graph like Figure 4. Note that one of the hidden nodes has been observed. 1) MRF without unidentified fraudsters: If none of the fraudster has been identified, we have a Markov random field shown in figure 3 and run the loopy belief propagation on it. Then we get the belief of each node, which is the probability of the node in each state. According to the belief of the hidden (white) node, we can classify the user as a fraudster or an honest user. If the probability of being an honest user (opposite to fraudster and accomplice) is bigger than 0.5, then we think this person is honest; otherwise a fraudster. In this way, the result of belief propagation shows that 42 users who are treated as fraudsters in the observations are actually honest. However, when we look at the transaction records, we find that these users have a lot of good transaction records but made a small number

of fraud transactions with a significant amount of money. So they are right the fraudsters we want to find! The fact that inference algorithm can’t identify them under the context of the transaction network partly explains the way these people hide themselves from being caught for fraud auctions. Accordingly, to make the inference more reasonable, we need to make the identified fraudsters exposed in the network, so we resort to the Markov random field in Figure 4.

Figure 3. Illustration for the transaction network

Figure 4. Illustration for the transaction network with identified fraudsters

5

Proceedings of the 41st Hawaii International Conference on System Sciences - 2008

2) MRF with identified fraudsters: As we have mentioned before, some users may have been identified as fraudsters by eBay, so we should make use of that information. We can simply do this by using a Markov random field in Figure 4. If the user has been identified as fraudster, we apply the information to the graph and substantiate the state of the hidden node corresponding to the user to ‘fraud’. Since we don’t have the information of the identified fraudsters now, in our experiment we set the identified fraudsters by hand. By looking at the transaction records, we think there are 40 users who are really fraudsters, then we treat them as identified fraudsters. After we run the belief propagation on this Markov random field, we find that 5,305 users who were thought honest are indeed accomplices. The values of εp and εo in the propagation and observation matrices do affect the result of belief propagation. Figure 5 and 6 show the variation of the number of found accomplice with respect to the variation of εp and εo. If we look at the design of the propagation and observation matrices, we will see that these two figures truly reflect the truth. For example, in Figure 5 when εp equals to 0.25, there are little users thought to be accomplices. In fact, when εp equals to 0.25, the probability for an accomplice conducting a transaction with an honest user is set to 0, so the group of fraudsters and accomplices and the group of honest users are supposed to be almost isolated in the transaction network and then the information won’t change much by propagating the information over the Markov random field. On the other hand, if we set εo to be 0.5, that is to say that everyone is a possible fraudster with a half chance by observation, so the observation information is meaningless and then an identified fraudster will make every connected user an accomplice by belief propagation.

Figure 6. Number of accomplices found (BP with varying o and p = 0.05)

5. Experimental results Through the analysis of our method, we can see the importance of choosing appropriate εp and εo. In Chau et al.’s [4], they proposed to set εp to 0.05 and εo to 0.2. Such method is heuristic. However, in order to do the parameter selection we need the ground truth data, which are unfortunately unavailable at this moment. But we still can do analysis on parameter selection in the following subsections based on the “ground truth” data we figured out. In Figure 7 we show the numbers of found accomplices while varying εp and εo simultaneously. The two plots in Figure 5 and 6 are the slices at εo = 0.2 and εp = 0.05.

Figure 7. Numbers of found accomplices (BP with varying p and o)

5.1. Initial values for the observation Figure 5. Number of accomplices found (BP with varying p and o = 0.2)

In this subsection, we will see whether changing the observation value will affect the result. We try

6

Proceedings of the 41st Hawaii International Conference on System Sciences - 2008

another three choices of the initial values of the observations: 1). setting all observed nodes as honest; 2). setting all observed nodes as fraudsters; 3). randomly setting observed nodes as fraudsters with half chance. Figure 8, 9 and 10 shows the numbers of found accomplices with the three assignments of observed nodes. We can see that setting all the observed nodes as honest will not change the results much. If setting all observed nodes as fraudsters we will classify almost all the nodes as fraudsters. It is surely reasonable since there are not many voices (evidences) for arguing for being honest. The situation shown by Figure 10 is more complicated, compared with those shown by Figure 8 and 9. In Figure 10, as we randomly set the state of observations, we will find either almost all the users or nearly a half of users are accomplices. This is reasonable. But classifying almost all or a half of users as accomplices doesn’t depend on the values of εp and εo in a simple way. εp and εo will have a joint effect on the number of accomplices identified.

Figure 8. Numbers of found accomplices (All observed nodes = honest)

Figure 10. Numbers of accomplices found. (Probability of observed nodes as fraudster = 50%)

5.2. Parameter selection As we have mentioned before, in order to set the parameters in an objective way, we need the ground truth data and then find the optimal parameters best classifying the data. But unfortunately, we don’t have the ground truth data of labeled fraudsters and accomplices. In our experiment, we label 40 fraudsters by ourselves, but we can’t label all the suspicious fraudsters. However, we know that the accomplices are friends of the fraudsters. They conduct good behaved transactions with the fraudsters and gave good feedback. So the accomplices should be in the group of users who connect to the fraudsters but never get hurt. We categorize those who have “friendly” transactions with the fraudsters as accomplices. We find that in the 55,869 users with 40 identified fraudsters, there are 29,787 users who have “friendly” transactions with the fraudsters. We also observe that among the 29,787 users, some of them really have frequent transactions with the fraudsters. In the following, we call the 29,787 users “suspicious accomplice”. Table 1 shows the distribution of the numbers of transactions of these suspicious accomplices with the fraudsters. Then we can see there are 7,942 users having multiple friendly transactions with the fraudsters, and 652 users among them having more than five friendly transactions with the fraudsters. Table 1. Number of transactions among the suspicious accomplices

Figure 9. Numbers of accomplices found. (All observed nodes = fraudster)

7

Proceedings of the 41st Hawaii International Conference on System Sciences - 2008

Since the labeled data is unavailable, we created a set of “labeled” data: •

•

Group I (the positive set) consists of 7,942 users who have conducted more than one transaction with fraudsters plus the 40 fraudsters. We regard this group of users as “accomplice” or “fraudster”. Group II (the negative set) is comprised of 26,114 users who have never conducted any transactions with fraudsters. We regard this group of uses as “honest”.

network. Second, in all of our experiments, belief propagation converges very quickly, usually less than 20 iterations, implying the algorithm is stable as well. Third, we find that parameters εp and εo both significantly influence converged beliefs. meaning that learning an optimal set of εp and εo is necessary. Although we don’t have the ground truth data, we have figured out a way to choose a possible ground truth set and find its mot significant parameters. The next step of our work is acquiring ebay’s fraudster list. We will also compare the performance of our system with other parallel systems.

Then we use Group I and II as our labeled data. Thus we only consider the two groups of users above, rather than all 58,869. With our “labeled” data we are able to perform parameter selection and select parameters that best classify users in Group I and II. Figure 11-13 shows the numbers of correctly classified fraudsters in Group I and honest users in Group II with different choices of εp and εo. Figure 11 shows the true positives, figure 12 shows the true negatives and figure 13 displays the model accuracy. The accuracy rate reaches its maximum value at the point where εp = 0.05 and εo = 0.35.

Figure 12. True negatives: correctly classified honest users in Group II

Figure 11. True positives: correctly classified fraudsters in Group I

6. Conclusion and future work In this paper, we used Markov random fields to model the transaction network on eBay and implemented the belief propagation algorithm to classify users in one of two categories: fraudsters or honest users. Based on our results, we can draw several conclusions. First, the identified information of the users is very helpful, even indispensable. Only based on that, we can do some significative analysis. Otherwise, if just depending on the observations, the fraudsters cannot be caught in the huge connection

Figure 13. Accuracy: correctly classified users in Group I and honest users in Group II

Acknowledgement The authors thank Mr. Duen Horng Chau the data he provided as well as his advice for this project.

8

Proceedings of the 41st Hawaii International Conference on System Sciences - 2008

7. References [1] Nancy K. Baym. The emergence of on-line community. In Steven G. Jones, editor, Cybersociety 2.0: revisiting computer-mediated communication and community, pages 35–68. Sage Publications, Inc., Thousand Oaks, CA, USA, 1998. [2] Josh Boyd. In Community We Trust: Online Security Communication at eBay. Journal of Computer-Mediated Communication, 7(3), April 2002. [3] Mary M. Calkins. My Reputation Always Had More Fun Than Me: The Failure of eBay’s Feedback Model to Effectively Prevent Online Auction Fraud. Richmond Journal of Law and Technolgy, 7(4), 2001. [4] Duen Horng Chau, Shashank Pandit, and Christos Faloutsos. Detecting fraudulent personalities in networks of online auctioneers. In Proceedings of PKDD 2006, pages 103–114, Berlin, Germany, September 18-22, 2006. [5] Roger Crockett. Going, Going...Richer, The wellheeled crowd at online auctions. Business Week, December 13th 1999. [6] P. L. Dobruschin. The description of a random field by means of conditional probabilities and conditions of its regularity. Society for Industrial and Applied Mathematics, 8(2):197–224, 1968. [7] Rong-Ruey Duh, Karim Jamal, and Shyam Sunder. Control and Assurance in E-Commerce: Privacy, Integrity, and Security at eBay. Taiwan Accounting Review, 3(1):1–27, October 2002. [8] Maynard Webb (eBay Chief Operation Officer). Serving the Net Generation. Presentation at CUMREC Conferences, May 2005. http://www.educause.edu/ir/library/powerpoint/CMR0 532.pps. [9] William T. Freeman and Egon C. Pasztor. Markov networks for lowlevel vision. In Workshop on Statistical and Computational Theories of Vision, volume MERL-TR99-08, 1999.

[11] Internet Crime Complaint Center. 2005 IC3 Annual Report, 2005. http://www.ic3.gov/media/annualreport/2005 IC3Report.pdf. [12] Sirkka L. Jarvenpaa and Noam Tractinsky. Consumer trust in an Internet store: A cross-cultural validation. Journal of Computer-Mediated Communication, 5(2), 1999. [13] Kevin P. Murphy, Yair Weiss, and Michael I. Jordan. Loopy belief propagation for approximate inference: An empirical study. Proceedings of Uncertainty in AI, pages 467–475, 1999. [14] Paul Resnick, Richard Zeckhauser, Eric Friedman, and Ko Kuwabara. Reputation systems. Communications of the ACM, 43(12):45–48, December 2000. [15] Paul Resnick, Richard Zeckhauser, John Swanson, and Kate Lockwood. The value of reputation on eBay: A controlled experiment. Experimental Economics, 9(2):79–101, June 2006. [16] US Census Bureau. 2004 E-commerce Multisector Report, May 2006. http://www.census.gov/eos/www/papers/2004/2004rep ortfinal.pdf. [17] M. J. Wainwright and M. I. Jordan. Graphical models, exponential families, and variational inference. Technical report, Department of Statistics, University of California, Berkeley, 2003. [18] Martin J. Wainwright and Michael I. Jordan. A variational principle for graphical models, chapter New Directions in Statistical Signal Processing: From Systems to Brain. MIT Press, Cambridge, MA, 2005. [19] Jonathan S. Yedidia, William T. Freeman, and Yair Weiss. Understanding Belief Propagation and Its Generalizations. Exploring Artificial Intelligence in the New Millennium. January 2003.

[10] Stuart Geman and Donald Geman. Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions of Pattern Analysis and Machine Intelligence, 6(6):721–741, 1984.

9