Fried Chicken Bucket ProcessesMary McGlohonMachine Learning DepartmentCarnegie Mellon [email protected]: Stochastic processes, graphical models, hierarchical modelsAbstractChinese restaurant processes are useful hierarchical models; how-ever, they make certain assumptions on finiteness that may not beappropriate for modeling some phenomena. Therefore, we introducefried chicken bucket processes (FCBP) that involve different samplingmethods. We also introduce spork notation as a simple way of repre-senting this model.1 IntroductionChinese restaurant processes and Indian buffet processes are useful in a num-ber of domains for statistical modeling. This work introduces a new model,the Fried chicken bucket process, and presents spork notation, a useful rep-resentation for FCBP’s and other graphical models.To the author’s knowledge this is the first restaurant-related model thatsamples from continuous distributions in addition to discrete ones.2 Related workIt is useful to describe work that has inspired this paper.12.1 Chinese restaurant processesThe Chinese restaurant process (CRP) is a stochastic process that producesa distribution on partitions of integers [1]. To visualize, one imagines aChinese restaurant with an infinite number of tables. Customers arrive oneat a time. As each arrives, he decides which table to sit at based on thefollowing distribution similar to a Dirichlet distribution:p(T ablei|n) =n(i)γ + n − 1p(NewT able|n) =γγ + n − 1where n is the number of previous customers and n(i) is the numberseated at table i.This may be extended into hierarchies such as the customers also choosingfrom an infinite number of Chinese restaurants [2]. This process also inspiredGriffiths and Ghahramani to describe the Indian buffet process for infinitelatent feature models, as shown in [5] and applied again by Thibaux andJordan in [6].2.2 Plate notationIn high dimensional problems, representation comes in the form of very largegraphical models with many nodes. Formerly researchers simply had theirgraduate students draw all the nodes. Then, in 1994, Buntine introducedplate notation [3], which drastically reduced the work required to draw agraphical model, and has made it possible for today’s machine learning grad-uate students to focus their efforts on maintaining statistics-related entrieson Wikipedia. The plate notation simply groups together nodes that areduplicated– that is, have the same interior-exterior links. An example, flipsof a thumb tack, is shown in Figure 1.3 Fried Chicken Bucket Processes3.1 Description of modelOn the top level, one imagines a fried chicken restaurant with a chicken gen-erating function (cgf): that is, a distribution of chicken parts from which the2Figure 1: A graphical model without plate notation (left) and with plate no-tation (right).buckets are made. The restaurant also serves homogeneous okra, coleslaw,or other side dishes which may be treated as continuous.A family orders a n-piece bucket of fried chicken, which begins the nextlevel. From the cgf, n pieces of fried chicken are drawn, making a muchcoarser distribution of chicken parts. The family also takes sides. Oncethe family drives home and spreads dinner on the table, each of k familymembers chooses chicken pieces from the bucket. Draws are random to avoidsquabbles, and the distribution is obviously without replacement1. Afterchicken is drawn, each family member chooses a continuous amount of sidedishes. It is well known that the fried chicken runs out while there are oftenleftover side dishes; therefore for this model we assume that coleslaw andokra are infinite as well as continuous. However, the amount of these dishesmay be conditional on the discrete pieces of chicken that were drawn fromthe bucket, as paper plates have finite capacity.1Sampling with replacement would be unsanitary.3Figure 2: A FCBP in spork notation.3.2 Illustration of modelWe can best illustrate the FCBP using a piece of hardware related to thefried chicken bucket: the spork. The cfg (suggested through the handle ofthe spork) generates the bucket in the reservoir of the spoonlike part. Fromthe bucket, the plates result (prongs), which then “pick” items from thecontinuous and infinite side dishes. This is shown in Figure 2.We propose that spork notation be used for any process where a discretesampling influences a subsequent continuous sampling.43.3 Instances of modelFor theatrical purposes one may choose to specify the cgf. The most obviouschoice is a multinomial distribution, with one pifor each chicken part thatmay go into the bucket, wherePipi= 1. For example, we might choose(pleg= .3, pbreast= .39, pwing= .3, pbeak= .01).4 Applications of FCBPMany phenomena may be modeled as an interaction between a discrete sam-pling that influences the way in which a continuous sampling behaves. Onemay think of mixture models in this fashion; the prongs of the spork maybe considered k classes from which different continuous distributions of vari-ables may result. This is significant because mixture models and the methodsare sometimes difficult to grasp, and machine learning concepts are easier tounderstand when they are presented using culinary examples [4].5 Future WorkIt would be desired to extend FCBP’s to yet another hierarchy. For instance,one might imagine a strip mall, college campus, or region of a country withan infinite number of fast food stands and allow mixing proportions on afamily’s dinner table. This, and other further applications of FCBP are leftas an exercise to the reader.6 ConclusionMachine learning researchers need to stop having meetings when they’rehungry.References[1] D. J. Aldous. Exchangeability and related topics.´Ecole ’e’ t´e de proba-bilit´es de Saint-Flour XIII-1983. Lecture Notes in Mathematics, 1117.5[2] D. Blei, T. Gri, M. Jordan, and J. Tenenbaum. Hierarchical topic modelsand the nested chinese restaurant process, 2004.[3] W. L. Buntine. Operations for learning with graphical models. Journalof Artificial Intelligence Research, 2:159–225, 1994.[4] K. El-Arini. Pizza delivery processes. In Machine learning office conver-sations, 2006.[5] T. Griffiths and Z. Ghahramani. Infinite latent feature models and theindian buffet process, 2005.[6] R. Thibaux and M. I. Jordan. Hierarchical beta processes and the indianbuffet process. Technical report, University of California,