The Chinese Restaurant Process - Computer Science
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用英语介绍中式餐馆的作文As an aficionado of Chinese cuisine, I am excited to introduce to you a renowned Chinese restaurant that embodies the essence of authentic Chinese dining - the Imperial Garden.Located in the heart of the city, the Imperial Garden offers a truly immersive experience into the world of Chinese culinary traditions. From the moment you step through its ornate doors, you are greeted with the rich aromas of traditional spices and the warm ambiance of classic Chinese décor.The menu at the Imperial Garden is a testament to the diversity and depth of Chinese cuisine. It features a wide array of regional specialties, from the fiery flavors of Sichuan to the delicate dim sum of Cantonese cuisine. Each dish is crafted with precision and skill, using only the freshest ingredients to ensure an unparalleled dining experience.One of the standout features of the Imperial Garden is its traditional tea service. The restaurant offers anextensive selection of premium Chinese teas, each with its own unique flavor profile and health benefits. Guests can enjoy the art of tea preparation as they savor the soothing brews that complement the bold flavors of the cuisine.In addition to its exquisite food and drink offerings, the Imperial Garden also prides itself on providing exceptional hospitality. The staff is attentive and knowledgeable, guiding guests through the menu and offering recommendations based on individual preferences. The restaurant's elegant yet comfortable setting provides the perfect backdrop for everything from intimate dinners to festive gatherings.The Imperial Garden is not just a place to eat, but a destination for cultural exploration and appreciation. It is a celebration of the rich culinary heritage of China, inviting guests to embark on a sensory journey through the flavors, aromas, and traditions of one of the world's most esteemed cuisines.对于我来说,中餐馆是一种文化的体现,同时也是一种美食的享受。
介绍中国餐馆英文作文英文:When it comes to Chinese cuisine, one thing that immediately comes to mind is the Chinese restaurant. Chinese restaurants can be found all over the world, and they offer a wide variety of dishes that are sure tosatisfy any palate. In this essay, I will introduce Chinese restaurants and share my personal experiences.Chinese restaurants are usually divided into two types: traditional and modern. Traditional Chinese restaurants often have a more formal atmosphere and serve classic dishes such as Peking duck, hot pot, and dim sum. Modern Chinese restaurants, on the other hand, have a more contemporary feel and serve dishes that are more fusion-oriented, such as sushi rolls with Chinese ingredients or Chinese-style tacos.One of my favorite Chinese restaurants is a traditionalone located in my hometown. Whenever I go there, I always order their Peking duck, which is roasted to perfection and served with thin pancakes, scallions, and hoisin sauce. The restaurant also serves a variety of dim sum, which are small dishes that are perfect for sharing. My personal favorites are the shrimp dumplings and the pork buns.Another Chinese restaurant that I enjoy is a modern one located in a nearby city. They offer a unique fusion menu that combines Chinese and Mexican flavors. One of their signature dishes is the Chinese-style tacos, which are made with crispy wonton shells and filled with spicy beef, cilantro, and avocado salsa. They also have a variety of sushi rolls that incorporate Chinese ingredients such as crispy noodles and hoisin sauce.Overall, Chinese restaurants offer a diverse range of dishes that cater to different tastes and preferences. Whether you prefer traditional or modern cuisine, there is always something for everyone to enjoy.中文:说起中国菜,一个立刻浮现在脑海中的就是中餐馆。
中国餐饮方面的英文表达The catering industry in China has a rich and diverse history, with a wide range of culinary styles and regional specialties. From traditional street food to high-end banquet cuisine, Chinese dining offers something for every palate. In recent years, the Chinese catering industry has experienced significant growth and innovation, driven by changing consumer preferences, technological advancements, and a growing global interest in Chinese cuisine.One of the most notable trends in the Chinese catering industry is the rise of food delivery and online ordering services. With the widespread use of smartphones and the internet, Chinese consumers have embraced the convenience of ordering meals from their favorite restaurants with just a few taps on their mobile devices. This has led to a boom inthe food delivery business, with a multitude of companies vying for market share in urban centers across China.Another important development in the Chinese catering industry is the increasing emphasis on quality and food safety. In response to several high-profile food safety scandals in recent years, Chinese consumers have become more discerning about the provenance and quality of the food they consume. This has led to greater scrutiny of food production processes and an emphasis on organic and locally sourced ingredients. Many restaurants and catering companies have responded to these concerns by implementing stringent quality control measures and transparent sourcing practices.In terms of culinary trends, Chinese catering has seen a resurgence of traditional and regional cuisines. While international flavors and fusion dishes continue to be popular, there is a growing appreciation for the authentic flavors and techniques of Chinese regional cooking. Fromspicy Sichuanese hotpot to delicate Cantonese dim sum, Chinese diners are rediscovering the diversity and complexity of their culinary heritage.In addition to traditional Chinese cuisine, the catering industry in China has also seen a burgeoning interest in plant-based and vegetarian options. With growing concerns about health and environmental sustainability, many Chinese consumers are seeking out meat-free alternatives and innovative plant-based dishes. This has prompted many restaurants and catering companies to develop creative and appealing vegetarian menus that cater to this evolving market demand.Internationally, Chinese catering has become an increasingly influential force in the global culinary scene. Chinese restaurants and chefs are garnering accolades and attention for their inventive and sophisticated approaches to traditional Chinese cooking. In cities around the world,Chinese fine dining establishments are redefining the perception of Chinese cuisine and attracting a new generation of food enthusiasts.At the same time, the Chinese catering industry is embracing technological innovation to improve efficiency and customer experience. From automated ordering systems to kitchen robotics, Chinese restaurants are at the forefront of adopting cutting-edge technologies to streamline operations and enhance the dining experience. This integration of technology has not only improved service levels but has also opened up new opportunities for restaurant owners and entrepreneurs.In conclusion, the Chinese catering industry is experiencing a period of dynamic change and growth, driven by evolving consumer preferences, culinary innovation, and technological advancements. As Chinese cuisine continues to captivate the global palate, the future of the Chinesecatering industry looks bright, with endless possibilities for creativity and expansion.。
中国餐厅英文作文I went to a Chinese restaurant last night and it was absolutely amazing. The food was so delicious and flavorful, and the atmosphere was lively and vibrant. I couldn't get enough of the authentic Chinese dishes that were served.The restaurant was beautifully decorated withtraditional Chinese elements, such as red lanterns and wooden furniture. The ambiance was warm and inviting, andit really made the dining experience feel special.One of the things that stood out to me the most was the variety of dishes available. From dim sum to Peking duck, there was something for everyone to enjoy. Each dish hadits own unique flavors and textures, and I loved trying a little bit of everything.The service at the restaurant was also top-notch. The staff were friendly and attentive, and they made sure that we had everything we needed throughout the meal. It reallyadded to the overall enjoyment of the dining experience.The highlight of the night was definitely the Peking duck. It was perfectly cooked with crispy skin and tender meat, and it was served with all the traditional accompaniments. It was a true feast for the senses.Overall, I had a fantastic time at the Chinese restaurant and I can't wait to go back again. The food, the atmosphere, and the service were all exceptional, and I would highly recommend it to anyone looking for a great dining experience.。
中国式餐厅作文英语题目,A Visit to a Chinese Restaurant。
Introduction。
Chinese cuisine is renowned worldwide for its rich flavors, diverse dishes, and cultural significance. Recently, I had the pleasure of visiting a local Chinese restaurant in my city. This experience provided me with a deeper insight into the culinary traditions and hospitality of Chinese culture.Ambiance。
Upon entering the restaurant, I was immediately struck by the vibrant ambiance. Red lanterns adorned the ceiling, casting a warm glow over the diners below. Intricate wooden carvings and traditional artwork adorned the walls, transporting me to another time and place. Soft instrumental music played in the background, creating asoothing atmosphere that complemented the dining experience.Menu Selection。
The menu presented a myriad of tantalizing options, ranging from familiar favorites to exotic delicacies. From classic dishes like sweet and sour chicken to more adventurous choices such as Peking duck, there was something to suit every palate. After much deliberation, I decided to start with a steaming bowl of wonton soup, followed by kung pao chicken and stir-fried vegetables.Service。
中国餐馆英文作文I recently went to a Chinese restaurant and it was a great experience. The atmosphere was cozy and the decor was very traditional. The menu was extensive and had a lot of options to choose from.The first dish that we ordered was hot and sour soup. It was the perfect starter as it was not too heavy and had a great balance of flavors. The soup had a nice tangy taste and was loaded with vegetables and tofu.Next, we ordered some dumplings that were filled with pork and vegetables. The dumplings were perfectly steamed and had a nice chewy texture. The dipping sauce that came with it was a perfect complement to the dumplings.For the main course, we ordered Kung Pao chicken and some fried rice. The Kung Pao chicken was spicy and had a nice crunch from the peanuts. The fried rice was fluffy and had a nice aroma of soy sauce.To end the meal, we ordered some fortune cookies and hot tea. The fortune cookies were a fun way to end the meal and the hot tea was very soothing.Overall, it was a great experience and I woulddefinitely recommend this restaurant to anyone who is looking for some authentic Chinese food. The service was also great and the staff was very friendly.。
中餐英文接待流程The process of hosting guests in a Chinese restaurant involves several key steps that ensure an exceptional dining experience.1. Greeting the Guests: Upon arrival, the first point of contact is the host or hostess. They greet guests warmly,often with a smile, and acknowledge their presence. A polite greeting such as "Welcome!" or "Hello! How can I assist you today?" sets a friendly tone for the evening.2. Seating the Guests: The host then leads the guests to their table, providing the option of different seating arrangements depending on their preferences. Offering achoice between a quieter corner or a lively section can enhance the overall experience. It's important to ensure that the table is clean and well-prepared before guests are seated.3. Presenting the Menu: Once seated, servers present the menu with attention to detail. They may highlight signature dishes or chef’s recommendations. Introducing traditional Chinese culinary concepts and serving styles helps guests appreciate the meal choices. Providing a wine or tea list can also complement the dining experience.4. Taking Orders: After allowing sufficient time for guests to peruse the menu, servers return to take orders.It’s essential to listen carefully, confirm each item, and offer assistance with selections if needed. For large parties, confirming orders can help avoid any misunderstandings.5. Serving the Meals: Meals are typically served in a family-style format, with several dishes placed in the center of the table for sharing. Servers should time the delivery of dishes so that they arrive once everyone is seated and readyto eat. Care should be taken to arrange the dishesaesthetically on the table.6. Checking in with Guests: Throughout the meal, servers should periodically check in with guests to ensure satisfaction. Asking if they need more dishes or drinks, or if there's anything they might need reflects excellent service standards.7. Dessert and Tea Service: After the main meal, servers can suggest dessert options along with traditional Chinese tea. Informing guests about the significance of certain desserts and teas can add a cultural appreciation to the dining experience.8. Presenting the Bill: Once the meal is finished, servers bring the bill discreetly. They thank the guests for dining and inquire if they enjoyed their experience. A positive farewell, such as "We hope to see you again!" leavesa lasting impression.9. Gathering Feedback: After guests have departed, staff should gather feedback on their experience. This can involvea brief discussion among the team or encouraging online reviews, which are vital for continuous improvement and customer engagement.By adhering to these steps, a Chinese restaurant can offer a comprehensive dining experience that is both respectful of cultural traditions and attentive to guest needs.。
中国餐厅的英文作文Chinese restaurants are known for their delicious food and rich cultural heritage. From traditional dishes like Peking duck and dim sum to modern fusion creations, Chinese cuisine offers a wide range of flavors and textures that appeal to food lovers all over the world.Chinese restaurants often feature traditional decor, with red lanterns, intricate wood carvings, and beautiful silk tapestries adorning the walls. The atmosphere islively and bustling, with the sound of clinking chopsticks and sizzling woks filling the air.One of the most popular dishes at Chinese restaurants is Peking duck, a crispy and succulent dish that is often carved tableside by skilled chefs. The duck is typically served with thin pancakes, hoisin sauce, and sliced cucumbers and scallions, allowing diners to create their own delicious wraps.Dim sum is another beloved Chinese culinary tradition, with small, bite-sized dishes served in steamer baskets or on small plates. From savory dumplings filled with pork andshrimp to sweet custard tarts and fluffy buns, dim sum offers a wide variety of flavors and textures to enjoy.In recent years, Chinese restaurants have also embraced modern fusion cuisine, incorporating ingredients and techniques from other culinary traditions to create innovative and exciting dishes. For example, dishes like General Tso's chicken and orange beef combine traditional Chinese flavors with a Western twist, appealing to a diverse range of tastes.中餐馆以其美味的食物和丰富的文化遗产而闻名。
介绍中国餐馆的英文作文English Answer:Chinese restaurants have become an integral part of the global culinary landscape, offering diners a delectable array of authentic and delectable dishes. Their popularity stems from the unique blend of flavors, textures, and aromas that characterize Chinese cuisine.Ambiance and Decor:Chinese restaurants often feature a vibrant andinviting atmosphere, adorned with traditional Chinese artwork, lanterns, and auspicious symbols. The décor may vary depending on the region of China the restaurant specializes in, ranging from the rustic charm of a Cantonese teahouse to the elegant opulence of a Sichuanesque establishment.Menu:The menu of a Chinese restaurant is typically extensive, featuring a diverse selection of dishes that cater to awide range of tastes and preferences. Diners can choosefrom an array of appetizers, such as spring rolls, pot stickers, and wontons; a variety of soups, including wonton, hot and sour, and egg drop soups; a plethora of main courses, ranging from stir-fries and braised dishes to steamed fish and roasted meats; and a delectable assortment of rice and noodles.Ingredients:Chinese cuisine is known for its use of fresh, high-quality ingredients. Vegetables, such as bok choy, gai lan, and bean sprouts, are often featured prominently in dishes. Meats, such as chicken, beef, pork, and seafood, are also commonly used. Soy sauce, rice wine, ginger, garlic, and sesame oil are essential ingredients in many Chinese dishes.Cooking Techniques:Chinese restaurants employ various cooking techniques to create their delicious dishes. Stir-frying, a method of cooking food quickly in a hot wok, is a popular technique used in Cantonese cuisine. Braising, a slow-cooking method that involves simmering food in a savory liquid, is often used in Sichuanese cuisine. Steaming, a method of cooking food over boiling water, is commonly used to prepare delicate items such as fish and vegetables.Service:Chinese restaurants typically offer attentive and friendly service. Servers are knowledgeable about the menu and can provide diners with recommendations based on their preferences. They are also quick to refill drinks and ensure that diners have a pleasant dining experience.Cultural Significance:Chinese restaurants are not only culinary destinations but also cultural touchstones. They offer diners an opportunity to immerse themselves in Chinese culturethrough food and ambiance. They serve as gathering places for families, friends, and business associates to socialize and enjoy good food.Conclusion:Chinese restaurants have gained worldwide recognition for their delicious food, vibrant atmosphere, and cultural significance. They offer diners a unique and memorable culinary experience that tantalizes the taste buds and enriches the soul.中文回答:中国餐馆已成为全球美食版图中不可或缺的一部分,为食客提供各种正宗且美味的菜肴。
中国餐饮英语作文Chinese cuisine is a culinary tradition that has captivated the world for centuries. With its rich history, diverse regional flavors, and innovative cooking techniques, it has become one of the most celebrated and influential gastronomic traditions globally. From the delicate balance of flavors in Cantonese dim sum to the bold and spicy Sichuan dishes, Chinese food offers a remarkable breadth of culinary experiences.At the heart of Chinese cuisine lies a deep respect for quality ingredients and a commitment to seasonality. The country's vast and varied landscapes provide an abundance of fresh produce, seafood, and meats, which are skillfully transformed into dishes that showcase the natural flavors. Chinese chefs have mastered the art of creating harmonious blends of sweet, sour, salty, bitter, and umami, resulting in dishes that are both satisfying and deeply nuanced.One of the most distinctive features of Chinese cuisine is its regional diversity. Each province and even individual cities have their own unique culinary traditions, influenced by local ingredients, culturalinfluences, and historical trade routes. The Cantonese cuisine of southern China, for example, is renowned for its delicate and refined flavors, with a focus on fresh seafood and the skilled use of wok cooking. In contrast, the Sichuan cuisine of western China is celebrated for its bold and spicy flavors, incorporating the famous Sichuan peppercorn and a range of chili peppers.The Hunan cuisine of central China is known for its intense and pungent flavors, with a heavy reliance on preserved and smoked ingredients. The Shangdong cuisine of eastern China, on the other hand, emphasizes the natural sweetness of ingredients, often featuring seafood and vegetables. The diversity of Chinese cuisine is a testament to the country's rich cultural heritage and the ingenuity of its people.Beyond the regional variations, Chinese cuisine also boasts a remarkable array of cooking techniques and preparation methods. From the delicate art of dim sum steaming to the high-heat wok frying, Chinese chefs have mastered a wide range of culinary skills. The use of the wok, in particular, is a hallmark of Chinese cooking, allowing for the rapid searing and stir-frying of ingredients to create dishes that are simultaneously crisp, tender, and bursting with flavor.Chinese cuisine also places a strong emphasis on the presentation of food, with dishes often arranged in visually striking and harmoniouscompositions. The use of colorful ingredients, intricate garnishes, and thoughtful plating techniques contribute to the overall aesthetic appeal of Chinese cuisine, making it a feast for both the palate and the eyes.One of the most captivating aspects of Chinese cuisine is its ability to evolve and adapt to changing times and global influences. While maintaining a strong connection to its traditional roots, Chinese food has also embraced new ingredients, cooking methods, and culinary trends. The rise of fusion cuisine, which blends Chinese techniques and flavors with those of other culinary traditions, has resulted in exciting and innovative dishes that push the boundaries of what we consider "Chinese" food.Moreover, the global popularity of Chinese cuisine has led to a greater appreciation and understanding of its nuances and complexities. Restaurants around the world have embraced Chinese culinary traditions, introducing diners to the depth and diversity of this rich gastronomic heritage. From the ubiquitous Chinese takeout to the Michelin-starred fine dining establishments, Chinese cuisine has firmly established itself as a beloved and integral part of the global culinary landscape.In conclusion, Chinese cuisine is a truly remarkable and captivating culinary tradition that continues to captivate and inspire foodenthusiasts around the world. With its regional diversity, innovative cooking techniques, and evolving nature, Chinese food offers a world of flavors and experiences that are both deeply rooted in tradition and constantly pushing the boundaries of culinary creativity. As we continue to explore and appreciate the wonders of Chinese cuisine, we can only imagine the exciting culinary journeys that lie ahead.。
Passage 1上海菜系是中国最年轻的地方菜系,通常被成为“本帮菜”,有着400多年的历史。
同中国其他菜系一样,“本帮菜”具有“色,香,味”三大要素。
//上海菜的特点是注重调料的使用,食物的质地和菜的原汁原味。
其中最著名的有特色点心“南翔小笼”和特色菜“松鼠鲑鱼”。
//“南翔小笼”是猪肉馅,个小味美,皮薄汁醇。
“松鼠鲑鱼”色泽黄亮,形如松鼠,外皮脆而内肉嫩,汤汁酸甜适口。
//在品尝过“松鼠鲑鱼”之后,我们常常惊讶于“松鼠”的形状,觉得在三大评价标准上在添加“形”这个标准才更合适。
//Shanghai cuisine, usually called Benbang cuisine, is the youngest among the major regional cuisines in China, with a history of more than 400 years. Like all other Chinese regional cuisines, Benbang cuisines takes “color, aroma and taste” as its essential quality elements.//Shanghai cuisine emphasizes in particular the expert use of seasonings, the selection of raw materials with quality texture, and original flavors. Shanghai cuisine is famous for a special snack known as Nanxiang Steamed Meat Dumplings and a special dish called “Squirrel-Shaped Mandarin Fish”.//Nanxiang Steamed Meat Dumplings are small in size, with thin and translucent wrappers, filled inside with ground pork and rich tasty soup. Squirrel-Shaped Mandarin Fish is yellow-colored and squirrel-shaped, with a crispy skin and tender meat, all covered with a sweet and sour source.//After tasting Squirrel-Shaped Mandarin Fish, we are alwaysamazed by the squirrel shape and think that it is more appropriate to plus “appearance” as the fourth element.Passage 2美国人强调效率、竞争和独创性,而中国人则将严谨规划放在首位,鼓励团队成员之间的密切合作和无私奉献。
大学英语作文:中国式烹饪(Chinese cooking)大学英语作文:中国式烹饪(Chinese cooking)在平平淡淡的学习、工作、生活中,许多人都写过作文吧,借助作文人们可以实现文化交流的`目的。
那么你有了解过作文吗?下面是小编精心整理的大学英语作文:中国式烹饪(Chinese cooking),欢迎大家借鉴与参考,希望对大家有所帮助。
meatballs with carrot slices in soup ale, since there is only one response and this one is not satisfying, i figured out something for fun. i learnt this from my roommate in college. it is very simply and you dont find it in any of the fashionable restaurants. but it is old-fashioned as you require. and it is delicious at that.buy a pound of minced meat in a supermarket. add salt, ginger powder, sliced onion, a little flour, and two spoonfuls of water. stir the thing vigorously for two minutes. boil a pot of water. make a dozen meatballs out of the miture youve prepared. put the balls into boiling water. when the water boils again, add sliced carrot. add salt and pepper to your taste. the soup is ready to serve.homemade and crude as it is, you never forget the taste!。
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文档下载后可定制随意修改,请根据实际需要进行相应的调整和使用,谢谢!并且,本店铺为大家提供各种各样类型的实用资料,如教育随笔、日记赏析、句子摘抄、古诗大全、经典美文、话题作文、工作总结、词语解析、文案摘录、其他资料等等,如想了解不同资料格式和写法,敬请关注!Download tips: This document is carefully compiled by theeditor. I hope that after you download them,they can help yousolve practical problems. The document can be customized andmodified after downloading,please adjust and use it according toactual needs, thank you!In addition, our shop provides you with various types ofpractical materials,such as educational essays, diaryappreciation,sentence excerpts,ancient poems,classic articles,topic composition,work summary,word parsing,copyexcerpts,other materials and so on,want to know different data formats andwriting methods,please pay attention!I love going to Chinese restaurants. The food is always so delicious. There are so many different dishes to choose from.The atmosphere in Chinese restaurants is usually really nice. It's cozy and makes you feel at home.The service is often great. The waitstaff is friendly and attentive.And the prices are usually reasonable. You can get a good meal without spending a fortune.。
COS597C:Bayesian nonparametricsLecturer:David Blei Lecture#1 Scribes:Peter Frazier,Indraneel Mukherjee September21,2007In thisfirst lecture,we begin by introducing the Chinese Restaurant Pro-cess.After a brief review offinite mixture models,we describe the Chinese Restaurant Process mixture,where the latent variables are distributed ac-cording to a Chinese Restaurant Process.We end by noting a connection between Dirchlet processes and the Chinese Restaurant Process mixture.The Chinese Restaurant ProcessWe will define a distribution on the space of partitions of the positive integers, N.This would induce a distribution on the partitions of thefirst n integers, for every n∈N.Imagine a restaurant with countably infinitely many tables,labelled1,2,.... Customers walk in and sit down at some table.The tables are chosen ac-cording to the following random process.1.Thefirst customer always chooses thefirst table.2.The n th customer chooses thefirst unoccupied table with probabilityα,and an occupied table with probability c,where c is the number of people sitting at that table.In the above,αis a scalar parameter of the process.One might check that the above does define a probability distribution.Let us denote by k n the number of different tables occupied after n customers have walked in. Then1≤k n≤n and it follows from the above description that precisely tables1,...,k n are occupied.Example A possible arrangement of10customers is shown in Figure1. Denote by z i the table occupied by the customer i.The probability of this arrangement is11,3,82,5,9,104,6,7...Figure 1:The Chinese restaurant process.Circles represent tables and the numbers around them are the customers sitting at that table.Pr(z 1,...,z 10)=Pr(z 1)Pr(z 2|z 1)...Pr(z 10|z 1,...,z 9)=ααα1+α12+αα3+α14+α15+α26+α27+α28+α39+αWe can make the following observations:1.The probability of a seating is invariant under permutations.Permut-ing the customers permutes the numerators in the above computation,while the denominators remains the same.This property is known as exchangeability .2.Any seating arrangement creates a partition.For example,the above seating arrangement partitions customers 1,...,10into the following three groups (1,3,8),(2,5,9,10),(4,6,7).Exchangeability now implies that two seating arrangements whose partitions consist of the same number of components with identical sizes will have the same proba-bility.For instance,the probability of any seating arrangement of ten customers where three tables are occupied,with three customers each on two of the tables and the remaining four on the third table,will have the same probability as the seating in our example.Thus we could define a distribution on the space of all partitions of the integer n ,where n is the total number of customers.The number of partitions is given by the partition function p (n ),which has no simple closed form.Asymptotically,p (n )=exp(O (√n )).23.The expected number of occupied tables for n customers grows loga-rithmically.In particularE [k n |α]=O (αlog n )This can be seen as follows:Let X i be the IRV for the event that the i th customer starts a new table.The probability of this happening is Pr[X i =1]=α/(i −1+α).Since k n = i X i ,by linearity of expectatin the summation is equal to i α/(α+i −1)which is upper bounded by O (αH n )where H n is the n th harmonic sum.By taking the limit as n goes to infinity,we could perhaps define a dis-tribution on the space of all natural numbers.However,technical difficulties might arise while dealing with distributions over infinite sequences,and ap-propriate sigma algebras have to be chosen.Chapters 1-4of [5],and the lecture notes from ORFE 551(sent to course mailing list)are recommended for reading up basic measure theory.Review:Finite mixture modelsFinite mixture models are latent variable models.To model data via finite mixtures of distributions,the basic steps are1.Posit hidden random variables2.Construct a joint distribution of observed and hidden random variablespute the posterior distribution of the hidden variables given the observed variables.Examples include Gaussian mixtures,Kalman filter,Factor analysis,Hid-den Markov models,etc.We review the Gaussian mixture model.A Gaussian mixture with K components,for fixed K ,can be described by the following generative process:1.Choose cluster proportions π∼Dir (α),where Dir (α)is a dirichlet prior,with parameter α,over distributions over k points.2.Choose K means µk ∼N (0,σ2µ)33.For each data point:(a)Choose cluster assignment z∼Mult(π)(b)Choose x∼N(µz,σ2X).The above gives a model for the joint distribution Pr(π,µ1:K,z1:N x1:N|α,σ2µ,σ2X).We will dropα,σ2µ,σ2Xfrom now on,and assume they are known andfixed.The posterior distribution,given data x1:N is Pr(π,µ1:K,z1:N|x1:N)and has the followin:1.This decomposed the data over the latent space,thus revealing under-lying structure when present.2.The posterior helps predict a new data point via the predictive distribu-tion Pr(x|x1:N).For simplicity,we show how to calculate this quantity when the cluster proportionsπisfixed.Pr(x|x1:N,π)=zµzPr(x,z,µz|x1:N,π)dµz=zµzπz Pr(x|µz)Pr(µz|x1:N)dµz=z πzµzPr(x|µz)Pr(µz|x1:N)dµzWe could compute Pr(µz|x1:N)dµz from the posterior Pr(µ1:K,z1:N|x1:N,π) by marginalizing out the z1:N and theµk for k=z.This would enableus to complete the above calculation to obtain the predictive distribu-tion.Chinese Restaurant Process MixtureA Chinese restaurant process mixture is constructed by the following proce-dure:1.Endow each table with a mean,µ∗k ∼N(0,σ2µ),k=1,2,....2a.Customer n sits down at table z n∼CRP(α;z1,...,z n−1).4µ∗11,3,8µ∗22,5,9,10µ∗34,6,7...Figure 2:The Chinese restaurant process mixture.Each table k has a mean µ∗k and customers sitting at it are distributed according to that mean.Tables 1through 3and customers 1through 10are pictured.2b.A datapoint is drawn x n ∼N (µ∗z n ,σ2x ).We will consider the posterior and predictive distributions.The hiddenvariables are the infinite collection of means,µ∗1,µ∗2,...,and the cluster as-signments z 1,z 2,....Consider the posterior on these hidden variables given the first N datapoints,p (µ∗1:N ,z 1:N |x 1:N ,θ),where we define θ=(σ2µ,σ2x ,α)to contain the fixed parameters of the model.Note that we only need to care about the means of the first N tables becausethe rest will have posterior distribution N (0,σ2µ)unchanged from the prior.Similarly,we only need to care about the cluser assignments of the first N customers because the rest will have posterior equal to the prior.The predictive distribution given the data and some additional hidden variables isp (x |x 1:N ,µ∗1:N +1,z 1:N ,θ)=1+K Nz =1p (z |z 1:N )p (x |z,µ∗z ).These hidden variables may be integrated out with the posterior on the hid-den variables to give the predictive distribution conditioned only on the data.Note that,whereas in the finite mixture model the cluster proportions were modeled explicitly,here the cluster proportions are within the z variables.Also note that permuting the x 1:N results in the same predictive distribution,so we have exchangeability here as we did earlier.Why is exchangeability important?Having exchangeability is as though we drew a parameter from a prior and then drew data independently and5identically from that prior.Thus,the data are independent conditioned on the parameter,but are not independent in general.This is weaker than assuming independence.Specifically,DeFinetti’s exchangeability theorem[1]states that the ex-changeability of a random sequence x1,x2,...is equivalent to having a pa-rameterθdrawn from a distribution F(·)and then choosing x n iid from the distribution implied byθ.That is,θ∼F(·)x n iid∼θ.We may apply this idea to the Chinese restaurant process mixture,whichis really a distribution on distributions,or a distribution on the(µz1,µz2,...).The random meansµz1,µz2,...are exchangeable,so this implies that theirdistribution may be expressed in the form given by DeFinetti’s exchangeabil-ity theorem.Their distribution is given byG∼DirichletProcess(αG0)µzi iid∼G.Here G0is the distribution of theµ∗on the reals,e.g.,N(0,σ2u ).Notethat we get repeated values when sampling the z i whenever a customer sitsat an already occupied table.Then customer i hasµzi identical to theµzjofall customers j previously seated at the table.In fact,G is an almost surely discrete distribution on the reals with a countably infinite number of atoms.The reading for next week,[2,4],is on how to compute the posterior of the hidden parameters given the data,and also includes two new perspectives on the Chinese restaurant process.One perspective is the one just described, of the Chinese restaurant process as a Dirichlet process,and the other is as an infinite limit offinite mixture models.In the reading,focus on[4].In addition,a good general reference on Bayesian statistics that may be helpful in the course is[3].References[1]Jose-M.Bernardo and Adrian F.M.Smith.Bayesian theory.Wiley Seriesin Probability and Mathematical Statistics:Probability and Mathemati-cal Statistics.John Wiley&Sons Ltd.,Chichester,1994.6[2]Michael D.Escobar and Mike West.Bayesian density estimation and in-ference using mixtures.J.Amer.Statist.Assoc.,90(430):577–588,1995.[3]Andrew Gelman,John B.Carlin,Hal S.Stern,and Donald B.Rubin.Bayesian data analysis.Texts in Statistical Science Series.Chapman& Hall/CRC,Boca Raton,FL,second edition,2004.[4]Radford M.Neal.Markov chain sampling methods for Dirichlet processmixture put.Graph.Statist.,9(2):249–265,2000.[5]David Williams.Probability with martingales.Cambridge MathematicalTextbooks.Cambridge University Press,Cambridge,1991.7。