Order Statistics

Python

Independence (probability theory)

Random variable

Joint Discrete Distributions

Correlation

Multinomial Distribution

Markov chain Monte Carlo (MCMC)

Beta distribution

Hypothesis Testing

Multi-Armed Bandits

Probability via Simulation

Discrete random variables

Continuous Random Variables

Joint Continuous Distributions

Multivariate Normal Distribution

Discrete Time Stochastic Process

Dirichlet Distribution

Bootstrapping

Markov's inequality

Convolution

Discrete Probability

Expectation

The Normal Distribution

Conditional distribution

Moment Generating Functions

Maximum a Posteriori Estimation

Confidence Intervals

The Chernoff Bound

Maximum Likelihood Estimation (MLE)

Counting

Conditional probability

Variance

Transforming Random Variables

Covariance

Limit Theorems

Central Limit Theorem

Method of Moments Estimation

Properties of Estimators

Credible Intervals

Chebyshev‘s inequality

CSE 312 Foundations of Computing II University of Washington Winter 2022 This course dives deep into the role of probability in the realm of computer science, exploring applications such as algorithms, systems, data analysis, machine learning, and more. Prerequisites include CSE 311, MATH 126, and a grasp of calculus, linear algebra, set theory, and basic proof techniques. Concepts covered range from discrete probability to hypothesis testing and bootstrapping. ## Why is CSE 312 Important?

While the initial foundations of computer science began in the world of discrete mathematics (after all, modern computers are digital in nature), recent years have seen a surge in the use of probability as a tool for the analysis and development of new algorithms and systems. As a result, it is becoming increasingly important for budding computer scientists to understand probability theory, both to provide new perspectives on existing ideas and to help further advance the field in new ways.

Probability is used in a number of contexts, including analyzing the likelihood that various events will happen, better understanding the performance of algorithms (which are increasingly making use of randomness), or modeling the behavior of systems that exist in asynchronous environments ruled by uncertainty (such as requests being made to a web server). Probability provides a rich set of tools for modeling such phenomena
and allowing for precise mathematical statements to be made about the performance of an algorithm or a system in such situations.

Furthermore, computers are increasingly often being used as data analysis tools to glean insights from the enormous amounts of data being gathered in a variety of fields; you’ve no doubt heard the phrase “big data” referring to this phenomenon. Probability theory and statistics are the foundational methods used for designing new algorithms to model such data, allowing, for example, a computer to make predictions about new or uncertain events. In fact, many of you have already been the users of such techniques. For example, most email systems now employ automated spam detection and filtering. Methods for being able to automatically infer whether or not an email message is spam are frequently rooted in probabilistic methods. Similarly, if you have ever seen online product recommendation (e.g., “customers who bought X are also likely to buy Y”), you’ve seen yet another application of probability in computer science. Even more subtly, answering detailed questions like how many buckets you should have in your a hash table or how many machines you should deploy in a data center (server farm) for an online application make use of probabilistic techniques to give precise formulations based on testable assumptions.
 Our goal in this course is to build foundational skills and give you experience in the following areas:

1. **Understanding the combinatorial nature of problems**: Many real problems are based on understanding the multitude of possible outcomes that may occur, and determining which of those outcomes satisfy some criteria we care about. Such an understanding is important both for determining how ikely an outcome is, but also for understanding what factors may affect the outcome (and which of those may be in our control).
2. **Working knowledge of probability theory and some of the key results in statistics**: Having a solid knowledge of probability theory and statistics is essential for computer scientists today. Such knowledge includes theoretical fundamentals as well as an appreciation for how that theory can be successfully applied in practice. We hope to impart both these concepts in this class.
3. **Appreciation for probabilistic statements**: In the world around us, probabilistic statements are often made, but are easily misunderstood. For example, when a candidate in an election is said to have a 53% likelihood of winning does this mean that the candidate is likely to get 53% of the vote, or that that if 100 elections were held today, the candidate would win 53% of them? Understanding the difference between these statements requires an understanding of the model in the underlying probabilistic analysis.
4. **Applications in machine learning and theoretical computer science**: We are not studying probability theory simply for the joy of drawing summation symbols (okay, maybe some people are, but that’s not what we’re really targeting in this class), but rather because there are a wide variety of applications where probability and statistics allow us to solve problems that might otherwise be out of reach (or would be solved more poorly without the tools that probability and statistics can bring to bear). We’ll look at examples of such applications throughout the class. For example, machine learning is a quickly growing subfield of artificial intelligence which has grown to impact many applications in computing. It focuses on analyzing large quantities of data to build models that can then be harnessed in real problems, such as filtering email, improving web search, understanding computer system performance, predicting financial markets, or analyzing DNA. Probability and statistics form the foundation of these systems. Another example application area is the use of randomized algorithms and probabilistic data structures. These usually have simpler and more elegant implementations than their deterministic counterparts, and have more efficient time and/or space complexity. We will be learning about some of these applications and you will have the opportunity to implement some of these algorithms.
 CSE 311 and MATH 126. Here is a quick rundown of some of the mathematical tools we’ll
be using in this class: calculus (integration and differentiation), linear algebra (basic operations on vectors and matrices), an understanding of the basics of set theory (subsets, complements, unions, intersections, cardinality, etc.), and familiarity with basic proof techniques (including induction). - (Recommended) Alex Tsun, Probability & Statistics with Applications to Computing, 2020. Available free online [here](http://www.alextsun.com/files/Prob_Stat_for_CS_Book.pdf).
- (Optional) Sheldon Ross, A First Course in Probability (10th Ed.), Pearson Prentice Hall, 2018.
- (Optional) Larsen and Marx, An Introduction to Mathematical Statistics (5th edition). Prentice-Hall.
- (Optional) Dimitri P. Bertsekas and John N. Tsitsiklis, Introduction to Probability, First Edition, Athena Scientific, 2000. Available free online [here](https://www.vfu.bg/en/e-Learning/Math--Bertsekas_Tsitsiklis_Introduction_to_probability.pdf).

CSE 312 Foundations of Computing II

Mathematical Foundations

Theoretical Computer Science

Game theory

Normal-form game

Nash equilibrium

Keynesian beauty contest

Auction theory

Incomplete-information game

Bayesian game

Generalized second-price auction

First-price Auctions

Revenue Equivalence

Revelation Principle

Auction Design Goals

Vickrey–Clarke–Groves (VCG) mechanism

Welfare Maximization

Myerson–Satterthwaite theorem

Myerson's Payment Characterization

Myerson's Optimal Auction

The Myerson mechanism

Posted-Price Mechanisms

Regular and MHR Distributions

Inverse Transform Sampling

Approximation algorithm

Prophet inequality

Bulow Klemperer Theorem

Iterative Auction

Combinatorial auction

Ascending Auction

Walrasian equilibrium

Unit-Demand Valuations

Crawford-Knoer (CK) Auction

CSCI 1440/2440 Topics in Algorithmic Game Theory Brown University Spring 2023 Offered by Brown University, this course intertwines game theory and computational considerations. With emphasis on strategic agent behavior, system design, and computational tractability, students delve into auction theory, bidding strategies, computational advertising, and automated negotiation. Knowledge in Java, Python, and certain mathematical areas is essential. This course examines topics in game theory and mechanism design through the lens of computation. Like such a course in an economics department, the focus is the design and analysis of systems utilized by self-interested agents. Students investigate how the potential for strategic agent behavior can/should influence system design, and the ramifications of conflicts of interest between system designers and participating agents. Unlike a traditional economics course, however, emphasis on computational tractability is paramount, so that simplicity may trump other design desiderata. Students will learn to analyze competing designs using the tools of theoretical computer science. They will also use artificial intelligence to build autonomous agents for computational advertising markets, wireless spectrum auctions, automated negotiation, and prediction markets.

There are two primary learning outcomes intended for students taking this course. Students should be able to:

- Reason about the design of multiagent systems taking into account agents' incentives, a perspective borrowed from economists, as well as computational performance, the bread and butter of computer scientists.
- Design and build effective autonomous agents for market domains that again incorporate both strategic and computational considerations.

  Mathematical maturity and programming experience are necessary for success in this course. The following are specific areas in which basic expertise is assumed, along with suggested courses for acquiring said expertise.

- Comfort with continuous mathematics: e.g., Math 0180, Math 0350, APMA 0350, or APMA 0360.
- Comfort with probability and statistics: e.g., CS 1450, APMA 1650, APMA 1655, or Math 1620.
- Comfort writing proofs: e.g., CS 22, CS 1010, CS 1550, or any 1000-level Math class.
- Comfort with programming: e.g., CS 4, CS 111, CS 15, CS 17, CS 19, or equivalent.

Some knowledge of Java and Python is assumed; neither language is taught.

For students wishing to enroll in the graduate section of this course, CSCI 2440, knowledge of Markov decision processes and linear programming is also assumed.

This course has no formal prerequisites. However, a sufficient level of mathematical maturity is required, as is knowledge of programming. For the most part, labs and homework assignments will be in Python, while the final project will be in Java. All assignments are programmed in pairs. If you worry that your programming skills are not up to snuff, take this opportunity to pair with someone who can help you get up to speed.  ### Topics

1. What is game theory?
1. What is mechanism design?
1. What is auction theory?
1. Simple, awesome, and EPIC auctions
1. Bidding strategies for combinatorial auctions
1. Empirical game-theoretic analysis and mechanism design
1. Application domains
    1. Computational advertising markets
    1. Wireless spectrum markets
    1. Automated negotiation
    1. Prediction markets

CSCI 1440/2440 Topics in Algorithmic Game Theory

Interdisciplinary

Order statistic

Order Statistics

2 courses cover this concept

CSE 312 Foundations of Computing II

CSCI 1440/2440 Topics in Algorithmic Game Theory