# Announcements. CS 188: Artificial Intelligence Fall Sampling. Prior Sampling. Likelihood Weighting. Rejection Sampling

Save this PDF as:

Size: px
Start display at page:

Download "Announcements. CS 188: Artificial Intelligence Fall Sampling. Prior Sampling. Likelihood Weighting. Rejection Sampling"

## Transcription

1 CS 188: Artificial Intelligence Fall 2008 Lecture 18: Decision Diagrams 10/30/2008 Announcements P4 EXTENDED to Tuesday 11/4 Midterms graded, pick up after lecture Midterm course evaluation up on web soon, please fill out! Dan Klein C Berkeley Final contest instructions out today! Prizes will be good 1 2 Sampling Basic idea: Draw N samples from a sampling distribution S Compute an approximate posterior probability Show this converges to the true probability P Prior Sampling Cloudy Outline: Sampling from an empty network Rejection sampling: reject samples disagreeing with evidence Likelihood weighting: use evidence to weight samples Sprinkler etgrass Rain 3 4 Rejection Sampling Likelihood eighting Let s say we want P(C) No point keeping all samples around Just tally counts of C outcomes Let s say we want P(C s) Same thing: tally C outcomes, but ignore (reject) samples which don t have S=s This is rejection sampling It is also consistent for conditional probabilities (i.e., correct in the limit) Sprinkler S Cloudy C etgrass Rain R c, s, r, w c, s, r, w c, s, r, w c, s, r, w c, s, r, w 7 Problem with rejection sampling: If evidence is unlikely, you reject a lot of samples You don t exploit your evidence as you sample Consider P(B a) Burglary Idea: fix evidence variables and sample the rest Burglary Alarm Alarm Problem: sample distribution not consistent! Solution: weight by probability of evidence given parents 8 1

2 Likelihood Sampling Likelihood eighting Sampling distribution if z sampled and e fixed evidence Cloudy Cloudy C Sprinkler Rain Now, samples have weights S Rain R etgrass Together, weighted sampling distribution is consistent 9 10 Likelihood eighting Pacman Contest Note that likelihood weighting doesn t solve all our problems Rare evidence is taken into account for downstream variables, but not upstream ones A better solution is Markov-chain Monte Carlo (MCMC), more advanced e ll return to sampling for robot localization and tracking in dynamic BNs S Cloudy C Rain R Recap: Inference Example Decision Networks Find P( F=bad) Restrict all factors P() No hidden vars to eliminate (this time!) Just join and normalize P(,F=bad) P(F=bad ) P() F P(F ) good 0.8 bad 0.2 F P(F ) good 0.1 bad 0.9 P( F=bad) ME: choose the action which maximizes the expected utility given the evidence Can directly operationalize this with decision diagrams Bayes nets with nodes for utility and actions Lets us calculate the expected utility for each action New node types: Chance nodes (just like BNs) Actions (rectangles, must be parents, act as observed evidence) tilities (depend on action and chance nodes) mbrella 16 2

3 Decision Networks Example: Decision Networks Action selection: Instantiate all evidence Calculate posterior over parents of utility node Set action node each possible way Calculate expected utility for each action Choose maximizing action mbrella 17 mbrella = leave mbrella = take Optimal decision = leave mbrella A (A,) P() leave leave take 20 take Example: Decision Networks Value of Information mbrella = leave mbrella = take Optimal decision = take mbrella =bad P( F=bad) A (A,) leave 100 leave 0 take 20 take Idea: compute value of acquiring each possible piece of evidence Can be done directly from decision network Example: buying oil drilling rights Two blocks A and B, exactly one has oil, worth k Prior probabilities 0.5 each, mutually exclusive Current price of each block is k/2 ME = 0 (either action is a maximizer) Solution: compute value of information a 1/2 = expected gain in ME from observing new information b 1/2 Probe gives accurate survey of A. Fair price? Survey may say oil in a or oil in b, prob 0.5 each If we know O, ME is k/2 (either way) Gain in ME? VPI(O) = k/2 Fair price: k/2 DrillLoc OilLoc O P D O a a k/2 a b -k/2 b a -k/2 b b k/2 20 Value of Information VPI Example Current evidence E=e, utility depends on S=s ME with no evidence mbrella Potential new evidence E : suppose we knew E = e ME if forecast is bad A BT E is a random variable whose value is currently unknown, so: Must compute expected gain over all possible values ME if forecast is good distribution leave 100 leave 0 take 20 take 70 F P(F) good 0.59 bad 0.41 (VPI = value of perfect information)

4 VPI Properties VPI Scenarios Nonnegative in expectation Imagine actions 1 and 2, for which 1 > 2 How much will information about E j be worth? Nonadditive ---consider, e.g., obtaining E j twice Order-independent 23 Little we re sure action 1 is better. A lot either could be much better Little info likely to change our action but not our utility 24 Reasoning over Time Often, we want to reason about a sequence of observations Speech recognition Robot localization ser attention Medical monitoring Need to introduce time into our models Basic approach: hidden Markov models (HMMs) More general: dynamic Bayes nets 25 Markov Models A Markov model is a chain-structured BN Each node is identically distributed (stationarity) Value of X at a given time is called the state As a BN: X 1 X 2 X 3 X 4 Parameters: called transition probabilities or dynamics, specify how the state evolves over time (also, initial probs) 26 [DEMO: Battleship] Conditional Independence X 1 X 2 X 3 X 4 Basic conditional independence: Past and future independent of the present Each time step only depends on the previous This is called the (first order) Markov property Note that the chain is just a (growing) BN e can always use generic BN reasoning on it (if we truncate the chain) Example: Markov Chain : States: X = {, } Transitions: Initial distribution: 1.0 hat s the probability distribution after one step? This is a CPT, not a BN!

6 Mini-Viterbi Algorithm Mini-Viterbi Better answer: cached incremental updates Define: Read best sequence off of m and a vectors

### CS 188: Artificial Intelligence. Probability recap

CS 188: Artificial Intelligence Bayes Nets Representation and Independence Pieter Abbeel UC Berkeley Many slides over this course adapted from Dan Klein, Stuart Russell, Andrew Moore Conditional probability

### Introduction to Markov Chain Monte Carlo

Introduction to Markov Chain Monte Carlo Monte Carlo: sample from a distribution to estimate the distribution to compute max, mean Markov Chain Monte Carlo: sampling using local information Generic problem

### Logic, Probability and Learning

Logic, Probability and Learning Luc De Raedt luc.deraedt@cs.kuleuven.be Overview Logic Learning Probabilistic Learning Probabilistic Logic Learning Closely following : Russell and Norvig, AI: a modern

### Lecture 2: Introduction to belief (Bayesian) networks

Lecture 2: Introduction to belief (Bayesian) networks Conditional independence What is a belief network? Independence maps (I-maps) January 7, 2008 1 COMP-526 Lecture 2 Recall from last time: Conditional

### Artificial Intelligence Mar 27, Bayesian Networks 1 P (T D)P (D) + P (T D)P ( D) =

Artificial Intelligence 15-381 Mar 27, 2007 Bayesian Networks 1 Recap of last lecture Probability: precise representation of uncertainty Probability theory: optimal updating of knowledge based on new information

### Machine Learning. CS 188: Artificial Intelligence Naïve Bayes. Example: Digit Recognition. Other Classification Tasks

CS 188: Artificial Intelligence Naïve Bayes Machine Learning Up until now: how use a model to make optimal decisions Machine learning: how to acquire a model from data / experience Learning parameters

### Lecture 6: The Bayesian Approach

Lecture 6: The Bayesian Approach What Did We Do Up to Now? We are given a model Log-linear model, Markov network, Bayesian network, etc. This model induces a distribution P(X) Learning: estimate a set

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note 11

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao,David Tse Note Conditional Probability A pharmaceutical company is marketing a new test for a certain medical condition. According

### Part 1: Link Analysis & Page Rank

Chapter 8: Graph Data Part 1: Link Analysis & Page Rank Based on Leskovec, Rajaraman, Ullman 214: Mining of Massive Datasets 1 Exam on the 5th of February, 216, 14. to 16. If you wish to attend, please

### The Basics of Graphical Models

The Basics of Graphical Models David M. Blei Columbia University October 3, 2015 Introduction These notes follow Chapter 2 of An Introduction to Probabilistic Graphical Models by Michael Jordan. Many figures

### Lecture Note 1 Set and Probability Theory. MIT 14.30 Spring 2006 Herman Bennett

Lecture Note 1 Set and Probability Theory MIT 14.30 Spring 2006 Herman Bennett 1 Set Theory 1.1 Definitions and Theorems 1. Experiment: any action or process whose outcome is subject to uncertainty. 2.

### Variable Elimination 1

Variable Elimination 1 Inference Exact inference Enumeration Variable elimination Junction trees and belief propagation Approximate inference Loopy belief propagation Sampling methods: likelihood weighting,

### Part III: Machine Learning. CS 188: Artificial Intelligence. Machine Learning This Set of Slides. Parameter Estimation. Estimation: Smoothing

CS 188: Artificial Intelligence Lecture 20: Dynamic Bayes Nets, Naïve Bayes Pieter Abbeel UC Berkeley Slides adapted from Dan Klein. Part III: Machine Learning Up until now: how to reason in a model and

### Decision Making under Uncertainty

6.825 Techniques in Artificial Intelligence Decision Making under Uncertainty How to make one decision in the face of uncertainty Lecture 19 1 In the next two lectures, we ll look at the question of how

### Lecture 10: Sequential Data Models

CSC2515 Fall 2007 Introduction to Machine Learning Lecture 10: Sequential Data Models 1 Example: sequential data Until now, considered data to be i.i.d. Turn attention to sequential data Time-series: stock

### 1 Maximum likelihood estimation

COS 424: Interacting with Data Lecturer: David Blei Lecture #4 Scribes: Wei Ho, Michael Ye February 14, 2008 1 Maximum likelihood estimation 1.1 MLE of a Bernoulli random variable (coin flips) Given N

### Bayesian Tutorial (Sheet Updated 20 March)

Bayesian Tutorial (Sheet Updated 20 March) Practice Questions (for discussing in Class) Week starting 21 March 2016 1. What is the probability that the total of two dice will be greater than 8, given that

### Model-based Synthesis. Tony O Hagan

Model-based Synthesis Tony O Hagan Stochastic models Synthesising evidence through a statistical model 2 Evidence Synthesis (Session 3), Helsinki, 28/10/11 Graphical modelling The kinds of models that

### November 28 th, Carlos Guestrin. Lower dimensional projections

PCA Machine Learning 070/578 Carlos Guestrin Carnegie Mellon University November 28 th, 2007 Lower dimensional projections Rather than picking a subset of the features, we can new features that are combinations

### The intuitions in examples from last time give us D-separation and inference in belief networks

CSC384: Lecture 11 Variable Elimination Last time The intuitions in examples from last time give us D-separation and inference in belief networks a simple inference algorithm for networks without Today

### The PageRank Citation Ranking: Bring Order to the Web

The PageRank Citation Ranking: Bring Order to the Web presented by: Xiaoxi Pang 25.Nov 2010 1 / 20 Outline Introduction A ranking for every page on the Web Implementation Convergence Properties Personalized

### Bayesian Phylogeny and Measures of Branch Support

Bayesian Phylogeny and Measures of Branch Support Bayesian Statistics Imagine we have a bag containing 100 dice of which we know that 90 are fair and 10 are biased. The

### Probability OPRE 6301

Probability OPRE 6301 Random Experiment... Recall that our eventual goal in this course is to go from the random sample to the population. The theory that allows for this transition is the theory of probability.

### Decision Trees and Networks

Lecture 21: Uncertainty 6 Today s Lecture Victor R. Lesser CMPSCI 683 Fall 2010 Decision Trees and Networks Decision Trees A decision tree is an explicit representation of all the possible scenarios from

### Artificial Intelligence

Artificial Intelligence ICS461 Fall 2010 Nancy E. Reed nreed@hawaii.edu 1 Lecture #13 Uncertainty Outline Uncertainty Probability Syntax and Semantics Inference Independence and Bayes' Rule Uncertainty

### Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22

CS 70 Discrete Mathematics for CS Fall 2006 Papadimitriou & Vazirani Lecture 22 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice, roulette

### Tagging with Hidden Markov Models

Tagging with Hidden Markov Models Michael Collins 1 Tagging Problems In many NLP problems, we would like to model pairs of sequences. Part-of-speech (POS) tagging is perhaps the earliest, and most famous,

### STA 4273H: Statistical Machine Learning

STA 4273H: Statistical Machine Learning Russ Salakhutdinov Department of Statistics! rsalakhu@utstat.toronto.edu! http://www.cs.toronto.edu/~rsalakhu/ Lecture 6 Three Approaches to Classification Construct

### Course: Model, Learning, and Inference: Lecture 5

Course: Model, Learning, and Inference: Lecture 5 Alan Yuille Department of Statistics, UCLA Los Angeles, CA 90095 yuille@stat.ucla.edu Abstract Probability distributions on structured representation.

### Simulation and Risk Analysis

Simulation and Risk Analysis Using Analytic Solver Platform REVIEW BASED ON MANAGEMENT SCIENCE What We ll Cover Today Introduction Frontline Systems Session Ι Beta Training Program Goals Overview of Analytic

### Artificial Intelligence. Conditional probability. Inference by enumeration. Independence. Lesson 11 (From Russell & Norvig)

Artificial Intelligence Conditional probability Conditional or posterior probabilities e.g., cavity toothache) = 0.8 i.e., given that toothache is all I know tation for conditional distributions: Cavity

### Bayesian Machine Learning (ML): Modeling And Inference in Big Data. Zhuhua Cai Google, Rice University caizhua@gmail.com

Bayesian Machine Learning (ML): Modeling And Inference in Big Data Zhuhua Cai Google Rice University caizhua@gmail.com 1 Syllabus Bayesian ML Concepts (Today) Bayesian ML on MapReduce (Next morning) Bayesian

### Bayesian Statistics in One Hour. Patrick Lam

Bayesian Statistics in One Hour Patrick Lam Outline Introduction Bayesian Models Applications Missing Data Hierarchical Models Outline Introduction Bayesian Models Applications Missing Data Hierarchical

### CMPSCI 683 Artificial Intelligence Questions & Answers

CMPSCI 683 Artificial Intelligence Questions & s 1. General Learning Consider the following modification to the restaurant example described in class, which includes missing and partially specified attributes:

### Introduction to Algorithmic Trading Strategies Lecture 2

Introduction to Algorithmic Trading Strategies Lecture 2 Hidden Markov Trading Model Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Outline Carry trade Momentum Valuation CAPM Markov chain

### Decision Making Under Uncertainty. Professor Peter Cramton Economics 300

Decision Making Under Uncertainty Professor Peter Cramton Economics 300 Uncertainty Consumers and firms are usually uncertain about the payoffs from their choices Example 1: A farmer chooses to cultivate

### Machine Learning: Fundamentals

Machine Learning: Fundamentals A model is a formal description of a belief about the world. Learning is the construction and/or revision of a model in response to observations of the world. The mathematical/statistical

### 6.042/18.062J Mathematics for Computer Science October 3, 2006 Tom Leighton and Ronitt Rubinfeld. Graph Theory III

6.04/8.06J Mathematics for Computer Science October 3, 006 Tom Leighton and Ronitt Rubinfeld Lecture Notes Graph Theory III Draft: please check back in a couple of days for a modified version of these

### Neural Networks for Machine Learning. Lecture 13a The ups and downs of backpropagation

Neural Networks for Machine Learning Lecture 13a The ups and downs of backpropagation Geoffrey Hinton Nitish Srivastava, Kevin Swersky Tijmen Tieleman Abdel-rahman Mohamed A brief history of backpropagation

### Machine Learning and Statistics: What s the Connection?

Machine Learning and Statistics: What s the Connection? Institute for Adaptive and Neural Computation School of Informatics, University of Edinburgh, UK August 2006 Outline The roots of machine learning

### Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091)

Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 5 Sequential Monte Carlo methods I February

### COS513 LECTURE 5 JUNCTION TREE ALGORITHM

COS513 LECTURE 5 JUNCTION TREE ALGORITHM D. EIS AND V. KOSTINA The junction tree algorithm is the culmination of the way graph theory and probability combine to form graphical models. After we discuss

### 11.2 POINT ESTIMATES AND CONFIDENCE INTERVALS

11.2 POINT ESTIMATES AND CONFIDENCE INTERVALS Point Estimates Suppose we want to estimate the proportion of Americans who approve of the president. In the previous section we took a random sample of size

### CS 331: Artificial Intelligence Fundamentals of Probability II. Joint Probability Distribution. Marginalization. Marginalization

Full Joint Probability Distributions CS 331: Artificial Intelligence Fundamentals of Probability II Toothache Cavity Catch Toothache, Cavity, Catch false false false 0.576 false false true 0.144 false

### Hidden Markov Models for biological systems

Hidden Markov Models for biological systems 1 1 1 1 0 2 2 2 2 N N KN N b o 1 o 2 o 3 o T SS 2005 Heermann - Universität Heidelberg Seite 1 We would like to identify stretches of sequences that are actually

### Section 2.1. Tree Diagrams

Section 2.1 Tree Diagrams Example 2.1 Problem For the resistors of Example 1.16, we used A to denote the event that a randomly chosen resistor is within 50 Ω of the nominal value. This could mean acceptable.

### 15-381: Artificial Intelligence. Probabilistic Reasoning and Inference

5-38: Artificial Intelligence robabilistic Reasoning and Inference Advantages of probabilistic reasoning Appropriate for complex, uncertain, environments - Will it rain tomorrow? Applies naturally to many

10-601 Machine Learning http://www.cs.cmu.edu/afs/cs/academic/class/10601-f10/index.html Course data All up-to-date info is on the course web page: http://www.cs.cmu.edu/afs/cs/academic/class/10601-f10/index.html

### Lecture 9: Introduction to Pattern Analysis

Lecture 9: Introduction to Pattern Analysis g Features, patterns and classifiers g Components of a PR system g An example g Probability definitions g Bayes Theorem g Gaussian densities Features, patterns

### Learning outcomes. Knowledge and understanding. Ability and Competences. Evaluation capability and scientific approach

Syllabus Master s Programme in Statistics and Data Mining 120 ECTS Credits Aim The rapid growth of databases provides scientists and business people with vast new resources. This programme meets the challenges

### DECISION ANALYSIS. Introduction. Prototype example. Decision analysis framework. Decision making without experimentation

Introduction DECISION ANALYSIS (Hillier & Lieberman Introduction to Operations Research, 8 th edition) Decision often must be made in uncertain environments Examples: Manufacturer introducing a new product

### Learning outcomes. Knowledge and understanding. Competence and skills

Syllabus Master s Programme in Statistics and Data Mining 120 ECTS Credits Aim The rapid growth of databases provides scientists and business people with vast new resources. This programme meets the challenges

### State Spaces Graph Search Searching. Graph Search. CPSC 322 Search 2. Textbook 3.4. Graph Search CPSC 322 Search 2, Slide 1

Graph Search CPSC 322 Search 2 Textbook 3.4 Graph Search CPSC 322 Search 2, Slide 1 Lecture Overview 1 State Spaces 2 Graph Search 3 Searching Graph Search CPSC 322 Search 2, Slide 2 State Spaces Idea:

### Compression algorithm for Bayesian network modeling of binary systems

Compression algorithm for Bayesian network modeling of binary systems I. Tien & A. Der Kiureghian University of California, Berkeley ABSTRACT: A Bayesian network (BN) is a useful tool for analyzing the

### Bayesian Statistics: Indian Buffet Process

Bayesian Statistics: Indian Buffet Process Ilker Yildirim Department of Brain and Cognitive Sciences University of Rochester Rochester, NY 14627 August 2012 Reference: Most of the material in this note

### Master s Theory Exam Spring 2006

Spring 2006 This exam contains 7 questions. You should attempt them all. Each question is divided into parts to help lead you through the material. You should attempt to complete as much of each problem

### Bayesian Networks Chapter 14. Mausam (Slides by UW-AI faculty & David Page)

Bayesian Networks Chapter 14 Mausam (Slides by UW-AI faculty & David Page) Bayes Nets In general, joint distribution P over set of variables (X 1 x... x X n ) requires exponential space for representation

### Machine Learning. CUNY Graduate Center, Spring 2013. Professor Liang Huang. huang@cs.qc.cuny.edu

Machine Learning CUNY Graduate Center, Spring 2013 Professor Liang Huang huang@cs.qc.cuny.edu http://acl.cs.qc.edu/~lhuang/teaching/machine-learning Logistics Lectures M 9:30-11:30 am Room 4419 Personnel

### Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091)

Monte Carlo and Empirical Methods for Stochastic Inference (MASM11/FMS091) Magnus Wiktorsson Centre for Mathematical Sciences Lund University, Sweden Lecture 6 Sequential Monte Carlo methods II February

### Lecture 4 : Bayesian inference

Lecture 4 : Bayesian inference The Lecture dark 4 energy : Bayesian puzzle inference What is the Bayesian approach to statistics? How does it differ from the frequentist approach? Conditional probabilities,

### Hidden Markov Models Chapter 15

Hidden Markov Models Chapter 15 Mausam (Slides based on Dan Klein, Luke Zettlemoyer, Alex Simma, Erik Sudderth, David Fernandez-Baca, Drena Dobbs, Serafim Batzoglou, William Cohen, Andrew McCallum, Dan

### ABC methods for model choice in Gibbs random fields

ABC methods for model choice in Gibbs random fields Jean-Michel Marin Institut de Mathématiques et Modélisation Université Montpellier 2 Joint work with Aude Grelaud, Christian Robert, François Rodolphe,

### I. Basics of Hypothesis Testing

Introduction to Hypothesis Testing This deals with an issue highly similar to what we did in the previous chapter. In that chapter we used sample information to make inferences about the range of possibilities

### Hidden Markov Models. Terminology and Basic Algorithms

Hidden Markov Models Terminology and Basic Algorithms Motivation We make predictions based on models of observed data (machine learning). A simple model is that observations are assumed to be independent

### Graph Theory. Directed and undirected graphs make useful mental models for many situations. These objects are loosely defined as follows:

Graph Theory Directed and undirected graphs make useful mental models for many situations. These objects are loosely defined as follows: Definition An undirected graph is a (finite) set of nodes, some

### GCSE Statistics Revision notes

GCSE Statistics Revision notes Collecting data Sample This is when data is collected from part of the population. There are different methods for sampling Random sampling, Stratified sampling, Systematic

CS787: Advanced Algorithms Lecture 14: Online algorithms We now shift focus to a different kind of algorithmic problem where we need to perform some optimization without knowing the input in advance. Algorithms

### A crash course in probability and Naïve Bayes classification

Probability theory A crash course in probability and Naïve Bayes classification Chapter 9 Random variable: a variable whose possible values are numerical outcomes of a random phenomenon. s: A person s

### STA 371G: Statistics and Modeling

STA 371G: Statistics and Modeling Decision Making Under Uncertainty: Probability, Betting Odds and Bayes Theorem Mingyuan Zhou McCombs School of Business The University of Texas at Austin http://mingyuanzhou.github.io/sta371g

### An Introduction to Using WinBUGS for Cost-Effectiveness Analyses in Health Economics

Slide 1 An Introduction to Using WinBUGS for Cost-Effectiveness Analyses in Health Economics Dr. Christian Asseburg Centre for Health Economics Part 1 Slide 2 Talk overview Foundations of Bayesian statistics

### Probability, Conditional Independence

Probability, Conditional Independence June 19, 2012 Probability, Conditional Independence Probability Sample space Ω of events Each event ω Ω has an associated measure Probability of the event P(ω) Axioms

### Lab 8: Introduction to WinBUGS

40.656 Lab 8 008 Lab 8: Introduction to WinBUGS Goals:. Introduce the concepts of Bayesian data analysis.. Learn the basic syntax of WinBUGS. 3. Learn the basics of using WinBUGS in a simple example. Next

### Hidden Markov Models in Bioinformatics. By Máthé Zoltán Kőrösi Zoltán 2006

Hidden Markov Models in Bioinformatics By Máthé Zoltán Kőrösi Zoltán 2006 Outline Markov Chain HMM (Hidden Markov Model) Hidden Markov Models in Bioinformatics Gene Finding Gene Finding Model Viterbi algorithm

### Monte Carlo-based statistical methods (MASM11/FMS091)

Monte Carlo-based statistical methods (MASM11/FMS091) Jimmy Olsson Centre for Mathematical Sciences Lund University, Sweden Lecture 5 Sequential Monte Carlo methods I February 5, 2013 J. Olsson Monte Carlo-based

### Probabilistic Models for Big Data. Alex Davies and Roger Frigola University of Cambridge 13th February 2014

Probabilistic Models for Big Data Alex Davies and Roger Frigola University of Cambridge 13th February 2014 The State of Big Data Why probabilistic models for Big Data? 1. If you don t have to worry about

### Web Search Engines: Solutions

Web Search Engines: Solutions Problem 1: A. How can the owner of a web site design a spider trap? Answer: He can set up his web server so that, whenever a client requests a URL in a particular directory

### Metropolis Light Transport. Samuel Donow, Mike Flynn, David Yan CS371 Fall 2014, Morgan McGuire

Metropolis Light Transport Samuel Donow, Mike Flynn, David Yan CS371 Fall 2014, Morgan McGuire Overview of Presentation 1. Description of necessary tools (Path Space, Monte Carlo Integration, Rendering

### Why Semantic Analysis is Better than Sentiment Analysis. A White Paper by T.R. Fitz-Gibbon, Chief Scientist, Networked Insights

Why Semantic Analysis is Better than Sentiment Analysis A White Paper by T.R. Fitz-Gibbon, Chief Scientist, Networked Insights Why semantic analysis is better than sentiment analysis I like it, I don t

### SYSM 6304: Risk and Decision Analysis Lecture 5: Methods of Risk Analysis

SYSM 6304: Risk and Decision Analysis Lecture 5: Methods of Risk Analysis M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October 17, 2015 Outline

### Some Examples of (Markov Chain) Monte Carlo Methods

Some Examples of (Markov Chain) Monte Carlo Methods Ryan R. Rosario What is a Monte Carlo method? Monte Carlo methods rely on repeated sampling to get some computational result. Monte Carlo methods originated

### AP Stats - Probability Review

AP Stats - Probability Review Multiple Choice Identify the choice that best completes the statement or answers the question. 1. I toss a penny and observe whether it lands heads up or tails up. Suppose

### Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10

CS 70 Discrete Mathematics and Probability Theory Fall 2009 Satish Rao, David Tse Note 10 Introduction to Discrete Probability Probability theory has its origins in gambling analyzing card games, dice,

### Introduction to Machine Learning

Introduction to Machine Learning Brown University CSCI 1950-F, Spring 2012 Instructor: Erik Sudderth Graduate TAs: Dae Il Kim & Ben Swanson Head Undergraduate TA: William Allen Undergraduate TAs: Soravit

### Coimisiún na Scrúduithe Stáit State Examinations Commission. Junior Certificate Examination 2015 Sample Paper. Mathematics

2015. S34 S Coimisiún na Scrúduithe Stáit State Examinations Commission Junior Certificate Examination 2015 Mathematics Paper 1 Higher Level Time: 2 hours, 30 minutes 300 marks Examination number Running

### Question 2 Naïve Bayes (16 points)

Question 2 Naïve Bayes (16 points) About 2/3 of your email is spam so you downloaded an open source spam filter based on word occurrences that uses the Naive Bayes classifier. Assume you collected the

### Gibbs Sampling and Online Learning Introduction

Statistical Techniques in Robotics (16-831, F14) Lecture#10(Tuesday, September 30) Gibbs Sampling and Online Learning Introduction Lecturer: Drew Bagnell Scribes: {Shichao Yang} 1 1 Sampling Samples are

### Bayesian Networks. Read R&N Ch. 14.1-14.2. Next lecture: Read R&N 18.1-18.4

Bayesian Networks Read R&N Ch. 14.1-14.2 Next lecture: Read R&N 18.1-18.4 You will be expected to know Basic concepts and vocabulary of Bayesian networks. Nodes represent random variables. Directed arcs

### EC 6310: Advanced Econometric Theory

EC 6310: Advanced Econometric Theory July 2008 Slides for Lecture on Bayesian Computation in the Nonlinear Regression Model Gary Koop, University of Strathclyde 1 Summary Readings: Chapter 5 of textbook.

### ECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2015

ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2015 These notes have been used before. If you can still spot any errors or have any suggestions for improvement, please let me know. 1

### 22c:145 Artificial Intelligence

22c:145 Artificial Intelligence Fall 2005 Uncertainty Cesare Tinelli The University of Iowa Copyright 2001-05 Cesare Tinelli and Hantao Zhang. a a These notes are copyrighted material and may not be used

### Final Exam, Spring 2007

10-701 Final Exam, Spring 2007 1. Personal info: Name: Andrew account: E-mail address: 2. There should be 16 numbered pages in this exam (including this cover sheet). 3. You can use any material you brought:

### Bayesian Methods. 1 The Joint Posterior Distribution

Bayesian Methods Every variable in a linear model is a random variable derived from a distribution function. A fixed factor becomes a random variable with possibly a uniform distribution going from a lower

### The Monty Hall Problem

Gateway to Exploring Mathematical Sciences (GEMS) 9 November, 2013 Lecture Notes The Monty Hall Problem Mark Huber Fletcher Jones Foundation Associate Professor of Mathematics and Statistics and George

### Sample Size Designs to Assess Controls

Sample Size Designs to Assess Controls B. Ricky Rambharat, PhD, PStat Lead Statistician Office of the Comptroller of the Currency U.S. Department of the Treasury Washington, DC FCSM Research Conference

### Common Core State Standards for Mathematics Accelerated 7th Grade

A Correlation of 2013 To the to the Introduction This document demonstrates how Mathematics Accelerated Grade 7, 2013, meets the. Correlation references are to the pages within the Student Edition. Meeting

### 4.4 Conditional Probability

4.4 Conditional Probability It is often necessary to know the probability of an event under restricted conditions. Recall the results of a survey of 100 Grade 12 mathematics students in a local high school.

### Chris Piech Handout #42 CS109 May 27, 2016 Practice Final Examination Final exam: Monday, June 6th, 3:30pm-6:30pm Location: CEMEX Auditorium

Chris Piech Handout #42 CS109 May 27, 2016 Practice Final Examination Final exam: Monday, June 6th, 3:30pm-6:30pm Location: CEMEX Auditorium This is an open note, closed calculator/computer exam. The last

### Government of Russian Federation. Faculty of Computer Science School of Data Analysis and Artificial Intelligence

Government of Russian Federation Federal State Autonomous Educational Institution of High Professional Education National Research University «Higher School of Economics» Faculty of Computer Science School