# Inference in Bayesian networks

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Inference in Bayesian networks hapter hapter

2 Outline Exact inference by enumeration Exact inference by variable elimination Approximate inference by stochastic simulation Approximate inference by Markov chain Monte arlo hapter

3 Inference tasks Simple queries: compute posterior marginal P(X i E = e) e.g., P (NoGas Gauge = empty, Lights = on, Starts = false) onjunctive queries: P(X i, X j E = e) = P(X i E = e)p(x j X i, E = e) Optimal decisions: decision networks include utility information; probabilistic inference required for P (outcome action, evidence) Value of information: which evidence to seek next? Sensitivity analysis: which probability values are most critical? Explanation: why do I need a new starter motor? hapter

4 Inference by enumeration Slightly intelligent way to sum out variables from the joint without actually constructing its explicit representation Simple query on the burglary network: P(B j, m) = P(B, j, m)/p (j, m) = αp(b, j, m) = α Σ e Σ a P(B, e, a, j, m) B J A E M Rewrite full joint entries using product of P entries: P(B j, m) = α Σ e Σ a P(B)P (e)p(a B, e)p (j a)p (m a) = αp(b) Σ e P (e) Σ a P(a B, e)p (j a)p (m a) Recursive depth-first enumeration: O(n) space, O(d n ) time hapter

5 Enumeration algorithm function Enumeration-Ask(X, e, bn) returns a distribution over X inputs: X, the query variable e, observed values for variables E bn, a Bayesian network with variables {X} E Y Q(X ) a distribution over X, initially empty for each value x i of X do extend e with value x i for X Q(x i ) Enumerate-All(Vars[bn], e) return Normalize(Q(X )) function Enumerate-All(vars, e) returns a real number if Empty?(vars) then return 1.0 Y irst(vars) if Y has value y in e then return P (y P a(y )) Enumerate-All(Rest(vars), e) else return y P (y P a(y )) Enumerate-All(Rest(vars), e y ) where e y is e extended with Y = y hapter

6 Evaluation tree P(b).001 P(e).002 P( e).998 P(a b,e).95 P( a b,e).05 P(a b, e).94 P( a b, e).06 P(j a) P(j a).05 P(j a) P(j a).05 P(m a) P(m a) P(m a) P(m a) Enumeration is inefficient: repeated computation e.g., computes P (j a)p (m a) for each value of e hapter

7 Inference by variable elimination Variable elimination: carry out summations right-to-left, storing intermediate results (factors) to avoid recomputation P(B j, m) = α P(B) } {{ } B Σ e P (e) }{{} E Σ a P(a B, e) }{{} A P (j a) } {{ } J = αp(b)σ e P (e)σ a P(a B, e)p (j a)f M (a) = αp(b)σ e P (e)σ a P(a B, e)f J (a)f M (a) = αp(b)σ e P (e)σ a f A (a, b, e)f J (a)f M (a) = αp(b)σ e P (e)f ĀJM (b, e) (sum out A) = αp(b)f ĒĀJM (b) (sum out E) = αf B (b) f ĒĀJM (b) P (m a) } {{ } M hapter

8 Variable elimination: Basic operations Summing out a variable from a product of factors: move any constant factors outside the summation add up submatrices in pointwise product of remaining factors Σ x f 1 f k = f 1 f i Σ x f i+1 f k = f 1 f i f X assuming f 1,..., f i do not depend on X Pointwise product of factors f 1 and f 2 : f 1 (x 1,..., x j, y 1,..., y k ) f 2 (y 1,..., y k, z 1,..., z l ) = f(x 1,..., x j, y 1,..., y k, z 1,..., z l ) E.g., f 1 (a, b) f 2 (b, c) = f(a, b, c) hapter

9 Variable elimination algorithm function Elimination-Ask(X, e, bn) returns a distribution over X inputs: X, the query variable e, evidence specified as an event bn, a belief network specifying joint distribution P(X 1,..., X n ) factors [ ]; vars Reverse(Vars[bn]) for each var in vars do factors [Make-actor(var, e) factors] if var is a hidden variable then factors Sum-Out(var, factors) return Normalize(Pointwise-Product(factors)) hapter

10 Irrelevant variables onsider the query P (Johnalls Burglary = true) P (J b) = αp (b) P (e) P (a b, e)p (J a) P (m a) e a m Sum over m is identically 1; M is irrelevant to the query B J A E M hm 1: Y is irrelevant unless Y Ancestors({X} E) Here, X = Johnalls, E = {Burglary}, and Ancestors({X} E) = {Alarm, Earthquake} so M aryalls is irrelevant (ompare this to backward chaining from the query in Horn clause KBs) hapter

11 Irrelevant variables contd. Defn: moral graph of Bayes net: marry all parents and drop arrows Defn: A is m-separated from B by iff separated by in the moral graph hm 2: Y is irrelevant if m-separated from X by E B E or P (Johnalls Alarm = true), both Burglary and Earthquake are irrelevant J A M hapter

12 L L L L omplexity of exact inference Singly connected networks (or polytrees): any two nodes are connected by at most one (undirected) path time and space cost of variable elimination are O(d k n) Multiply connected networks: can reduce 3SA to exact inference NP-hard equivalent to counting 3SA models #P-complete A B D 1. A v B v 2. v D v A B v v D AND hapter

13 Inference by stochastic simulation Basic idea: 1) Draw N samples from a sampling distribution S 2) ompute an approximate posterior probability ˆP 3) Show this converges to the true probability P 0.5 oin Outline: Sampling from an empty network Rejection sampling: reject samples disagreeing with evidence Likelihood weighting: use evidence to weight samples Markov chain Monte arlo (MM): sample from a stochastic process whose stationary distribution is the true posterior hapter

14 Sampling from an empty network function Prior-Sample(bn) returns an event sampled from bn inputs: bn, a belief network specifying joint distribution P(X 1,..., X n ) x an event with n elements for i = 1 to n do x i a random sample from P(X i parents(x i )) given the values of P arents(x i ) in x return x hapter

15 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

16 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

17 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

18 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

19 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

20 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

21 Example P() loudy P(S ).10 P(R ) S R P(W S,R) hapter

22 Sampling from an empty network contd. Probability that PriorSample generates a particular event S P S (x 1... x n ) = Π n i = 1P (x i parents(x i )) = P (x 1... x n ) i.e., the true prior probability E.g., S P S (t, f, t, t) = = = P (t, f, t, t) Let N P S (x 1... x n ) be the number of samples generated for event x 1,..., x n hen we have lim N ˆP (x 1,..., x n ) = lim N P S(x 1,..., x n )/N N = S P S (x 1,..., x n ) = P (x 1... x n ) hat is, estimates derived from PriorSample are consistent Shorthand: ˆP (x1,..., x n ) P (x 1... x n ) hapter

23 Rejection sampling ˆP(X e) estimated from samples agreeing with e function Rejection-Sampling(X, e, bn, N) returns an estimate of P (X e) local variables: N, a vector of counts over X, initially zero for j = 1 to N do x Prior-Sample(bn) if x is consistent with e then N[x] N[x]+1 where x is the value of X in x return Normalize(N[X]) E.g., estimate P( = true) using 100 samples 27 samples have = true Of these, 8 have = true and 19 have = false. ˆP( = true) = Normalize( 8, 19 ) = 0.296, Similar to a basic real-world empirical estimation procedure hapter

24 Analysis of rejection sampling ˆP(X e) = αn P S (X, e) (algorithm defn.) = N P S (X, e)/n P S (e) (normalized by N P S (e)) P(X, e)/p (e) (property of PriorSample) = P(X e) (defn. of conditional probability) Hence rejection sampling returns consistent posterior estimates Problem: hopelessly expensive if P (e) is small P (e) drops off exponentially with number of evidence variables! hapter

25 Likelihood weighting Idea: fix evidence variables, sample only nonevidence variables, and weight each sample by the likelihood it accords the evidence function Likelihood-Weighting(X, e, bn, N) returns an estimate of P (X e) local variables: W, a vector of weighted counts over X, initially zero for j = 1 to N do x, w Weighted-Sample(bn) W[x] W[x] + w where x is the value of X in x return Normalize(W[X ]) function Weighted-Sample(bn, e) returns an event and a weight x an event with n elements; w 1 for i = 1 to n do if X i has a value x i in e then w w P (X i = x i parents(x i )) else x i a random sample from P(X i parents(x i )) return x, w hapter

26 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = 1.0 hapter

27 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = 1.0 hapter

28 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = 1.0 hapter

29 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = hapter

30 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = hapter

31 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = hapter

32 Likelihood weighting example P() loudy P(S ).10 P(R ) S R P(W S,R) w = = hapter

33 Likelihood weighting analysis Sampling probability for WeightedSample is S W S (z, e) = Π l i = 1P (z i parents(z i )) Note: pays attention to evidence in ancestors only somewhere in between prior and posterior distribution loudy Weight for a given sample z, e is w(z, e) = Π m i = 1P (e i parents(e i )) Weighted sampling probability is S W S (z, e)w(z, e) = Π l i = 1P (z i parents(z i )) Π m i = 1P (e i parents(e i )) = P (z, e) (by standard global semantics of network) Hence likelihood weighting returns consistent estimates but performance still degrades with many evidence variables because a few samples have nearly all the total weight hapter

34 Approximate inference using MM State of network = current assignment to all variables. Generate next state by sampling one variable given Markov blanket Sample each variable in turn, keeping evidence fixed function MM-Ask(X, e, bn, N) returns an estimate of P (X e) local variables: N[X ], a vector of counts over X, initially zero Z, the nonevidence variables in bn x, the current state of the network, initially copied from e initialize x with random values for the variables in Y for j = 1 to N do for each Z i in Z do sample the value of Z i in x from P(Z i mb(z i )) given the values of MB(Z i ) in x N[x] N[x] + 1 where x is the value of X in x return Normalize(N[X ]) an also choose a variable to sample at random each time hapter

35 he Markov chain With = true, W et = true, there are four states: loudy loudy loudy loudy Wander about for a while, average what you see hapter

36 MM example contd. Estimate P( = true, W et = true) Sample loudy or given its Markov blanket, repeat. ount number of times is true and false in the samples. E.g., visit 100 states 31 have = true, 69 have = false ˆP( = true, W et = true) = Normalize( 31, 69 ) = 0.31, 0.69 heorem: chain approaches stationary distribution: long-run fraction of time spent in each state is exactly proportional to its posterior probability hapter

37 Markov blanket sampling Markov blanket of loudy is and Markov blanket of is loudy,, and W et loudy Probability given the Markov blanket is calculated as follows: P (x i mb(x i )) = P (x i parents(x i ))Π Zj hildren(x i )P (z j parents(z j )) Easily implemented in message-passing parallel systems, brains Main computational problems: 1) Difficult to tell if convergence has been achieved 2) an be wasteful if Markov blanket is large: P (X i mb(x i )) won t change much (law of large numbers) hapter

38 Summary Exact inference by variable elimination: polytime on polytrees, NP-hard on general graphs space = time, very sensitive to topology Approximate inference by LW, MM: LW does poorly when there is lots of (downstream) evidence LW, MM generally insensitive to topology onvergence can be very slow with probabilities close to 1 or 0 an handle arbitrary combinations of discrete and continuous variables hapter

### 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

### Bayesian Networks. Mausam (Slides by UW-AI faculty)

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

### 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

### 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

### 13.3 Inference Using Full Joint Distribution

191 The probability distribution on a single variable must sum to 1 It is also true that any joint probability distribution on any set of variables must sum to 1 Recall that any proposition a is equivalent

### 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

### 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

### 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

### 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

### 3. The Junction Tree Algorithms

A Short Course on Graphical Models 3. The Junction Tree Algorithms Mark Paskin mark@paskin.org 1 Review: conditional independence Two random variables X and Y are independent (written X Y ) iff p X ( )

### 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

### 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.

### Data Modeling & Analysis Techniques. Probability & Statistics. Manfred Huber 2011 1

Data Modeling & Analysis Techniques Probability & Statistics Manfred Huber 2011 1 Probability and Statistics Probability and statistics are often used interchangeably but are different, related fields

### 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

### Querying Joint Probability Distributions

Querying Joint Probability Distributions Sargur Srihari srihari@cedar.buffalo.edu 1 Queries of Interest Probabilistic Graphical Models (BNs and MNs) represent joint probability distributions over multiple

### 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

### Incorporating Evidence in Bayesian networks with the Select Operator

Incorporating Evidence in Bayesian networks with the Select Operator C.J. Butz and F. Fang Department of Computer Science, University of Regina Regina, Saskatchewan, Canada SAS 0A2 {butz, fang11fa}@cs.uregina.ca

### Big Data, Machine Learning, Causal Models

Big Data, Machine Learning, Causal Models Sargur N. Srihari University at Buffalo, The State University of New York USA Int. Conf. on Signal and Image Processing, Bangalore January 2014 1 Plan of Discussion

### 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

### Probabilistic Graphical Models

Probabilistic Graphical Models Raquel Urtasun and Tamir Hazan TTI Chicago April 4, 2011 Raquel Urtasun and Tamir Hazan (TTI-C) Graphical Models April 4, 2011 1 / 22 Bayesian Networks and independences

### Bayesian networks - Time-series models - Apache Spark & Scala

Bayesian networks - Time-series models - Apache Spark & Scala Dr John Sandiford, CTO Bayes Server Data Science London Meetup - November 2014 1 Contents Introduction Bayesian networks Latent variables Anomaly

### Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-Normal Distribution

Lecture 8: Random Walk vs. Brownian Motion, Binomial Model vs. Log-ormal Distribution October 4, 200 Limiting Distribution of the Scaled Random Walk Recall that we defined a scaled simple random walk last

### Life of A Knowledge Base (KB)

Life of A Knowledge Base (KB) A knowledge base system is a special kind of database management system to for knowledge base management. KB extraction: knowledge extraction using statistical models in NLP/ML

### 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

### 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

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

### Tutorial on Markov Chain Monte Carlo

Tutorial on Markov Chain Monte Carlo Kenneth M. Hanson Los Alamos National Laboratory Presented at the 29 th International Workshop on Bayesian Inference and Maximum Entropy Methods in Science and Technology,

### IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS

IEOR 6711: Stochastic Models, I Fall 2012, Professor Whitt, Final Exam SOLUTIONS There are four questions, each with several parts. 1. Customers Coming to an Automatic Teller Machine (ATM) (30 points)

### The rule for computing conditional property can be interpreted different. In Question 2, P B

Question 4: What is the product rule for probability? The rule for computing conditional property can be interpreted different. In Question 2, P A and B we defined the conditional probability PA B. If

### Similarity Search and Mining in Uncertain Spatial and Spatio Temporal Databases. Andreas Züfle

Similarity Search and Mining in Uncertain Spatial and Spatio Temporal Databases Andreas Züfle Geo Spatial Data Huge flood of geo spatial data Modern technology New user mentality Great research potential

### Tutorial on variational approximation methods. Tommi S. Jaakkola MIT AI Lab

Tutorial on variational approximation methods Tommi S. Jaakkola MIT AI Lab tommi@ai.mit.edu Tutorial topics A bit of history Examples of variational methods A brief intro to graphical models Variational

### Logic in general. Inference rules and theorem proving

Logical Agents Knowledge-based agents Logic in general Propositional logic Inference rules and theorem proving First order logic Knowledge-based agents Inference engine Knowledge base Domain-independent

### Pull versus Push Mechanism in Large Distributed Networks: Closed Form Results

Pull versus Push Mechanism in Large Distributed Networks: Closed Form Results Wouter Minnebo, Benny Van Houdt Dept. Mathematics and Computer Science University of Antwerp - iminds Antwerp, Belgium Wouter

### Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations

56 Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations Stochastic Processes and Queueing Theory used in Cloud Computer Performance Simulations Florin-Cătălin ENACHE

### Message-passing sequential detection of multiple change points in networks

Message-passing sequential detection of multiple change points in networks Long Nguyen, Arash Amini Ram Rajagopal University of Michigan Stanford University ISIT, Boston, July 2012 Nguyen/Amini/Rajagopal

### 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

### p if x = 1 1 p if x = 0

Probability distributions Bernoulli distribution Two possible values (outcomes): 1 (success), 0 (failure). Parameters: p probability of success. Probability mass function: P (x; p) = { p if x = 1 1 p if

### Review the following from Chapter 5

Bluman, Chapter 6 1 Review the following from Chapter 5 A surgical procedure has an 85% chance of success and a doctor performs the procedure on 10 patients, find the following: a) The probability that

### Section 6.1 Joint Distribution Functions

Section 6.1 Joint Distribution Functions We often care about more than one random variable at a time. DEFINITION: For any two random variables X and Y the joint cumulative probability distribution function

### Inference on Phase-type Models via MCMC

Inference on Phase-type Models via MCMC with application to networks of repairable redundant systems Louis JM Aslett and Simon P Wilson Trinity College Dublin 28 th June 202 Toy Example : Redundant Repairable

### Dirichlet Processes A gentle tutorial

Dirichlet Processes A gentle tutorial SELECT Lab Meeting October 14, 2008 Khalid El-Arini Motivation We are given a data set, and are told that it was generated from a mixture of Gaussian distributions.

### Confindence Intervals and Probability Testing

Confindence Intervals and Probability Testing PO7001: Quantitative Methods I Kenneth Benoit 3 November 2010 Using probability distributions to assess sample likelihoods Recall that using the µ and σ from

### Relational Dynamic Bayesian Networks: a report. Cristina Manfredotti

Relational Dynamic Bayesian Networks: a report Cristina Manfredotti Dipartimento di Informatica, Sistemistica e Comunicazione (D.I.S.Co.) Università degli Studi Milano-Bicocca manfredotti@disco.unimib.it

### LECTURE 4. Last time: Lecture outline

LECTURE 4 Last time: Types of convergence Weak Law of Large Numbers Strong Law of Large Numbers Asymptotic Equipartition Property Lecture outline Stochastic processes Markov chains Entropy rate Random

### Intelligent Systems: Reasoning and Recognition. Uncertainty and Plausible Reasoning

Intelligent Systems: Reasoning and Recognition James L. Crowley ENSIMAG 2 / MoSIG M1 Second Semester 2015/2016 Lesson 12 25 march 2015 Uncertainty and Plausible Reasoning MYCIN (continued)...2 Backward

### Introduction to Mobile Robotics Bayes Filter Particle Filter and Monte Carlo Localization

Introduction to Mobile Robotics Bayes Filter Particle Filter and Monte Carlo Localization Wolfram Burgard, Maren Bennewitz, Diego Tipaldi, Luciano Spinello 1 Motivation Recall: Discrete filter Discretize

### Sampling via Moment Sharing: A New Framework for Distributed Bayesian Inference for Big Data

Sampling via Moment Sharing: A New Framework for Distributed Bayesian Inference for Big Data (Oxford) in collaboration with: Minjie Xu, Jun Zhu, Bo Zhang (Tsinghua) Balaji Lakshminarayanan (Gatsby) Bayesian

### CNFSAT: Predictive Models, Dimensional Reduction, and Phase Transition

CNFSAT: Predictive Models, Dimensional Reduction, and Phase Transition Neil P. Slagle College of Computing Georgia Institute of Technology Atlanta, GA npslagle@gatech.edu Abstract CNFSAT embodies the P

### m (t) = e nt m Y ( t) = e nt (pe t + q) n = (pe t e t + qe t ) n = (qe t + p) n

1. For a discrete random variable Y, prove that E[aY + b] = ae[y] + b and V(aY + b) = a 2 V(Y). Solution: E[aY + b] = E[aY] + E[b] = ae[y] + b where each step follows from a theorem on expected value from

### CONTINGENCY (CROSS- TABULATION) TABLES

CONTINGENCY (CROSS- TABULATION) TABLES Presents counts of two or more variables A 1 A 2 Total B 1 a b a+b B 2 c d c+d Total a+c b+d n = a+b+c+d 1 Joint, Marginal, and Conditional Probability We study methods

### 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

### 2.1 Complexity Classes

15-859(M): Randomized Algorithms Lecturer: Shuchi Chawla Topic: Complexity classes, Identity checking Date: September 15, 2004 Scribe: Andrew Gilpin 2.1 Complexity Classes In this lecture we will look

### What Is Probability?

1 What Is Probability? The idea: Uncertainty can often be "quantified" i.e., we can talk about degrees of certainty or uncertainty. This is the idea of probability: a higher probability expresses a higher

### Guessing Game: NP-Complete?

Guessing Game: NP-Complete? 1. LONGEST-PATH: Given a graph G = (V, E), does there exists a simple path of length at least k edges? YES 2. SHORTEST-PATH: Given a graph G = (V, E), does there exists a simple

### ANALYTICS IN BIG DATA ERA

ANALYTICS IN BIG DATA ERA ANALYTICS TECHNOLOGY AND ARCHITECTURE TO MANAGE VELOCITY AND VARIETY, DISCOVER RELATIONSHIPS AND CLASSIFY HUGE AMOUNT OF DATA MAURIZIO SALUSTI SAS Copyr i g ht 2012, SAS Ins titut

### Linear Threshold Units

Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear

### Stochastic Processes and Advanced Mathematical Finance. Laws of Large Numbers

Steven R. Dunbar Department of Mathematics 203 Avery Hall University of Nebraska-Lincoln Lincoln, NE 68588-0130 http://www.math.unl.edu Voice: 402-472-3731 Fax: 402-472-8466 Stochastic Processes and Advanced

### Towards running complex models on big data

Towards running complex models on big data Working with all the genomes in the world without changing the model (too much) Daniel Lawson Heilbronn Institute, University of Bristol 2013 1 / 17 Motivation

### PMR5406 Redes Neurais e Lógica Fuzzy Aula 3 Multilayer Percetrons

PMR5406 Redes Neurais e Aula 3 Multilayer Percetrons Baseado em: Neural Networks, Simon Haykin, Prentice-Hall, 2 nd edition Slides do curso por Elena Marchiori, Vrie Unviersity Multilayer Perceptrons Architecture

### Neural networks. Chapter 20, Section 5 1

Neural networks Chapter 20, Section 5 Chapter 20, Section 5 Outline Brains Neural networks Perceptrons Multilayer perceptrons Applications of neural networks Chapter 20, Section 5 2 Brains 0 neurons of

### 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

### M/M/1 and M/M/m Queueing Systems

M/M/ and M/M/m Queueing Systems M. Veeraraghavan; March 20, 2004. Preliminaries. Kendall s notation: G/G/n/k queue G: General - can be any distribution. First letter: Arrival process; M: memoryless - exponential

### Math 2015 Lesson 21. We discuss the mean and the median, two important statistics about a distribution. p(x)dx = 0.5

ean and edian We discuss the mean and the median, two important statistics about a distribution. The edian The median is the halfway point of a distribution. It is the point where half the population has

### Pooling and Meta-analysis. Tony O Hagan

Pooling and Meta-analysis Tony O Hagan Pooling Synthesising prior information from several experts 2 Multiple experts The case of multiple experts is important When elicitation is used to provide expert

### Chapter 14: 1-6, 9, 12; Chapter 15: 8 Solutions When is it appropriate to use the normal approximation to the binomial distribution?

Chapter 14: 1-6, 9, 1; Chapter 15: 8 Solutions 14-1 When is it appropriate to use the normal approximation to the binomial distribution? The usual recommendation is that the approximation is good if np

### Poisson and Normal Distributions

Poisson and Normal Distributions Lectures 7 Spring 2002 Poisson Distribution The Poisson distribution can be derived as a limiting form of the binomial distribution in which n is increased without limit

### CMPSCI611: Approximating MAX-CUT Lecture 20

CMPSCI611: Approximating MAX-CUT Lecture 20 For the next two lectures we ll be seeing examples of approximation algorithms for interesting NP-hard problems. Today we consider MAX-CUT, which we proved to

### 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

### The Joint Probability Distribution (JPD) of a set of n binary variables involve a huge number of parameters

DEFINING PROILISTI MODELS The Joint Probability Distribution (JPD) of a set of n binary variables involve a huge number of parameters 2 n (larger than 10 25 for only 100 variables). x y z p(x, y, z) 0

### Probability and statistics; Rehearsal for pattern recognition

Probability and statistics; Rehearsal for pattern recognition Václav Hlaváč Czech Technical University in Prague Faculty of Electrical Engineering, Department of Cybernetics Center for Machine Perception

### Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab

Monte Carlo Simulation: IEOR E4703 Fall 2004 c 2004 by Martin Haugh Overview of Monte Carlo Simulation, Probability Review and Introduction to Matlab 1 Overview of Monte Carlo Simulation 1.1 Why use simulation?

### Monte Carlo methods in PageRank computation: When one iteration is sufficient

Monte Carlo methods in PageRank computation: When one iteration is sufficient K.Avrachenkov, N. Litvak, D. Nemirovsky, N. Osipova Abstract PageRank is one of the principle criteria according to which Google

### Normal approximation to the Binomial

Chapter 5 Normal approximation to the Binomial 5.1 History In 1733, Abraham de Moivre presented an approximation to the Binomial distribution. He later (de Moivre, 1756, page 242 appended the derivation

### 8. Machine Learning Applied Artificial Intelligence

8. Machine Learning Applied Artificial Intelligence Prof. Dr. Bernhard Humm Faculty of Computer Science Hochschule Darmstadt University of Applied Sciences 1 Retrospective Natural Language Processing Name

### 5 Directed acyclic graphs

5 Directed acyclic graphs (5.1) Introduction In many statistical studies we have prior knowledge about a temporal or causal ordering of the variables. In this chapter we will use directed graphs to incorporate

### 1 Formulating The Low Degree Testing Problem

6.895 PCP and Hardness of Approximation MIT, Fall 2010 Lecture 5: Linearity Testing Lecturer: Dana Moshkovitz Scribe: Gregory Minton and Dana Moshkovitz In the last lecture, we proved a weak PCP Theorem,

### 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

### 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

### 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

### Continuous Random Variables. and Probability Distributions. Continuous Random Variables and Probability Distributions ( ) ( ) Chapter 4 4.

UCLA STAT 11 A Applied Probability & Statistics for Engineers Instructor: Ivo Dinov, Asst. Prof. In Statistics and Neurology Teaching Assistant: Neda Farzinnia, UCLA Statistics University of California,

### Hidden Markov Models

8.47 Introduction to omputational Molecular Biology Lecture 7: November 4, 2004 Scribe: Han-Pang hiu Lecturer: Ross Lippert Editor: Russ ox Hidden Markov Models The G island phenomenon The nucleotide frequencies

### ECE302 Spring 2006 HW3 Solutions February 2, 2006 1

ECE302 Spring 2006 HW3 Solutions February 2, 2006 1 Solutions to HW3 Note: Most of these solutions were generated by R. D. Yates and D. J. Goodman, the authors of our textbook. I have added comments in

### Structured Learning and Prediction in Computer Vision. Contents

Foundations and Trends R in Computer Graphics and Vision Vol. 6, Nos. 3 4 (2010) 185 365 c 2011 S. Nowozin and C. H. Lampert DOI: 10.1561/0600000033 Structured Learning and Prediction in Computer Vision

### 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

### An Empirical Evaluation of Bayesian Networks Derived from Fault Trees

An Empirical Evaluation of Bayesian Networks Derived from Fault Trees Shane Strasser and John Sheppard Department of Computer Science Montana State University Bozeman, MT 59717 {shane.strasser, john.sheppard}@cs.montana.edu

### Probabilistic Networks An Introduction to Bayesian Networks and Influence Diagrams

Probabilistic Networks An Introduction to Bayesian Networks and Influence Diagrams Uffe B. Kjærulff Department of Computer Science Aalborg University Anders L. Madsen HUGIN Expert A/S 10 May 2005 2 Contents

### CHAPTER 2 Estimating Probabilities

CHAPTER 2 Estimating Probabilities Machine Learning Copyright c 2016. Tom M. Mitchell. All rights reserved. *DRAFT OF January 24, 2016* *PLEASE DO NOT DISTRIBUTE WITHOUT AUTHOR S PERMISSION* This is a

### Real Time Traffic Monitoring With Bayesian Belief Networks

Real Time Traffic Monitoring With Bayesian Belief Networks Sicco Pier van Gosliga TNO Defence, Security and Safety, P.O.Box 96864, 2509 JG The Hague, The Netherlands +31 70 374 02 30, sicco_pier.vangosliga@tno.nl

### 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

### Gambling and Data Compression

Gambling and Data Compression Gambling. Horse Race Definition The wealth relative S(X) = b(x)o(x) is the factor by which the gambler s wealth grows if horse X wins the race, where b(x) is the fraction

### 1 Review of Newton Polynomials

cs: introduction to numerical analysis 0/0/0 Lecture 8: Polynomial Interpolation: Using Newton Polynomials and Error Analysis Instructor: Professor Amos Ron Scribes: Giordano Fusco, Mark Cowlishaw, Nathanael

### 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

### Statistical Machine Translation: IBM Models 1 and 2

Statistical Machine Translation: IBM Models 1 and 2 Michael Collins 1 Introduction The next few lectures of the course will be focused on machine translation, and in particular on statistical machine translation

### 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

### 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

### Performance Analysis of a Telephone System with both Patient and Impatient Customers

Performance Analysis of a Telephone System with both Patient and Impatient Customers Yiqiang Quennel Zhao Department of Mathematics and Statistics University of Winnipeg Winnipeg, Manitoba Canada R3B 2E9

### 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

### MATH 110 Spring 2015 Homework 6 Solutions

MATH 110 Spring 2015 Homework 6 Solutions Section 2.6 2.6.4 Let α denote the standard basis for V = R 3. Let α = {e 1, e 2, e 3 } denote the dual basis of α for V. We would first like to show that β =