Bayesian Networks

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Used in Spring 2012, Spring 2013, Winter 2014
(partially)

1. Bayesian Networks
2. Conditional Independence
3. Creating Tables
4. Notations for Bayesian Networks
5. Calculating conditional probabilities from the
   tables
6. Calculating conditional independence
7. Markov Chain Monte Carlo
8. Markov Models.
9. Markov Models and Probabilistic methods in vision

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Introduction to Probabilistic Robotics

1. Probabilities
2. Bayes rule
3. Bayes filters
4. Bayes networks
5. Markov Chains

new
next
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Bayesian Networks and Markov Models

• Bayesian networks and Markov models
• Applications in User Modeling
• Applications in Natural Language Processing
• Applications in robotic control
• Applications in robot Vision

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Bayesian Networks (BNs) Overview

• Introduction to BNs
• Nodes, structure and probabilities
• Reasoning with BNs
• Understanding BNs
• Extensions of BNs
• Decision Networks
• Dynamic Bayesian Networks (DBNs)

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Definition of Bayesian Networks

• A data structure that represents the dependence
  between variables
• Gives a concise specification of the joint
  probability distribution
• A Bayesian Network is a directed acyclic graph
  (DAG) in which the following holds
• A set of random variables makes up the nodes in
  the network
• A set of directed links connects pairs of nodes
• Each node has a probability distribution that
  quantifies the effects of its parents

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Conditional Independence

• The relationship between
• conditional independence
• and BN structure
• is important for understanding how BNs work

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Conditional Independence Causal Chains

• Causal chains give rise to conditional
  independence
• Example Smoking causes cancer, which causes
  dyspnoea

smoking
cancer
dyspnoea
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Conditional Independence Common Causes

• Common Causes (or ancestors) also give rise to
  conditional independenceExample Cancer
  is a common cause of the two symptoms a positive
  Xray and dyspnoea

cancer
B
dyspnoea
Xray
A
C
(
)
? (A indep C) B
I have dyspnoea (C ) because of cancer (B) so I
do not need an Xray test
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Conditional Dependence Common Effects

• Common effects (or their descendants) give rise
  to conditional dependence
• Example Cancer is a common effect of pollution
  and smokingGiven cancer, smoking explains
  away pollution

pollution
A
C
???
smoking
B
cancer
pollution
cancer
(
)
?
We know that you smoke and have cancer, we do not
need to assume that your cancer was caused by
pollution
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Joint Distributions for describing uncertain
worlds

• Researchers found already numerous and dramatic
  benefits of Joint Distributions for describing
  uncertain worlds
• Students in robotics and Artificial
  Intelligence have to understand problems with
  using Joint Distributions
• You should discover how Bayes Net methodology
  allows us to build Joint Distributions in
  manageable chunks

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Bayes Net methodology
Why Bayesian methods matter?

1. Bayesian Methods are one of the most important
   conceptual advances in the Machine Learning / AI
   field to have emerged since 1995.
2. A clean, clear, manageable language and
   methodology for expressing what the robot
   designer is certain and uncertain about
3. Already, many practical applications in
   medicine, factories, helpdesks for instance
4. P(this problem these symptoms) // we will use
   P as probability
5. anomalousness of this observation
6. choosing next diagnostic test these
   observations

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Problem 1 Creating Joint Distribution Table

• Joint Distribution Table is an important concept

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Probabilistic truth table

• You can guess this table, you can take data from
  some statistics,
• You can build this table based on some partial
  tables

Truth table of all combinations of Boolean
Variables
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• Idea use decision diagrams to represent these
  data.

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• Use of independence while creating the tables

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• Wet Sprinkler Rain Example

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W
S
R
Wet-Sprinkler-Rain Example
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• Problem 1
• Creating the Joint Table

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Our Goal is to derive this table
Let us observe that if I know 7 of these, the
eight is obviously unique , as their sum 1
So I need to guess or calculate or find 2n-1 7
values
But the same data can be stored explicitely or
implicitely, not necessarily in the form of a
table!!
What extra assumptions can help to create this
table?
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Wet-Sprinkler-Rain Example
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Sprinkler on under condition that it rained
You need to understand causation when you create
the table
Wet-Sprinkler-Rain Example
Understanding of causation
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Independence simplifies probabilities
We use independence of variables S and R
P(SR) Sprinkler on under condition that it
rained
We can use these probabilities to create the table
S and R are independent
Wet-Sprinkler-Rain Example
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Wet-Sprinkler-Rain Example
We create the CPT for S and R based on our
knowledge of the problem
Conditional Probability Table (CPT)
It rained
Sprinkler was on
Grass is wet
What about children playing or a dog pissing? It
is still possible by this value 0.1
This first step shows the collected data
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Full joint for only S and R
Independence of S and R is used
0.95
0.90
0.90
0.01
Wet-Sprinkler-Rain Example
Use chain rule for probabilities
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Chain Rule for Probabilities
Random variables
0.95
0.90
0.90
0.01
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Full joint probability

• You have a table
• You want to calculate some probability

P(W)
Wet-Sprinkler-Rain Example
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Independence of S and R implies calculating fewer
numbers to create the complete Joint Table for W,
S and R
Six numbers
We reduced only from seven to six numbers
Wet-Sprinkler-Rain Example
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• Explanation of Diagrammatic Notations
• such as Bayes Networks

You do not need to build the complete table!!
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You can build a graph of tables or nodes which
correspond to certain types of tables
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Wet-Sprinkler-Rain Example
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Wet-Sprinkler-Rain Example
It rained
Sprinkler was on
Grass is wet
This first step shows the collected data
Conditional Probability Table (CPT)
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Full joint probability

• You have a table
• You want to calculate some probability

When you have this table you can modify it, you
can also calculate everything!!
P(W)
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• Problem 2
• Calculating conditional probabilities from the
  Joint Distribution Table

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Wet-Sprinkler-Rain Example
Probability that ST and WT
Probability that grass is wet under assumption
that sprinkler was on
Probability that ST
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Wet-Sprinkler-Rain Example
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We showed examples of both causal inference and
diagnostic inference
We will use this in next slide
Wet-Sprinkler-Rain Example
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Explaining Away the facts from the table
Calculated earlier from this table
lt
Wet-Sprinkler-Rain Example
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Conclusions on this problem

1. Table can be used for Explaining Away
2. Table can be used to calculate conditional
   independence.
3. Table can be used to calculate conditional
   probabilities
4. Table can be used to determine causality

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• Problem 3 What if S and R are dependent?
• Calculating
• conditional independence

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Conditional Independence of S and R
Wet-Sprinkler-Rain Example
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Diagrammatic notation for conditional
Independence of two variables
Wet-Sprinkler-Rain Example extended
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Conditional Independence formalized for sets of
variables
S3
S1
S2
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Now we will explain conditional independence
CLOUDY - Wet-Sprinkler-Rain Example
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Example Lung Cancer Diagnosis
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Example Lung Cancer Diagnosis

1. A patient has been suffering from shortness of
   breath (called dyspnoea) and visits the doctor,
   worried that he has lung cancer.
2. The doctor knows that other diseases, such as
   tuberculosis and bronchitis are possible causes,
   as well as lung cancer.
3. She also knows that other relevant information
   includes whether or not the patient is a smoker
   (increasing the chances of cancer and bronchitis)
   and what sort of air pollution he has been
   exposed to.
4. A positive Xray would indicate either TB or lung
   cancer.

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Nodes and Values in Bayesian Networks

• Q What are the nodes to represent and what
  values can they take?
• A Nodes can be discrete or continuous
• Boolean nodes represent propositions taking
  binary valuesExample Cancer node represents
  proposition the patient has cancer
• Ordered valuesExample Pollution node with
  values low, medium, high
• Integral valuesExample Age with possible values
  1-120

Lung Cancer
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Lung Cancer Example Nodes and Values
Node name Type Values
Pollution Binary low,high
Smoker Boolean T,F
Cancer Boolean T,F
Dyspnoea Boolean T,F
Xray Binary pos,neg
Shortness of breath
Example of variables as nodes in BN
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Lung Cancer Example Bayesian Network Structure
Pollution
Smoker
Cancer
Xray
Dyspnoea
Lung Cancer
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Conditional Probability Tables (CPTs) in
Bayesian Networks
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Conditional Probability Tables (CPTs) in Bayesian
Networks

• After specifying topology, we must specify
• the CPT for each discrete node
• Each row of CPT contains the conditional
  probability of each node value for each possible
  combination of values in its parent nodes
• Each row of CPT must sum to 1
• A CPT for a Boolean variable with n Boolean
  parents contains 2n1 probabilities
• A node with no parents has one row (its prior
  probabilities)

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Lung Cancer Example example of CPT
Smoking is true
Probability of cancer given state of variables P
and S
Pollution low
C cancer P pollution S smoking X Xray D
Dyspnoea
Bayesian Network for cancer
Lung Cancer
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• Several small CPTs are used to create larger JDTs.

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The Markov Property for Bayesian Networks

• Modelling with BNs requires assuming the Markov
  Property
• There are no direct dependencies in the system
  being modelled which are not already explicitly
  shown via arcs
• Example smoking can influence dyspnoea only
  through causing cancer

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Software NETICA for Bayesian Networks and
joint probabilities
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Reasoning with Numbers Using Netica software
Here are the collected data
Lung Cancer
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Representing the Joint Probability Distribution
Example
We want to calculate this
P pollution S smoking X Xray D Dyspnoea
This graph shows how we can calculate the joint
probability from other probabilities in the
network
Lung Cancer
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Problem 4Determining Causality and Bayes Nets
Advertisement example
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Causality and Bayes Nets Advertisement example

• Bayes nets allow one to learn about causal
  relationships
• One more Example
• Marketing analysts want to know whether to
  increase, decrease or leave unchanged the
  exposure of some advertisement in order to
  maximize profit from the sale of some product
• Advertised (A) and Buy (B) will be variables for
  someone having seen the advertisement or
  purchased the product

Advertised-Buy Example
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Causality Example

1. So we want to know the probability that Btrue
   given that we force Atrue, or Afalse
2. We could do this by finding two similar
   populations and observing B based on Atrue for
   one and Afalse for the other
3. But this may be difficult or expensive to find
   such populations

• Advertised (A) seen the advertisement

• Buy (B) will be variables for someone purchased
  the product

Advertised-Buy Example
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How causality can be represented in a graph?
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Markov condition and Causal Markov Condition

• But how do we learn whether or not A causes B at
  all?
• The Markov Condition states
• Any node in a Bayes net is conditionally
  independent of its non-descendants given its
  parents
• The CAUSAL Markov Condition (CMC) states
• Any phenomenon in a causal net is independent of
  its non-effects given its direct causes

Advertised (A) and Buy (B)
Advertised-Buy Example
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Acyclic Causal Graph versus Bayes Net

• Thus, if we have a directed acyclic causal graph
  C for variables in X, then, by the Causal Markov
  Condition, C is also a Bayes net for the joint
  probability distribution of X
• The reverse is not necessarily truea network may
  satisfy the Markov condition without depicting
  causality

Advertised-Buy Example
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Causality Example when we learn that p(ba) and
p(ba) are not equal

• Given the Causal Markov Condition CMC, we can
  infer causal relationships from conditional
  (in)dependence relationships learned from the
  data
• Suppose we learn with high Bayesian probability
  that p(ba) and p(ba) are not equal
• Given the CMC, there are four simple causal
  explanations for this (more complex ones too)

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Causality Example four causal explanations

• A causes B

If they advertise more you buy more
B causes A If you buy more, they have more money
to advertise
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Causality Example four causal explanations
selection bias

1. Hidden common cause of A and B (e.g. income)

• A and B are causes for data selection
• (a.k.a. selection bias,
• perhaps if database didnt record false instances
  of A and B)

In rich country they advertise more and they buy
more
If you increase information about Ad in database,
then you increase also information about Buy in
the database
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Causality Example continued

• But we still dont know if A causes B
• Suppose
• We learn about the Income (I) and geographic
  Location (L) of the purchaser
• And we learn with high Bayesian probability the
  network on the right

Advertised (AAd) and Buy (B)
Advertised-Buy Example
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Causality Example - using CMC

• Given the Causal Markov Condition CMC, the ONLY
  causal explanation for the conditional
  (in)dependence relationships encoded in the Bayes
  net is that Ad is a cause for Buy
• That is, none of the other relationships or
  combinations thereof produce the probabilistic
  relationships encoded here

Advertised (Ad) and Buy (B)
Advertised-Buy Example
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Causality in Bayes Networks

• Thus, Bayes Nets allow inference of causal
  relationships by the Causal Markov Condition (CMC)

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Problem 5Determine D-separation in Bayesian
Networks
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D-separation in Bayesian Networks

• We will formulate a Graphical Criterion of
  conditional independence
• We can determine whether a set of nodes X is
  independent of another set Y, given a set of
  evidence nodes E, via the Markov property
• If every undirected path from a node in X to a
  node in Y is d-separated by E, then X and Y are
  conditionally independent given E

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Determining D-separation (cont)

• A set of nodes E d-separates two sets of nodes X
  and Y, if every undirected path from a node in X
  to a node in Y is blocked given E
• A path is blocked given a set of nodes E, if
  there is a node Z on the path for which one of
  three conditions holds
• Z is in E and Z has one arrow on the path leading
  in and one arrow out (chain)
• Z is in E and Z has both path arrows leading out
  (common cause)
• Neither Z nor any descendant of Z is in E, and
  both path arrows lead into Z (common effect)

Chain Common cause Common effect
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Another Example of Bayesian Networks Alarm
Alarm Example

• Let us draw BN from these data

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Bayes Net Corresponding to Alarm-Burglar problem
Alarm Example
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Compactness, Global Semantics, Local Semantics
and Markov Blanket

• Compactness of Bayes Net

Earthquake
Burglar
John calls
Mary calls
Alarm Example
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Global Semantics, Local Semantics and Markov
Blanket for BNs

• Useful concepts

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Alarm Example
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• Markovs blanket are
• Parents
• Children
• Childrens parent

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Problem 6How to systematically Build a Bayes
Network -- Example
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Alarm Example
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Alarm Example
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Alarm Example
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So we add arrow
Alarm Example
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Alarm Example
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Alarm Example
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Bayes Net for the car that does not want to start
Such networks can be used for robot diagnostics
or diagnostic of a human done by robot
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Inference in Bayes Nets and how to simplify it
Alarm Example
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First method of simplification Enumeration
Alarm Example
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Alarm Example
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Second method Variable Elimination
Alarm Example
Variable A was eliminated
Variable E was eliminated
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Polytrees are better
3SAT Example
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IDEA Convert DAG to polytrees
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Clustering is used to convert non-polytree BNs
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EXAMPLE Clustering is used to convert
non-polytree BNs
Not a polytree
Is a polytree
Alarm Example
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• Approximate Inference
• Direct sampling methods
• Rejection sampling
• Likelihood weighting
• Markov chain Monte Carlo

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• 1. Direct Sampling Methods

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Direct Sampling
Direct Sampling generates minterms with their
probabilities
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We start from top
Wwet C cloudy R rain S sprinkler
Wet Sprinkler Rain Example
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Wwet C cloudy R rain S sprinkler
Cloudy yes
Wet Sprinkler Rain Example
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Wwet C cloudy R rain S sprinkler
Cloudy yes
Wet Sprinkler Rain Example
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Wwet C cloudy R rain S sprinkler
sprinkler no
Wet Sprinkler Rain Example
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Wwet C cloudy R rain S sprinkler
Wet Sprinkler Rain Example
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Wwet C cloudy R rain S sprinkler
Wet Sprinkler Rain Example
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Wwet C cloudy R rain S sprinkler
We generated a sample minterm C S R W
Wet Sprinkler Rain Example
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• 2. Rejection Sampling Methods

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Rejection Sampling

• Reject inconsistent samples

Wet Sprinkler Rain Example
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• 3. Likelihood weighting methods

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Wwet C cloudy R rain S sprinkler
Wet Sprinkler Rain Example
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Wwet C cloudy R rain S sprinkler
Wet Sprinkler Rain Example
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Wwet C cloudy R rain S sprinkler
Wet Sprinkler Rain Example
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Wwet C cloudy R rain S sprinkler
Wet Sprinkler Rain Example
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Wwet C cloudy R rain S sprinkler
Wet Sprinkler Rain Example
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Wwet C cloudy R rain S sprinkler
Wet Sprinkler Rain Example
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Wwet C cloudy R rain S sprinkler
Wet Sprinkler Rain Example
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Likelihood weighting vs. rejection sampling

1. Both generate consistent estimates of the joint
   distribution conditioned on the values of the
   evidence variables
2. Likelihood weighting converges faster to the
   correct probabilities
3. But even likelihood weighting degrades with many
   evidence variables because a few samples will
   have nearly all the total weight

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Sources

• Prof. David Page
• Matthew G. Lee
• Nuria Oliver,
• Barbara Rosario
• Alex Pentland
• Ehrlich Av
• Ronald J. Williams
• Andrew Moore tutorial with the same title
• Russell Norvigs AIMA site
• Alpaydins Introduction to Machine Learning
  site.