Probabilistic Reasoning and Bayesian Networks

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Probabilistic Reasoning and Bayesian Networks

• Lecture Prepared For
• COMP 790-058
• Yue-Ling Wong

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Probabilistic Robotics

• A relatively new approach to robotics
• Deals with uncertainty in robot perception and
  action
• The key idea is to represent uncertainty
  explicitly using the calculus of probability
  theory
• i.e. represent information by probability
  distributions over a whole space of guesses,
  instead of relying on a single "best guess"

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3 Parts of this Lecture

• Part 1. Acting Under Uncertainty
• Part 2. Bayesian Networks
• Part 3. Probabilistic Reasoning in Robotics

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Reference for Part 3

• Sebastian Thrun, et. al. (2005) Probabilistic
  Robotics
• The book covers major techniques and algorithms
  in localization, mapping, planning and control
• All algorithms in the book are based on a single
  overarching mathematical foundations
• Bayes rule
• its temporal extension known as Bayes filters

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Goals of this lecture

• To introduce this overarching mathematical
  foundations Bayes rule and its temporal
  extension known as Bayes filters
• To show how Bayes rule and Bayes filters are used
  in robotics

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Preliminaries

• Part 1
• Probability theory
• Bayes rule
• Part 2
• Bayesian Networks
• Dynamic Bayesian Networks

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Outline of this lecture

• Part 1. Acting Under Uncertainty (October 20)
• To go over fundamentals on probability theory
  that is necessary to understand the materials of
  Bayesian reasoning
• Start with AI perspective and without adding the
  temporal aspect of robotics
• Part 2. Bayesian Networks (October 22)
• DAG representation of random variables
• Dynamic Bayesian Networks (DBN) to handle
  uncertainty and changes over time
• Part 3. Probabilistic Reasoning in Robotics
  (October 22)
• To give you general ideas of how DBN is used in
  robotics to handle the changes of sensor and
  control data over time in making inferences
• Demonstrate use of Bayes rule and Bayes filter in
  a simple example of mobile robot monitoring the
  status (open or closed) of doors

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Historical Background and Applications of
Bayesian Probabilistic Reasoning

• Bayesian probabilistic reasoning has been used in
  AI since 1960, especially in medical diagnosis
• One system outperformed human experts in the
  diagnosis of acute abdominal illness(de Dombal,
  et. al. British Medical Journal, 1974)

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Historical Background and Applications of
Bayesian Probabilistic Reasoning

• Directed Acyclic Graph (DAG) representation for
  Bayesian reasoning started in the 1980's
• Example systems using Bayesian networks
  (1980's-1990's)
• MUNIN system diagnosis of neuromuscular
  disorders
• PATHFINDER system pathology

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Historical Background and Applications of
Bayesian Probabilistic Reasoning

• NASA AutoClass for data analysishttp//ti.arc.nas
  a.gov/project/autoclass/autoclass-c/finds the
  set of classes that is maximally probable with
  respect to the data and model
• Bayesian techniques are utilized to calculate the
  probability of a call being fraudulent at ATT

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Historical Background and Applications of
Bayesian Probabilistic Reasoning

• By far the most widely used Bayesian network
  systems
• The diagnosis-and-repair modules (e.g. Printer
  Wizard) in Microsoft Windows(Breese and
  Heckerman (1996). Decision-theoretic
  troubleshooting A framework for repair and
  experiment. In Uncertainty in Artificial
  Intelligence Proceedings of the Twelfth
  Conference, pp. 124-132)
• Office Assistant in Microsoft Office(Horvitz,
  Breese, Heckerman, and Hovel (1998). The Lumiere
  project Bayesian user modeling for inferring the
  goals and needs of software users. In Uncertainty
  in Artificial Intelligence Proceedings of the
  Fourteenth Conference, pp. 256-265.http//researc
  h.microsoft.com/horvitz/lumiere.htm)
• Bayesian inference for e-mail spam filtering

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Historical Background and Applications of
Bayesian Probabilistic Reasoning

• An important application of temporal probability
  models Speech recognition

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References and Sources of Figures

• Part 1Stuart Russell and Peter Norvig,
  Artificial Intelligence A Modern Approach, 2nd
  ed., Prentice Hall, Chapter 13
• Part 2Stuart Russell and Peter Norvig,
  Artificial Intelligence A Modern Approach, 2nd
  ed., Prentice Hall, Chapters 14 15
• Part 3Sebastian Thrun, Wolfram Burgard, and
  Dieter Fox, Probabilistic Robotics, Chapter 2

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Part 1 of 3 Acting Under Uncertainty
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Uncertainty Arises

• The agent's sensors give only partial, local
  information about the world
• Existence of noise of sensor data
• Uncertainty in manipulators
• Dynamic aspects of situations (e.g. changes over
  time)

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Degree of Belief

• An agent's knowledge can at best provide only a
  degree of belief in the relevant sentences.
• One of the main tools to deal with degrees of
  belief will be probability theory.

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Probability Theory

• Assigns to each sentence a numerical degree of
  belief between 0 and 1.

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In Probability Theory

• You may assign 0.8 to the a sentence"The
  patient has a cavity."
• This means you believe"The probability that the
  patient has a cavity is 0.8."
• It depends on the percepts that the agent has
  received to date.
• The percepts constitute the evidence on which
  probability assessments are based.

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Versus In Logic

• You assign true or false to the same
  sentence.True or false depends on the
  interpretation and the world.

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Terminology

• Prior or unconditional probability
• The probability before the evidence is obtained.
• Posterior or conditional probability
• The probability after the evidence is obtained.

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Example

• Suppose the agent has drawn a card from a
  shuffled deck of cards.
• Before looking at the card, the agent might
  assign a probability of 1/52 to its being the ace
  of spades.
• After looking at the card, the agent has obtained
  new evidence. The probability for the same
  proposition (the card being the ace of spades)
  would be 0 or 1.

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Terminology and Basic Probability Notation
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Terminology and Basic Probability Notation

• PropositionAscertain that such-and-such is the
  case.

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Terminology and Basic Probability Notation

• Random variableRefers to a "part" of the world
  whose "status" is initially unknown.Example
  Cavity might refer to whether the patient's lower
  left wisdom tooth has a cavity.Convention used
  here Capitalize the names of random variables.

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Terminology and Basic Probability Notation

• Domain of a random variableThe collection of
  values that a random variable can take
  on.Example The domain of Cavity might be
  ?true, false?The domain of Weather might be
  ?sunny, rainy, cloudy, snow?

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Terminology and Basic Probability Notation

• Abbreviations used here
• cavity to represent Cavity true
• ?cavity to represent Cavity false
• snow to represent Weather snow
• cavity ? ?toothache to represent Cavitytrue ?
  Toothachefalse

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Terminology and Basic Probability Notation

• cavity ? ?toothache
• or
• Cavitytrue ? Toothachefalse
• is a proposition that may be assigned with a
  degree of belief

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Terminology and Basic Probability Notation

• Prior or unconditional probabilityThe degree of
  belief associated with a proposition in the
  absence of any other information.Examplep(Cavi
  tytrue) 0.1 or p(cavity) 0.1

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Terminology and Basic Probability Notation

• p(Weathersunny) 0.7
• p(Weatherrain) 0.2
• p(Weathercloudy) 0.08
• p(Weathersnow) 0.02
• or we may simply write
• P(Weather) ?0.7, 0.2, 0.08, 0.02?

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Terminology and Basic Probability Notation

• Prior probability distributionA vector of values
  for the probabilities of each individual state of
  a random variableExample This denotes a prior
  probability distribution for the random variable
  Weather.P(Weather) ?0.7, 0.2, 0.08, 0.02?

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Terminology and Basic Probability Notation

• Joint probability distributionThe probabilities
  of all combinations of the values of a set of
  random variables.P(Weather, Cavity)
• denotes the probabilities of all combinations of
  the values of a set of random variables Weather
  and Cavity.

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Terminology and Basic Probability Notation

• P(Weather, Cavity)
• can be represented by a 4x2 table of
  probabilities.

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Terminology and Basic Probability Notation

• Full joint probability distributionThe
  probabilities of all combinations of the values
  of the complete set of random variables.

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Terminology and Basic Probability Notation

• Example Suppose the world consists of just the
  variables Cavity, Toothache, and
  Weather.P(Cavity, Toothache, Weather)
• denotes the full joint probability distribution
  which can be represented as a 2x2x4 table with 16
  entries.

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Terminology and Basic Probability Notation

• Posterior or conditional probabilityNotation
  p(ab)Read as "The probability of proposition
  a, given that all we know is proposition b."

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Terminology and Basic Probability Notation

• Examplep(cavitytoothache) 0.8Read as"If
  a patient is observed to have a toothache and no
  other information is yet available, then the
  probability of the patient's having a cavity will
  be 0.8."

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Terminology and Basic Probability Notation

• Equation
• where p(b) gt 0

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Terminology and Basic Probability Notation

• Product rule
• which is rewritten from the previous equation

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Terminology and Basic Probability Notation

• Product rule
• can also be written the other way around

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Intuition
Terminology and Basic Probability Notation
cavity ? toothache
cavity
toothache
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Intuition
Terminology and Basic Probability Notation
cavity ? toothache
cavity
toothache
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Derivation of Bayes' Rule
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Terminology and Basic Probability Notation

• Bayes' rule, Bayes' law, or Bayes' theorem

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Bayesian Spam Filtering

• Given that it has certain words in an email, the
  probability that the email is spam is equal to
  the probability of finding those certain words in
  spam email, times the probability that any email
  is spam, divided by the probability of finding
  those words in any email

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Speech Recognition

• Given the acoustic signal, the probability that
  the signal corresponds to the words is equal to
  the probability of getting the signal with the
  words, times the probability of finding those
  words in any speech, times a normalization
  coefficient

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Terminology and Basic Probability Notation

• Conditional distributionNotation P(XY)It
  gives the values of p(Xxi Yyj) for each
  possible i, j.

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Terminology and Basic Probability Notation

• Conditional distributionExample P(X,Y)
  P(XY)P(Y)denotes a set of equationsp(Xx1 ?
  Yy1) p(Xx1 Yy1)p(Yy1)p(Xx1 ? Yy2)
  p(Xx1 Yy2)p(Yy2)...

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Probabilistic InferenceUsing Full Joint
Distributions
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Terminology and Basic Probability Notation

• Simple dentist diagnosis example.
• 3 Boolean variables
• Toothache
• Cavity
• Catch (the dentist's steel probe catches in the
  patient's tooth)

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A full joint distribution for the Toothache,
Cavity, Catch world
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Getting information from the full joint
distribution
p(cavity ? toothache) 0.108 0.012 0.072
0.008 0.016 0.064 0.28
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Getting information from the full joint
distribution
p(cavity) 0.108 0.012 0.072 0.008 0.2
unconditional or marginal probability
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Marginalization, Summing Out, Theorem of Total
Probability, and Conditioning
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Getting information from the full joint
distribution
p(cavity) 0.108 0.012 0.072 0.008 0.2
p(cavity, catch, toothache) p(cavity, ?catch,
toothache) p(cavity, catch, ?toothache)
p(cavity, ?catch, ?toothache)
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Marginalization Rule

• Marginalization rule
• For any sets of variables Y and Z,
• A distribution over Y can be obtained by summing
  out all the other variables from any joint
  distribution containing Y.

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A variant of the rule after applying the product
rule

• Conditioning
• For any sets of variables Y and Z,
• Read as Y is conditioned on the variable Z.
• Often referred to as Theorem of total probability.

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Getting information from the full joint
distribution
The probability of a cavity, given evidence of a
toothache
conditional probability
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Getting information from the full joint
distribution
The probability of a cavity, given evidence of a
toothache
conditional probability
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Getting information from the full joint
distribution
The probability of a cavity, given evidence of a
toothache
conditional probability
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Getting information from the full joint
distribution
The probability of a cavity, given evidence of a
toothache
conditional probability
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Independence
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Independence

• If the propositions a and b are independent, then
• p(ab) p(a)
• p(ba) p(b)
• p(a?b) p(a,b) p(a)p(b)
• Think about the coin flipping example.

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Independence Example

• Suppose Weather and Cavity are independent.
• p(cavity Weathercloudy) p(cavity)
• p(Weathercloudy cavity) p(Weathercloudy)
• p(cavity, Weathercloudy) p(cavity)p(Weatherclo
  udy)

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Similarly

• If the variables X and Y are independent, then
• P(XY) P(X)
• P(YX) P(Y)
• P(X,Y) P(X)P(Y)

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Normalization
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Previous Example
The probability of a cavity, given evidence of a
toothache
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Previous Example
The probability of a cavity, given evidence of a
toothache
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Normalization

• The term
• remains constant, no matter which value of Cavity
  we calculate.
• In fact, it can be viewed as a normalization
  constant for the distribution P(Cavitytoothache),
  ensuring that it adds up to 1.

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Recall this example
The probability of a cavity, given evidence of a
toothache
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Now, normalization simplifies the calculation
The probability distribution of Cavity, given
evidence of a toothache
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Now, normalization simplifies the calculation
The probability distribution of Cavity, given
evidence of a toothache
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Example of Probabilistic InferenceWumpus World
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OK
OK
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OK
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Pit? Wumpus
OK
Pit? Wumpus
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Pit? Wumpus
OK
Pit? Wumpus
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Pit? Wumpus
Pit? Wumpus
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Pit? Wumpus
Pit? Wumpus
Pit? Wumpus
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Now what??
Pit? Wumpus
Pit? Wumpus
Pit? Wumpus
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By applying Bayes' rule, you can calculate the
probabilities of these cells having a pit, based
on the known information.
0.31
Pit? Wumpus
Pit? Wumpus
0.86
Pit? Wumpus
0.31
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To Calculate the Probability Distribution for
Wumpus Example
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Let unknown be a composite variable consisting of
Pi,j variables for squares other than Known
squares and the query square 1,3
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