Part%202%20of%203:%20Bayesian%20Network%20and%20Dynamic%20Bayesian%20Network

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Part 2 of 3 Bayesian Network and Dynamic
Bayesian Network
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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
  25Sebastian Thrun, Wolfram Burgard, and Dieter
  Fox, Probabilistic Robotics, Chapters 2 7
• Part 3Sebastian Thrun, Wolfram Burgard, and
  Dieter Fox, Probabilistic Robotics, Chapter 2

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Bayesian Network

• A data structure to represent dependencies among
  variables
• A directed graph in which each node is annotated
  with quantitative probability information

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An Example of a Simple Bayesian Network
Cavity
Weather
Toothache
Catch
Weather is independent of the other three
variables. Toothache and Catch are conditionally
independent, given Cavity.
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Bayesian Network

• Full specifications
• A set of random variables makes up the nodes of
  the network
• A set of directed links or arrows connects pairs
  of nodes. parent?child
• Each node Xi has a conditional probability
  distribution P(XiParents(Xi)) that quantifies
  the effect of the parents on the node
• Directed acyclic graph (DAG), i.e. no directed
  cycles

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Implementation of BN

• Open source BN software Java Bayes
• Commercial BN software MS Bayes, Netica

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Teaching and Research Tools in Academic
Environments

• GeNIe
• Developed at the Decision Systems Laboratory,
  University of Pittsburgh
• Runs only on Windows computers

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Demo
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An Example of Bayesian Network
Burglary
Lightning
Alarm
MaryCalls
JohnCalls
Demo using GeNIe
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An Application of Bayesian Network
Horvitz, et. al. (Microsoft Research) The Lumiere
Project Bayesian User Modeling for Inferring the
Goals and Needs of Software Users ftp//ftp.resear
ch.microsoft.com/pub/ejh/lum.pdf
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An Application of Bayesian Network
Horvitz, et. al. (Microsoft Research) The Lumiere
Project Bayesian User Modeling for Inferring the
Goals and Needs of Software Users ftp//ftp.resear
ch.microsoft.com/pub/ejh/lum.pdf
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Dynamic Bayesian NetworkProbabilistic Reasoning
Over Time
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Basic Ideas

• The process of change can be viewed as a series
  of snapshots
• Each snapshot (called a time slice) describes the
  state of the world at a particular time
• Each time slice contains a set of random
  variables, some of which are observable and some
  of which are not

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(No Transcript)
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DBN with Evolution of States, Controls, and
Measurements
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Terminology
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• State
• Environments are characterized by state
• Think of the state as the collection of all
  aspects of the robot and its environment that can
  impact the future

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• State
• Examples
• the robot's pose (location and orientation)
• variables for the configuration of the robot's
  actuators (e.g. joint angles)
• robot velocity and velocities of its joints
• location and features of surrounding objects in
  the environment
• location and velocities of moving objects and
  people
• whether or not a sensor is broken, the level its
  battery charge

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• State
• denoted as x
• xt the state at time t

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• Environment measurement data
• evidence
• information about a momentary state of the
  environment
• Examples
• camera images
• range scans

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• Environment measurement data
• Denoted as z
• zt the measurement data at time t
• denotes the set of all
• measurements acquired
• from time t1 to time t2

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• Control data
• convey information regarding change of state in
  the environment
• related to actuation
• Examples
• velocity of a robot (suggests the robot's pose
  after executing a motion command)
• odometry (measure of the effect of a control
  action)

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• Control data
• Denoted as u
• ut the change of state in the time interval
  (t-1t
• denotes a sequence of
• control data from
• time t1 to time t2

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Scenario

• a mobile robot uses its camera to detect the
  state of the door (open or closed)
• camera is noisy
• if the door is in fact open
• the probability of detecting it open is 0.6
• if the door is in fact closed
• the probability of detecting it closed is 0.8

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Scenario

• the robot can use its manipulator to push open
  the door
• if the door is in fact closed
• the probability of robot opening it is 0.8

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Scenario

• At time t0, the probability of the door being
  open is 0.5
• Suppose at t1 the robot takes no control action
  but it senses an open door, what is the
  probability of the door is open?

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Scenario

• Using Bayes Filter, we will see that
• at time t1 the probability of the door is open
  is
• 0.75 after taking a measurement
• at time t2 the probability of the door is open is
• 0.984 after the robot pushes open the door and
  takes another measurement

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DBN with Evolution of States, Controls, and
Measurements for the Mobile Robot Example
xt state of the door (open or closed) at time
t ut control data (robot's manipulator pushes
open or does nothing) at time t zt evidence or
measurement by sensors at time t
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Demo using GeNIe
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Basic Idea of the Algorithm of Bayes Filter

• Bayes_filter(bel(xt-1), ut, zt)
• for all xt do
• endfor
• return bel(xt)

Predict xt after exerting ut
Update belief of xt after making a measurement zt
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The subsequent slides explain how the Bayes' rule
is applied in this filter.
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Inference in Temporal Models
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Basic Inference Tasks in Probabilistic Reasoning

• Filtering or monitoring
• Prediction
• Smoothing or hindsight
• Most likely explanation

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Filtering, or monitoring

• The task of computing the belief statethe
  posterior distribution over the current state,
  given all evidence to date
• i.e. compute P(Xtz1t)

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Prediction

• The task of computing the posterior distribution
  over the future state, given all evidence to date
• i.e. compute P(Xtkz1t), for some k gt 0or
  P(Xt1z1t) for one-step prediction

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Smoothing, or hindsight

• The task of computing the posterior distribution
  over the past state, given all evidence to date
• i.e. compute P(Xkz1t), for some 0 ? k lt t

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Most likely explanation

• Given a sequence of observations, we want to find
  the sequence of states that is most likely to
  have generated those observations
• i.e. compute

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Review

• Bayes' rule

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Review

• 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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DBN with Evolution of States and MeasurementsTo
be used in the explanation of filtering and
prediction tasks in the subsequent slides
xt state of the door (open or closed) at time
t zt evidence or measurement by sensors at time
t
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The Task of Filtering

• To update the belief state
• By computing
• from the current state

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The Task of Filtering
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The Task of Filtering
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The Task of Filtering
b
c
a
a
b
c
b
c
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The Task of Filtering
The probability distribution of the state at t1,
given the measurements (evidence) to date i.e. it
is a one-step prediction for the next state
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The Task of Filtering
By conditioning on the current state xt, this
term becomes
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The Task of Filtering

• To update the belief state
• By computing
• from the current state

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The Task of Filtering
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The Task of Filtering
The robot's belief state The posterior over the
state variables X at time t1 is calculated
recursively from the corresponding estimate one
time step earlier
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The Task of Filtering
Most modern localization algorithms use one of
two representations of the robot's belief Kalman
filter and particle filter called Monte Carol
localization (MCL).
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The Task of Filtering
Kalman filter represent this belief state as a
single multivariate Gaussian
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The Task of Filtering
particle filter represent this belief state as a
collection of particles that correspond to states
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The Task of Filtering in Localization

• To update the belief state
• By computing
• from the current state

m
shaded nodes information that is given white
nodes information you want to find
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The Task of Filtering in Mapping

• To update the belief state
• By computing
• from the current state

m
shaded nodes information that is given white
nodes information you want to find
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The Task of Filtering
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An Example Graphical Solution of Extended Kalman
Filter
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An Example Graphical Solution of Particle Filter
Example
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An Example Implementation of EKF