DD2380 Artificial Intelligence: Probabilistic Reasoning
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What is one key advantage of knowing the structure and conditional probability distributions in Bayesian Networks?

  • It enables the performance of various inferential tasks. (correct)
  • It allows for easy visualization of complex data.
  • It simplifies the process of data collection.
  • It guarantees accurate predictions in all scenarios.

    • Which example demonstrates the need to reason about a sequence of observations?

  • Robot localization (correct)
  • Data sorting
  • Image recognition
  • Database management

    • In the context of sequential data, what is the measurement of time series associated with speech recognition primarily concerned with?

  • Translating audio signals into words or sentences. (correct)
  • Analyzing visual patterns.
  • Tracking user attention over time.
  • Interpreting gestures for sign language.

    • Which type of data measurement is specifically mentioned in relation to sign recognition?

    <p>Drawn path measurements.</p> Signup and view all the answers

    What is the primary focus of the recommended study material in relation to Bayesian Networks?

    <p>Detailed explanations of upcoming lecture exercises.</p> Signup and view all the answers

    What is the calculated probability of finding a zebra when an object is detected as a zebra (p(Z|O))?

    <p>0.1404</p> Signup and view all the answers

    What does p(O|Z) represent in the context of this example?

    <p>The probability of detecting an object given it is a zebra.</p> Signup and view all the answers

    Why might a person intuitively overestimate the probability of detecting a zebra in an image?

    <p>They consider the detector's accuracy without regard for base rates.</p> Signup and view all the answers

    What is a critical point regarding conditional independence mentioned in this content?

    <p>Understanding conditional independence helps in reasoning about uncertainties.</p> Signup and view all the answers

    How does the false positive rate influence the detection of zebras in images?

    <p>It leads to a significant increase in perceived zebra detections.</p> Signup and view all the answers

    What does the expression $P(Z|O)$ represent in the context of the vision system for detecting zebras?

    <p>The probability that there is a zebra if the detector gives a positive result</p> Signup and view all the answers

    If the prior probability of a zebra being present is $P(Z) = 0.02$, how does this influence the posterior probability calculation?

    <p>It serves as a baseline for evaluating the likelihood of positive observations</p> Signup and view all the answers

    What is the false positive probability $P(O|¬Z)$ in the zebra detection example?

    <p>0.1</p> Signup and view all the answers

    Which of the following equations accurately represents Bayes' Rule as applied to the zebra example?

    <p>$P(Z|O) = \frac{P(O|Z) P(Z)}{P(O)}$</p> Signup and view all the answers

    In the zebra detection system, what does $P(O|Z)$ equal?

    <p>0.8</p> Signup and view all the answers

    What role does normalization play in the application of Bayes Rule?

    <p>It adjusts the result to account for all possible outcomes</p> Signup and view all the answers

    How does a high false positive rate affect the detection of zebras?

    <p>It can lead to more incorrect conclusions about zebra presence</p> Signup and view all the answers

    What happens to the posterior probability $P(Z|O)$ if the prior $P(Z)$ is increased significantly?

    <p>The posterior probability will likely increase</p> Signup and view all the answers

    What is the relationship between variables B and C given A in a Bayesian network?

    <p>B and C are conditionally independent given A</p> Signup and view all the answers

    Which statement correctly describes the influence of A on C in a Bayesian network?

    <p>C depends on A</p> Signup and view all the answers

    How does knowing variable A affect the relationship between B and C?

    <p>Knowing A means knowing B gives no additional information about C</p> Signup and view all the answers

    What captures all the relevant information in A to determine E in a Bayesian network?

    <p>Variable C</p> Signup and view all the answers

    Which formula represents the Joint Probability Distribution (JPD) in relation to A, B, and C?

    <p>$P(B,C|A)$ is equal to $P(B,C,A)$</p> Signup and view all the answers

    What does the compactness of Bayesian networks refer to?

    <p>It factors the JPD into local, conditional distributions for each variable</p> Signup and view all the answers

    Which statement is true about E in relation to A and C?

    <p>E is conditionally independent of A given C</p> Signup and view all the answers

    What role does variable A play in influencing the relationship between D and E?

    <p>A affects E's relationship with C and indirectly affects D</p> Signup and view all the answers

    What does the factorization of $P(A, B, C, D)$ imply about the relationships between the variables?

    <p>D is conditionally independent of A and B given C.</p> Signup and view all the answers

    Which of the following statements correctly describes the role of prior probability in the zebra detection example?

    <p>Prior probability indicates how often zebras appear in images.</p> Signup and view all the answers

    In Bayesian networks, what does it mean when one variable is said to influence another?

    <p>The influencing variable affects the observations made about the influenced variable.</p> Signup and view all the answers

    What is the significance of using the chain rule in the factorization of joint distributions?

    <p>It allows for any order of multiplication without loss of meaning.</p> Signup and view all the answers

    Which of the following best describes the observations made by a detector in the zebra detection scenario?

    <p>Observations consist of both true positives and false positives.</p> Signup and view all the answers

    In the equation $P(X_1, X_2, ext{...}, X_n) = P(X_i) P(X_i)$, what does each term represent?

    <p>The product of the individual probabilities of each variable.</p> Signup and view all the answers

    What factorization pattern is suggested when working with joint distributions involving conditional independencies?

    <p>Work from the top and factor out variables in a descending order.</p> Signup and view all the answers

    Which of the following correctly identifies the relationship between the alarm and earthquakes?

    <p>The alarm can be falsely triggered by small earthquakes.</p> Signup and view all the answers

    What role does conditional independence play in the factorization of joint probabilities?

    <p>It allows for simplified calculations by reducing dependencies.</p> Signup and view all the answers

    When analyzing the factorization of $P(A, B, C, D)$, which option represents a valid step?

    <p>Factor out each variable based on its dependencies.</p> Signup and view all the answers

    What does it mean if two variables are conditionally independent given a third variable?

    <p>The probability of one variable does not change with the additional information of the third variable.</p> Signup and view all the answers

    Given $P(X, Y | Z) = P(X | Z) P(Y | Z)$, what does this imply about the relationship between X and Y?

    <p>X and Y are conditionally independent given Z.</p> Signup and view all the answers

    In terms of Bayesian networks, what does a directed edge (arrow) from node A to node B represent?

    <p>A direct influence of A on B.</p> Signup and view all the answers

    Which scenario best illustrates conditional independence among three variables X, Y, and Z?

    <p>Knowing Y gives no information about X if Z is known.</p> Signup and view all the answers

    If two variables A and B are conditionally independent given C, which of the following statements is true?

    <p>The independence holds for all possible values of C.</p> Signup and view all the answers

    What is the expression for the joint probability of three variables X, Y, and Z in the presence of conditional independence?

    <p>$P(X, Y, Z) = P(X | Z) P(Y | Z) P(Z)$</p> Signup and view all the answers

    In the context of probabilistic graphical models, what is the primary purpose of using directed acyclic graphs (DAGs)?

    <p>To encode conditional independence assumptions.</p> Signup and view all the answers

    If the variables in a Bayesian network are arranged such that A is a parent of B and C, how does this influence their relationships?

    <p>B and C's probabilities are influenced by A.</p> Signup and view all the answers

    Which statement is correct regarding the joint probability of independent variables X and Y?

    <p>$P(X, Y) = P(X) P(Y)$</p> Signup and view all the answers

    In Bayesian networks, which of the following would NOT represent a conditional independence assumption?

    <p>A is dependent on B given C.</p> Signup and view all the answers

    What is the implication of stating that variable U is conditionally independent of variable T given variable R?

    <p>R provides no information about the relationship between U and T.</p> Signup and view all the answers

    If a probe catches in the cavity only in relation to other factors, which example illustrates conditional independence?

    <p>The probe's interaction does not depend on the presence of a toothache.</p> Signup and view all the answers

    What does it mean for two variables A and C to be leaf nodes in a Bayesian network?

    <p>They have no children nodes.</p> Signup and view all the answers

    In a Bayesian network, if A influences both B and C, how do we express the relationship mathematically?

    <p>P(B, C | A) = P(B | A) P(C | A)</p> Signup and view all the answers

    Study Notes

    Course Information

    • Course Title: DD2380 Artificial Intelligence
    • Topic: Probabilistic Reasoning
    • Instructor: André Pereira
    • Start Time: 15:15
    • Required Reading: Chapters 13-15, Russel & Norvig

    Slide Credits

    • Based on original slides from Patric Jensfelt and Iolanda Leite, KTH
    • Materials from: http://ai.berkeley.edu
    • Kevin Murphy, MIT, UBC, Google
    • Danica Kragic, KTH
    • W. Burgard, C. Stachniss, M. Benewitz and K. Arras, when at Albert-Ludwigs-Universität Freiburg

    Outline

    • Probabilities
      • Motivation
      • Notation and Recap
      • Bayes Rule
      • Conditional Independence
    • Probabilistic Graphical Models
      • Bayesian Networks
      • Sequential Data
        • Markov Models (next lecture)
        • Hidden Markov Models (next lecture)

    Motivation

    • Probability quantifies the likelihood of an event happening in uncertain situations.
    • Uncertainty plays a critical role in:
      • Sensor interpretation
      • Sensor fusion
      • Map making
      • Path planning
      • Self-localization
      • Control

    Real-World Examples (Autonomous Car)

    • Cross intersection safely
      • Observations from car sensors
        • Sensor models
        • Statistics from different roads
        • Weather models
      • Observations from other vehicles
        • Can I cross safely with 99% or 99.99999% safety?

    Diagnose Diseases

    • Doctors use prior knowledge of disease prevalence and connections to factors like age, sex, habits, and symptoms (e.g., temperature).
    • Observe symptoms, evaluate against known possibilities.
    • Diagnose.

    Probability Recap 1/3

    • Probability of event X: p(X)
    • p(X) ∈ [0, 1] (0 ≤ p(X) ≤ 1)
    • 1 = Σall x p(X)
    • p(¬X): Probability that X is false.
    • p(X) = 1 - p(¬X)
    • Joint probability of X AND Y: p(X, Y)
    • Conditional probability of X GIVEN Y: p(X|Y)

    Probability Recap 2/3

    • Product rule: p(X, Y) = p(Y|X)p(X)
    • Sum rule (marginalization): p(X) = Σ yp(X, Y)

    Sum Rule (Marginalization)

    • Calculates the probability of an event by summing probabilities over all possible values of other variables

    Law of Total Probability (conditioning)

    • Combines probabilities using sum and product rules:
    • p(X) = Σy p(X,Y) (sum rule)
    • p(X,Y) = p(X|Y)p(Y) (product rule)

    Conditional Probability

    • P(A|B) = P(A∩B) / P(B).
      • P(A∩B) - Intersection of A and B events
      • P(B) - probability B event occurs

    Conditional Probability (Weather Example)

    • P(W = s | T = c) = P(W = s,T = c) / P(T = c)

    Conditional Dependence

    • Applications in Artificial Intelligence, Natural Language Processing, Robotics, Computer Vision

    Recognizing Street Signs Example

    • Understanding what street signs look like is based on prior experiences

    Probabilistic Inference

    • Compute desired probabilities from known probabilities (e.g., conditional from joint)
    • Conditional probabilities represent an agent's beliefs given evidence.
    • Observations update beliefs

    Bayes' Rule

    • P(A|B) = [P(B|A)P(A)] / P(B)
      • P(A|B): Posterior probability of A given B
      • P(B|A): Likelihood of observing B given A
      • P(A): Prior probability of A
      • P(B): Probability of observing B

    Bayes' Rule Derivation

    -Derivation of Bayes rule formula

    Bayes Rule using Normalization

    • P(A|B) = [P(B|A)P(A)]/P(B)
    • Conditional formula

    Bayes Rule Example

    • Understanding application of Bayes Rules to a detection task

    Bayes Rule Example Solution

    • A solution to a probability scenario showing how the Bayes Rule formula can be applied

    Bayes Rule Example Discussion

    • Intuition of Bayes Rules
    • Example of a vision system for detecting zebras and applying conditional probability

    Conditional Independence

    • Unconditional Independence - rare
    • Conditional Independence - more common in uncertain environments

    Conditional Independence Formulas

    • If X is conditionally independent of Y given Z: P(X|Y,Z) = P(X|Z).

    Conditional Independence Example (Toothache Example)

    • Catch is conditionally independent of Toothache given Cavity

    Probability Recap 3/3

    • Conditional Probability: P(x|y) = p(x,y) / p(y)
    • Product Rule: p(x,y) = p(y|x)p(x)
    • Chain Rule: P(X₁...Xₙ) = Σi=1..n P(Xi|X1..i-1)

    Break

    • 15-minute break

    Probabilistic Graphical Models

    • Compact Representation of joint distribution
    • Graphical representation for analyzing/structuring probability information
    • Variables are encoded as nodes, and conditional independence encoded with arcs.

    Bayesian Network

    • A special type of probabilistic graphical model

    Bayesian Network (continued)

    • Properties of a Bayesian network (e.g., root node, leaf nodes, parent, children)
    • Interpretation of relationships in a Bayesian Network

    Bayesian Network (continued)

    • Interpretation of relationships (e.g., "causes") in the network

    Bayesian Network (continued)

    • Conditional Independence in the network and the role evidence plays to these relationships

    Bayesian Network (continued)

    • How Bayesian networks enable reasoning about sequences of observations in time or space or "measurements of time series")

    Sequential Data-Example 1 and 2

    • Examples in the area of recognition in time or space
    • Sign recognition, speech recognition

    Next Lecture

    • Hidden Markov Models (HMM)

    Additional Study Material

    • Online learning resource
    • Additional content and tutorials
    • Quiz on Bayesian Networks

    End of Taming Uncertainty Part 1/2

    • Conclusion of the current presentation segment

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    Lecture 6 - Probabilistic Reasoning PDF

    Description

    This quiz covers the essential concepts of probabilistic reasoning as explored in chapters 13-15 of Russell & Norvig. Key topics include Bayes Rule, Bayesian Networks, and their applications in AI. Test your understanding and apply these principles to real-world scenarios.

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