Uncertainty 1

56 flashcards · Mathematics · by J M COOPER

1. What do probabilistic assertions summarise?

   -laziness: failure to enumerate exceptions, qualifications, etc. -ignorance: lack of relevant facts, initial conditions, etc.

2. What risks are inherent in applying a logical approach in uncertainty?

   - risks falsehood - or leads to conclusions that are too weak for decision making:“A25 will get me there on time if there’s no accident on the bridgeand it doesn’t rain and my tires remain intact etc etc...

3. What is Subjective or Bayesian probability?

   Probabilities relate propositions to one’s own state of knowledgee.g., P(A25|no reported accidents) = 0.06

4. What does ω ∈ Ω mean?

   ω 'is an element of the set' Ω

5. What is utility theory? What is Decision theory?

   an agent's preferences over possible outcomes can be captured by a utility function that maps these outcomes to a real number; the higher the number the more that agent likes that outcome. The function is called a utility function. Combining utility theory and probability theory to make decisions

6. What is Ω?

   Omega (capital)

7. What is ω

   Omega (lowercase)

8. What is Σ?

   Sigma (uppercase) means to sum up

9. What is a probability model? (Sometimes called probability space or more generally, sample space)

   A sample space with an assignment P(ω) for every ω ∈ Ω, where 0 ≤ P(ω) ≤ 1 ΣωP(ω) = 1 e.g., P(1) = P(2) = P(3) = P(4) = P(5) = P(6) = 1/6.

10. Event A is any subset of the sample space. What does P(A) = Σ{ω∈A}P(ω) mean?

    The probability of event A is the sum of the probabilities of all events where that event is an element of A. e.g. , P(die roll < 4) = P(1) + P(2) + P(3) = 1/6 + 1/6 + 1/6 = 1/2

11. What is a random variable?

    a function from an outcome (sample point) to some range e.g., Odd(1) = true. e.g. X = 1 if heads, or 0 if tails

12. What is a probability distribution?

    Provides the probabilities of occurrence of different possible outcomes. It gives values for all possible assignments: e.g. P(Weather) = ⟨ 0.72, 0.1, 0.08, 0.1⟩ (normalized, i.e., sums to 1)

13. Explain: P(X = xi) = Σ{ω:X(ω) = xi}P(ω)

    The probability that random variable X has value xi is the sum of the probabilities of each world where X equals xi. e.g. P(Odd = true) = P(1) + P(3) + P(5) = 1/6 + 1/6 + 1/6 = 1/2

14. What is a proposition?

    A proposition is a sentence which is either true or false.

15. What does ≡ mean?

    logical equivalence

16. What are Kolmogorov’s Axioms?

    All probabilities are between 0 and 1 0≤P(A) ≤1 Truth has probability 1, false has probability 0 P(true) = 1P(false = 0) The probability of disjunction is given by: P(a∨b) =P(a) +P(b) −P(a∧b)

17. What is a prior or unconditional probability of a proposition?

    - belief prior to arrival of any (new) evidence - e.g., P(Cavity = true) = 0.1 and P(Weather = sunny) = 0.72

18. What did de Finetti say about Kolmogorov’s Axioms?

    "an agent who bets according to probabilities that violate these axioms can be forced to bet so as to lose money regardless of outcome." • Agent 2 bets according to the Agent 1’s inconsistent beliefs – Agent 1 thinks these bets are all fair, so is prepared to make them – But Agent 1 will lose whatever happens

19. What is a joint probability distribution?

    Joint probability distribution for a set of r.v.s gives the probability of every atomic event on those r.v.s (i.e., every sample point) P(W eather, Cavity) = a 4 × 2 matrix of values: Every question about a domain can be answered by the joint distribution because every event is a sum of sample points

20. What is the product rule?

    An alternative formulation of conditional probability P(a∧b) = P(a|b)P(b) = P(b|a)P(a) A general version holds for whole distributions, e.g., P(Weather, Cavity) = P(Weather|Cavity)P(Cavity)

21. What are conditional or posterior probabilities?

    e.g.,P(cavity|toothache) = 0.8 i.e., given that toothache is all I know. NOT “if toothache then 80% chance of cavity”

22. What is the definition of conditional probability?

23. What is P(A|B)?

    //is this right? P(A|B) = P(B|A)P(A) The probability of A given B, is the probability of B occurring given that A has occurred, times the probability of A occurring in the first place. If P and B are unconditional (eg if P(B|A) = P(B)) then it's just P(A)P(B)

24. How is the formula for conditional probability related to the chain rule?

    Conditional probability formula: P(A|B) = P(A∧B)/P(B) We can rearrange the formula for conditional probability to get the product rule: P(A,B) = p(A|B) p(B) We can extend this for three variables: P(A,B,C) = P(A| B,C) P(B,C) = P(A|B,C) P(B|C) P(C) and in general to n variables: P(A1, A2, ..., An) = P(A1| A2, ..., An) P(A2| A3, ..., An) P(An-1|An) P(An) We refer to this as the chain rule.

25. Explain P(φ) =Σω:ω|=φP(ω)

    The probability of proposition φ is the sum of the probabilities of all events where that event entails φ.

26. What is inference by enumeration?

    Start with the joint distribution For any proposition φ, sum the probabilities of all atomic events where φ is true: P(φ) =Σω:ω|=φP(ω)

27. What is the Chain rule?

    conditional probability: P(A|B) = P(A&B)/P(B) can be reformulated as the product rule: P(A&B) = P(A|B)P(B) chain rule = repeated application of the product rule eg product rule = P(A&B) = P(A|B)P(B) so P(X1,X2,X3,X4) = P(X4|X1,X2,X3)P(X1,X2,X3) (and then apply the product rule again on the second term) = P(X4|X1,X2,X3)P(X3|X1,X2)P(X1,X2) (and again and so on...) = P(X4|X1,X2,X3)P(X3|X1,X2)P(X2|X1)P(X1) ET VOILA: P(X1,X2,X3,X4) = P(X4|X1,X2,X3)P(X3|X1,X2)P(X2|X1)P(X1)

28. What does : mean?

    "such that"

29. What does ⊨ mean?

    Entails. When one statement logically follows from one or more others.

30. Infer by enumeration: P(toothache)

    P(toothache) = 0.108 + 0.012 + 0.016 + 0.064 = 0.2

31. Infer by enumeration P(cavity∨toothache)

    P(cavity∨toothache) = 0.108+0.012+0.072+0.008+0.016+0.064 = 0.28

32. Infer by enumeration P(¬cavity|toothache)

    P(¬cavity|toothache) = P(¬cavity ∧ toothache)/ P(toothache) = 0.016 + 0.064 / 0.108 + 0.012 + 0.016 + 0.064 = 0.4

33. Why is non-monotonic (default) logic problematic in uncertainty?

    - Don't know how to tell what assumptions are reasonable - Don't know how to handle contradiction

34. What is the Cartesian product?

    The Cartesian product of two sets (A and B) is the set of all points (a,b) where a ∈ A and b ∈ B.

35. Infer by enumeration P(cavity|toothache) using normalisation.

    Denominator can be viewed as a normalisation constant α So instead of P(cavity ∧ toothache)/ P(toothache) we can do: P(Cavity|toothache) = α P(Cavity, toothache) So P(Cavity, toothache) is 0.108 + 0.016. We also need the complement ie P(¬Cavity, toothache) which is 0.012 + 0.064 P(Cavity, toothache) = 0.12 P(¬Cavity, toothache) = 0.08 To get the normalising constant for P(Cavity, toothache): (I put the 1/ in there so we can multiply by α instead of dividing by it.) α = 1/ ( 0.12 + 0.08 ) = 5 so…

36. Given a joint distribution, how can one compute conditional probabilities by inference by enumeration?

    Compute distribution on query variable by fixing evidence (observed) variables and summing over hidden (unobserved) variables.

37. What problems exist with inference by enumeration?

    - Worst-case time complexity O(dn) where d is the largest arity - Space complexity O(dn) to store the joint distribution - How to find the numbers for O(dn) entries?

38. What does arity mean?

    The arity of a function or operation is the number of arguments or operands that the function takes.

39. When are A and B independent?

    A and B are independent iff (if and only if) P(A|B) = P(A) or P(B|A) = P(B) or P(A, B) = P(A)P(B)

40. What equivalent statements show that Catch is conditionally independent of Toothache given Cavity.

    P(Catch|Toothache, Cavity) =P(Catch|Cavity) P(Toothache|Catch, Cavity) =P(Toothache|Cavity) P(Toothache, Catch|Cavity) =P(Toothache|Cavity)P(Catch|Cavity)

41. What is Bayes' Rule?

42. Catch is conditionally independent of Toothache given Cavity. Write out the full joint distribution using the chain rule: P(Toothache, Catch, Cavity)

    P(Toothache, Catch, Cavity) =P(Toothache|Catch, Cavity)P(Catch, Cavity) =P(Toothache|Catch,Cavity)P(Catch|Cavity)P(Cavity) =P(Toothache|Cavity)P(Catch|Cavity)P(Cavity) I.e., 2 + 2 + 1 = 5 independent numbers (equations 1 and 2 remove 2)

43. What does the use of conditional independence do to the the size of the representation of the joint distribution in most cases? Why do we care about conditional independence?

    - In most cases, the use of conditional independence reduces the size of the representation of the joint distribution from exponential in n to linear in n. - Conditional independence is our most basic and robust form of knowledge about uncertain environments.

44. What is Bayes' Rule useful for?

    Useful for assessing diagnostic probability from causal probability: P(Cause|Effect) = P(Effect|Cause)P(Cause)/P(Effect)

45. What is Bayes' Rule in distribution form? (normalisation)

46. What does ∪ mean?

    the union (denoted by ∪) of a collection of sets is the set of all elements in the collection.

47. Unknown = Fringe ∪ Other Observations are conditionally independent of other hidden squares given neighbouring hidden squares. Show why.

    Unknown = Fringe ∪ Other Making an observation about the squares in the known area will not affect any of the squares in the unknown area except the squares neighbouring them - so we can forget about the others for now and look at the fringe only. P(b|P1,3, Known, Unknown) = P(b|P1,3, Known, Fringe)

48. Explain! :-o

    We want to find the probability of there being a pit in 1,3, given what we know about the location of known pits (known) and breezes (b) Apply the normalisation factor to the sum of the probabilities of P1,3 and unknown, known and b for each element in unknown. TODO needs work :-/

49. Specify the probability model: Pij = true iff [i, j] contains a pit Bij = true iff [i, j] is breezy pits are placed randomly, probability 0.2 per square Include only B1,1, B1,2, B2,1 in the probability model

    TODO what are first and second terms then used for????

50. What is the problem using inference by enumeration in the Wumpus world?

    Grows exponentially with number of squares!

51. What is probability?

    Probability is a rigorous formalism for uncertain knowledge

52. What is joint probability distribution and how may queries be answered using it?

    Joint probability distribution specifies probability of every atomic event. Queries can be answered by summing over atomic events

53. How may we reduce the joint probability distribution size for nontrivial domains?

    Reason with independence and conditional independence.

54. What are the problems with assuming conditional independence?

    Doesn’t really hold true most of the time. Easy when x were categorical - but data is frequently not. However turns out empirically (and for theoretical reasons) that we frequently get good performance even if this independence assumption is nonsense.

55. What is the conditional independence assumption and why is it useful?

    Assuming that the variables in question are independent makes it a lot easier and quicker to calculate the probabilities.

56. What is the difference between marginal, conditional and joint probabilities and how are they related? give example.

    P(H=hit) P(H=hit) is the marginal probability. It reads "The probability of getting hit.". It is the proportion of people that got hit crossing the street, irrespective of traffic light. P(H=hit|L=red) P(H=hit|L=red) is the conditional probability. It reads "The probability that you get hit, given that the light is red". It is the proportion of hits among the people that cross the street in red light. Finally, P(H=hit,L=red) P(H=hit,L=red) is the joint probability. It reads "the probability tha…