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CS3105 AI > 4: Bayesian Reasoning > Flashcards

4: Bayesian Reasoning Flashcards

1
Q

What is Bayesian Reasoning?

A

Bayesian reasoning is an application of probability theory to inductive reasoning. Bayes’ rule/theorem states that P(A | B) = [P(B | A) * P(A)] / P(B)

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2
Q

What is probabilistic reasoning?

A

Using the power of probability theory to handle uncertainty and still facilitate conclusions from deductive logic from a formal argument.

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3
Q

What are Bayesian Networks?

A

A directed acyclic graph that models the conditional dependencies of different variables in a statistical system.

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4
Q

What is Bayesian probabilistic reasoning?

A

The use of Bayes’ rule and Bayesian networks in application to probabilistic reasoning (and in turn deductive logic).

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5
Q

What are some sources of uncertainty?

A
  1. Partial observability
    • To give a complete model we would have to include
    many facts we don’t know, such as road state, other
    drivers’ plans, etc.
  2. Noisy sensors
    • Information we can get can be inaccurate, e.g. a traffic report claiming free roads when there is a jam
  3. Uncertainty in action outcomes
    • The road bridge might close at short notice
  4. immense complexity of modelling and
    predicting traffic
    • Even if we had complete information, it may be
    intractable to compute a valid prediction
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6
Q

Why do we use probability methods rather than logical ones to deal with uncertainty?

A

???

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7
Q

What is a sample space? How is it notated?

A

The set of possible values of a random variable, notated as Ω.

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8
Q

What is an atomic event? How is it notated?

A

???

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9
Q

What is a probability space? How is it notated?

A

???

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10
Q

What is an event? How is it defined?

A

???

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11
Q

How can you find the probability of an event?

A

???

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12
Q

What is a random variable?

A

A variable (set of different possible values) whose value depends on the outcome of a random (unpredictable, without pattern) phenomenon.

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13
Q

What is a probability distribution?

A

???

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14
Q

What are Kolmogorov’s Axioms for Probability?

A

For any event A

  1. 0 <= P(A) <= 1
  2. P(true) = 1, P(false) = 0
  3. P(¬A) = 1 - P(A)
  4. P(A ∨ B) = P(A) + P(B) - P(A ⋀ B)
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15
Q

What is De Finetti’s Argument?

A

???

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16
Q

What is a joint probability distribution?

A

???

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17
Q

What is a complete joint probability distribution?

A

???

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18
Q

What can a complete joint probability distribution do?

A

Calculate any atomic event probability

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19
Q

What is a complex event probability?

A

???

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20
Q

What are Conditional Probabilities?

A

Probabilities of events happen given that other event/s have happened beforehand.

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21
Q

What does the vertical bar | notate?

A

Conditional probabilities.

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22
Q

How can we mathematically define Conditional Probabilities?

A

P(A | B) = P(A ⋀ B) ÷ P(B)

23
Q

What is the Product Rule?

A

P(A ⋀ B) = P(A | B) * P(B)

24
Q

How do you generalise the Product Rule to multiple conditional probabilities?

A

P(A ⋀ B) = P(A | B) * P(B)

TO

P(A ⋀ B ⋀ C) = P(A | B ⋀ C) * P(B ⋀ C) = P(A | B ⋀ C) * P(B | C) * P(C)

25
Q

What does it mean for two events to be independent? How can you define it mathematically?

A

Two events are independent when the outcome of one does not affect the outcome of the other.

Events A and B are independent if P(A | B) = P(A) and P(B | A) = P(B).

26
Q

How does the product rule develop for two independent events?

A

Initially, P(A ⋀ B) = P(A | B) * P(B)

If A and B are independent, P(A | B) = P(A)

Therefore P(A ⋀ B) = P(A | B) * P(B) = P(A) * P(B)

In general, P(A ⋀ B ⋀ C ⋀ … ⋀ Z) = P(A) * P(B) * P(C)…P(Z)
if that all of A, B, C,…Z are all independent

27
Q

Does the conditional probability link to causation?

A

No.

28
Q

What does it mean for events to be atomic?

A

???

29
Q

What is a Cartesian product? How do you find it?

A

How you multiply two sets or matrices. Just make a new set or matrix of the needed number of rows and columns, and in each cell multiply the row and columns of each they come from.

30
Q

How do you find the sample space of multiple events?

A

Find the Cartesian product of the sample space for each individual event. This is usually just multiplying the size of the sample space of each event.

31
Q

When may we assume events or event repetition are not independent?

A

When we are presented with:

  • a maths reason to believe so
  • a non-maths reason to believe so
  • a situation in which it is a reasonable assumption that makes our calculations easier
32
Q

How do you find the probability of a non-atomic event, A?

A

P(A) = Σ(ω∊A) * P(ω) = Σ(ω∊A)

33
Q

What is conditional independence? How can you define it mathematically?

A

When two events are independent given that another event has a certain outcome.

A and B are conditionally independent given B for:
P(A , C | B) = P(A | B) * P(C | B)

34
Q

How do we get to Bayes Rule?

A

P(A | B) = P(A ⋀ B) ÷ P(B)

SO

P(A ⋀ B) = P(A | B) * P(B)

AND

P(A ⋀ B) = P(B | A) * P(A)

SO

P(B) * P(A | B) = P(B | A) * P(A)

SO

P(A | B) = [P(B | A) * P(A)] ÷ P(B)

35
Q

What are causal conditional probabilities?

A

???

36
Q

What are diagnostic conditional probabilities?

A

???

37
Q

What is the difference between causal conditional probabilities and diagnostic conditional probabilities?

A

???

38
Q

What is the significance of Baye’s rule?

A

???

39
Q

How can we apply Baye’s rule to medical diagnoses?

A

???

40
Q

How do you build a Bayesian Network?

A

???

41
Q

What is a conditional probability table? How can one be built from a Bayesian Network?

A

???

42
Q

How many numbers do you need to fully fill out a conditional probability table? How does the process work?

A

5 (?) ??? something to do with conditional independence

43
Q

How does conditional independence work in Bayesian networks?

A

Each node is independent of all nondescendants (other nodes that are not its children or children of its children, etc) given that all the parent nodes of the node are true.

44
Q

What are domain values of nodes in Bayesian networks?

A

???

45
Q

How big is the probability distribution of a Bayesian network?

A

The full probability distribution has (d ^ n) - 1

numbers, where n = the number of nodes in the network, and each node has d domain values.

46
Q

How many numbers do you need to define the full probability distribution of a Bayesian network? Why is this significant?

A

At most n * d ^ p numbers

Since each node has at most p parents, and the full probability distribution has (d ^ n ) - 1
numbers, since there are n nodes in a network, and each node has d domain values.

This is significant because it allows a large reduction in data structure sizes and the resultant complexity of computation.

47
Q

What is the full probability distribution of a Bayesian network?

A

???

48
Q

What is a directed acyclic graph?

A

???

49
Q

What type of graph are Bayesian networks?

A

Directed acyclic graphs.

50
Q

How do you construct Bayesian Networks?

A

???

51
Q

What is Causal Ordering?

A

???

52
Q

What is a Markov Blanket for a node?

A

The area in a graph (could be thought of a subgraph) which contains all other nodes which alter the state of node to whom the Markov Blanket is associated. This means nodes outside the Markov Blanket have no effect on the node in question no matter their states.

53
Q

How do Markov Blankets apply to independence in Bayesian Networks?

A

If all the nodes in a Markov Blanket for a node are true, the node is conditionally independent of all nodes outside the Markov Blanket.

54
Q

What are Inference tasks, and how do they relate to Bayesian networks?

A

???

CS3105 AI (8 decks)

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