Cognitive Systems 300: Probability and Causality (cont.)
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1 Cognitive Systems 300: Probability and Causality (cont.) David Poole and Peter Danielson University of British Columbia Fall David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
2 The story so far... Agents act in environments. A controller considers: what should the agent do now what should the agent remember or believe as a function of percepts and previous memory. Memories are symbol structure; reasoning is search. Hierarchical systems reduce complexity. Acting is gambling. probabilities: possible worlds + conditioning Belief networks are a representation of conditional (in)dependence. 2 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
3 Learning Objectives At the end of the week you should be able to: know how to compute marginals and apply Bayes theorem build a belief network for a domain predict the inferences for a belief network explain the predictions of a causal model 3 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
4 Outline 1 Belief network review 2 Understanding Dependence 3 Reading 4 Causality 4 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
5 Components of a belief network A belief network consists of: a directed acyclic graph with nodes labeled with random variables a domain for each random variable a set of conditional probability tables for each variable given its parents (including prior probabilities for nodes with no parents). Represents the conditional (in)dependence assumption: a variable is independent of its non-descendants given its parents. 5 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
6 Constructing belief networks To represent a domain in a belief network, consider: What are the relevant variables? What will be observed? What would we like to find out (query)? What other features make the model simpler? What values should these variables take? What is the relationship between the variables? What independencies are appropriate? Direct dependencies become parent-child arcs. How does each variable depend on its parents? This is expressed in terms of the conditional probabilities. 6 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
7 Example: wet grass on a summer s day Variables: Shoes wet after walking on grass Sprinkler was on last night Grass wet Rained last night Grass shiny and appears to be wet 7 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
8 Example: wet grass on a summer s day Rained Sprinkler on Grass Wet Grass Shiny Shoes Wet 8 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
9 Example: wet grass Variables: Shoes wet after walking on grass Sprinkler was on last night Grass wet Rained last night Grass shiny and appears to be wet What if we didn t know the season? 9 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
10 Example: wet grass Variables: Shoes wet after walking on grass Sprinkler was on last night Grass wet Rained last night Grass shiny and appears to be wet What if we didn t know the season? Season (winter or summer) David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
11 Example: wet grass Season Rained Sprinkler on Grass Wet Grass Shiny Shoes Wet 10 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
12 Outline 1 Belief network review 2 Understanding Dependence 3 Reading 4 Causality 11 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
13 Example: fire alarm belief network Variables: Fire: there is a fire in the building Tampering: someone has been tampering with the fire alarm Smoke: what appears to be smoke is coming from an upstairs window Alarm: the fire alarm goes off Leaving: people are leaving the building en masse. Report: a colleague says that people are leaving the building en masse. (A noisy sensor for leaving.) 12 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
14 Common descendants tampering alarm fire Are tampering and fire dependent (given no observations)? Are tampering and fire dependent given alarm? 13 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
15 Common descendants tampering alarm fire Are tampering and fire dependent (given no observations)? Are tampering and fire dependent given alarm? Intuitively, tampering can explain away fire 13 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
16 Common ancestors fire Are alarm and smoke dependent? Are alarm and smoke dependent given fire? alarm smoke 14 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
17 Common ancestors Are alarm and smoke dependent? Are alarm and smoke dependent given fire? alarm fire smoke Intuitively, fire can explain alarm and smoke; learning one can affect the other by changing the belief in fire. 14 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
18 Chain alarm leaving Are alarm and report dependent? Are alarm and report are dependent given leaving? report 15 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
19 Chain alarm leaving report Are alarm and report dependent? Are alarm and report are dependent given leaving? Intuitively, the only way that the alarm affects report is by affecting leaving. 15 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
20 Understanding independence: example A B C D E F G H I J K L M N O P R Q S 16 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
21 Determining if X is independent of Y given variables Z (D-separation) Given observations of set of variables Z, variables X and Y are dependent given Z if there a sequence of variables V 0, V 1, V 2,..., V n where X = V 0 and Y = V n and there is an arc between each V i 1 and V i, with the following directions of arcs for each V i : V i 1 V i V i+1 where V i is not in Z V i 1 V i V i+1 where V i is not in Z V i 1 V i V i+1 where V i is not in Z V i 1 V i V i+1 where V i is in Z or a descendent of V i is in Z If there is no such sequence, X and Y are independent given Z 17 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
22 Understanding independence: example A B C D E F G H I J K L M N O P R Are G and H dependent given {}? Are G and H dependent given {A}? Are B and C dependent given {}? Are B and C dependent given {R}? Are B and Q dependent given {}? Are B and Q dependent given {N}? Are B and E dependent given {Q}? Are B and F dependent given {Q, P}? 18 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.) Q S
23 Understanding independence: example A B C D E F G H I J K L M N O P R Are G and H dependent given {}? Yes Are G and H dependent given {A}? No Are B and C dependent given {}? No Are B and C dependent given {R}? Yes Are B and Q dependent given {}? Yes Are B and Q dependent given {N}? No Are B and E dependent given {Q}? Yes Are B and F dependent given {Q, P}? Yes 18 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.) Q S
24 Outline 1 Belief network review 2 Understanding Dependence 3 Reading 4 Causality 19 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
25 Outline 1 Belief network review 2 Understanding Dependence 3 Reading 4 Causality 20 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
26 Causality An intervention on a variable changes its value by some mechanism outside of the model. A causal model is a directed model which predicts the effects of interventions. A variable only affects its descendants. The parents of a node are its direct causes. We would expect that a causal model to obey the independence assumption of a belief network. All causal networks are belief networks. Not all belief networks are causal networks. An intervention has a different effect than an observation. 21 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
27 Sprinkler Example Season Rained Sprinkler on Grass Wet Grass Shiny Shoes Wet 22 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
28 Which of the following is a causal model? A) Light is on Switch is up B) Light is on Switch is up C) both D) neither 23 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
29 Which of the following is a causal model? A) Light is on Switch is up B) Light is on Switch is up C) both D) neither Answer: B 23 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
30 Which of the following is a causal model? A) Takes Marijuana Takes Hard Drugs B) Takes Marijuana Takes Hard Drugs C) both D) neither 24 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
31 Which of the following is a causal model? A) Takes Marijuana Takes Hard Drugs B) Takes Marijuana Takes Hard Drugs C) both D) neither Answer: D (Neither makes a correct prediction about interventions) 24 David Poole and Peter Danielson Cognitive Systems 300: Probability and Causality (cont.)
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