Uncertain Reasoning

1
Uncertain Reasoning

• CPSC 315 Programming Studio
• Spring 2009
• Project 2, Lecture 6

2
Reasoning in Complex Domains or Situations

• Reasoning often involves moving from evidence
  about the world to decisions
• Systems almost never have access to the whole
  truth about their environment
• Reasons for lack of knowledge
• Cost/benefit trade-off in knowledge engineering
• Less likely, less influential factors often not
  included in model
• No complete theory of domain
• Complete theories are few and far between
• Incomplete knowledge of situation
• Acquiring all knowledge of situation is
  impractical

3
Forms of Uncertain Reasoning

• Partially-believed domain features
• E.g. chance of rain 80
• Probability (focus of todays lecture)
• Other (we will return to this)
• Partially-true domain features
• E.g. cloudy .8
• Fuzzy logic (outside scope of this class)

4
Making Decisions to Meet Goals

• Decision theory Probability theory Utility
  theory
• Decisions the outcome of systems reasoning,
  actions to take or avoid
• Probability how system reasons
• Utility systems goals / preferences

5
Quick Question

• You go to the doctor and are tested for a
  disease. The test is 98 accurate if you have
  the disease. 3.6 of the population has the
  disease while 4 of the population tests
  positive.
• How likely is it you have the disease?

6
Quick Question 2

• You go to the doctor and are tested for a
  disease. The test is 98 accurate if you have
  the disease. 3.6 of the population has the
  disease while 7 of the population tests
  positive.
• How likely is it you have the disease?

7
Basics of Probability

• Unconditional or prior probability
• Degree of belief of something being true in
  absence of any information
• P (cavity true) 0.1 or P (cavity) 0.1
• Implies P (not cavity) 0.9

8
Basics of Probability

• Unconditional or prior probability
• Can be for a set of values
• P (Weather sunny) 0.7
• P (Weather rain) 0.2
• P (Weather cloudy) .08
• P (Weather snow) .02
• Note Weather can have only a single value
  system must know that rain and snow implies clouds

9
Basics of Probability

• Conditional or posterior probability
• Degree of belief of something being true given
  knowledge about situation
• P (cavity toothache) 0.8
• Mathematically, we know P (a b) P (a b) /
  P (b)
• Requires system to know unconditional probability
  of combinations of features
• This knowledge becomes exponential relative to
  the size of the feature set

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Bayes Rule

• Remember P (a b) P (a b) / P (b)
• Can be rewritten
• P (a b) P (a b) P (b)
• Swapping a and b features yields
• P (a b) P (b a) P (a)
• Thus
• P (b a) P (a) P (a b) P (b)
• Rewriting we get Bayes Rule
• P (b a) P (a b) P (b) / P (a)

11
Reasoning with Bayes Rule

• Bayes Rule
• P (b a) P (a b) P (b) / P (a)
• Example
• Lets take
• P (disease) 0.036
• P (test) 0.04
• P (test disease) 0.98
• P (disease test) ?

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Reasoning with Bayes Rule

• Bayes Rule
• P (b a) P (a b) P (b) / P (a)
• Example
• P (disease) 0.036
• P (test) 0.04
• P (test disease) 0.98
• P (disease test) ?
• P (test disease) P
  (disease) / P (test)
• 0.98 0.036 / 0.04
• 88.2

13
Reasoning with Bayes Rule

• What if test has more false positives
• Still 98 accurate for those with disease
• Example
• P (disease) 0.036
• P (test) 0.07
• P (test disease) 0.98
• P (disease test) ?
• P (test disease) P
  (disease) / P (test)
• 0.98 0.036 / 0.07
• 50.4

14
Reasoning with Bayes Rule

• What if test has more false negatives
• Now 90 accurate for those with disease
• Example
• P (disease) 0.036
• P (test) 0.04
• P (test disease) 0.90
• P (disease test) ?
• P (test disease) P
  (disease) / P (test)
• 0.90 0.036 / 0.04
• 81

15
Combining Evidence

• What happens when we have more than one piece of
  evidence
• Example toothache and tool catches on tooth
• P (cavity toothache catch) ?
• Problem toothache and catch are not independent
• If someone has a toothache there is a greater
  chance they will have a catch and vice-versa

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Independence of Events

• Independence of features / events
• Features / events cannot be used to predict each
  other
• Example values rolled on two separate die
• Example hair color and food preference
• Probabilistic reasoning works because systems
  divide domain into independent sub-domains
• Do not need the exponentially increasing data to
  understand interactions
• Unfortunately, non-independent sub-domains can
  still be huge (have many interacting features)

17
Conditional Independence

• What happens when we have more than one piece of
  evidence
• Example toothache and tool catches on tooth
• P (cavity toothache catch) ?
• Conditional independence
• Assume indirect relationship
• Example toothache and catch are both caused by
  cavity but not any other feature
• Then P (toothache catch cavity) P
  (toothache cavity) P (catch cavity)

18
Conditional Independence

• This lets us say
• P (toothache catch cavity)
• P (toothache cavity) P (catch cavity)
• P (cavity toothache catch) ?
• P (toothache catch cavity) P (cavity)
• P (toothache cavity) P (catch cavity)
  P (cavity)
• Avoids requiring system to have data on all
  permutations
• Difficulty How true?
• What about a chipped or cracked tooth?

19
Human Reasoning

• Studies show people, without training and
  prompting, do not reason probabilistically
• People make incorrect inferences when confronted
  with probabilities like those of the last few
  slides
• If asked for all prior and posterior
  probabilities then they will posit systems with
  rather large inconsistencies

20
Human Reasoning

• Studies show people, without training, do not
  reason probabilistically
• Some systems have used non-probabilistic forms of
  uncertain reasoning
• Qualitative categories rather than numbers
• Must be true, highly likely, likely, some chance,
  unlikely, virtually impossible, impossible
• Rules for how these combine based on human
  reasoning
• Value depends on where belief values come from
• If belief values from external evidence about
  world then use probability
• If belief values provided by user then
  non-probabilistic approach may do better