Download Reasoning under Uncertainty

Uncertain Reasoning
CPSC 315 – Programming Studio
Spring 2009
Project 2, Lecture 6
Reasoning in Complex
Domains or Situations
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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
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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
Forms of Uncertain Reasoning
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Partially-believed domain features
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E.g. chance of rain = 80%
Probability (focus of today’s lecture)
Other (we will return to this)
Partially-true domain features
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E.g. cloudy = .8
Fuzzy logic (outside scope of this class)
Making Decisions to Meet
Goals
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Decision theory =
Probability theory +
Utility theory
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Decisions – the outcome of system’s reasoning,
actions to take or avoid
Probability – how system reasons
Utility – system’s goals / preferences
Quick Question
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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?
Quick Question 2
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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?
Basics of Probability
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Unconditional or prior probability
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Degree of belief of something being true in
absence of any information
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P (cavity = true) = 0.1 or P (cavity) = 0.1
Implies P (not cavity) = 0.9
Basics of Probability
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Unconditional or prior probability
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Can be for a set of values
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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
Basics of Probability
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Conditional or posterior probability
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Degree of belief of something being true given
knowledge about situation
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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
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This knowledge becomes exponential relative to the
size of the feature set
Bayes’ Rule
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Remember: P (a | b) = P (a ^ b) / P (b)
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Can be rewritten
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Swapping a and b features yields
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P (a ^ b) = P (b | a) * P (a)
Thus
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P (a ^ b) = P (a | b) * P (b)
P (b | a) * P (a) = P (a | b) * P (b)
Rewriting we get Bayes’ Rule
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P (b | a) = P (a | b) * P (b) / P (a)
Reasoning with Bayes’ Rule
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Bayes’ Rule
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P (b | a) = P (a | b) * P (b) / P (a)
Example
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Let’s take
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P (disease) = 0.036
P (test) = 0.04
P (test | disease) = 0.98
P (disease | test) = ?
Reasoning with Bayes’ Rule
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Bayes’ Rule
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P (b | a) = P (a | b) * P (b) / P (a)
Example
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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 %
Reasoning with Bayes’ Rule
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What if test has more false positives
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Still 98% accurate for those with disease
Example
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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 %
Reasoning with Bayes’ Rule
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What if test has more false negatives
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Now 90% accurate for those with disease
Example
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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 %
Combining Evidence
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What happens when we have more than one
piece of evidence
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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
Independence of Events
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Independence of features / events
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Features / events cannot be used to predict each
other
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Probabilistic reasoning works because systems
divide domain into independent sub-domains
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Example: values rolled on two separate die
Example: hair color and food preference
Do not need the exponentially increasing data to
understand interactions
Unfortunately, non-independent sub-domains can
still be huge (have many interacting features)
Conditional Independence
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What happens when we have more than one piece
of evidence
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Conditional independence
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Example: toothache and tool catches on tooth
P (cavity | toothache ^ catch) = ?
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)
Conditional Independence
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This let’s us say
P (toothache ^ catch | cavity)
= P (toothache | cavity) * P (catch | cavity)
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P (cavity | toothache ^ catch) = ?
= P (toothache ^ catch | cavity) * P (cavity)
= P (toothache | cavity) * P (catch | cavity) * P (cavity)
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Avoids requiring system to have data on all
permutations
Difficulty: How true?
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What about a chipped or cracked tooth?
Human Reasoning
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Studies show people, without training and
prompting, do not reason probabilistically
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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
Human Reasoning
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Studies show people, without training, do not reason
probabilistically
Some systems have used non-probabilistic forms of
uncertain reasoning
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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