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Uncertain Reasoning
CPSC 315 – Programming Studio
Spring 2009
Project 2, Lecture 6
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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
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Forms of Uncertain Reasoning
Partially-believed domain features E.g. chance of rain = 80% Probability (focus of today’s lecture) Other (we will return to this)
Partially-true domain features E.g. cloudy = .8 Fuzzy logic (outside scope of this class)
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Making Decisions to Meet Goals
Decision theory = Probability theory +Utility theory
Decisions – the outcome of system’s reasoning, actions to take or avoid
Probability – how system reasons Utility – system’s goals / preferences
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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?
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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?
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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
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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
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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 knowP (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)
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Reasoning with Bayes’ Rule
Bayes’ Rule P (b | a) = P (a | b) * P (b) / P (a)
Example Let’s 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 %
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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 %
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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 %
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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)
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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)
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Conditional Independence
This let’s us sayP (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?
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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
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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