CS 416 Artificial Intelligence

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# CS 416 Artificial Intelligence - PowerPoint PPT Presentation

CS 416 Artificial Intelligence. Lecture 14 Uncertainty Chapter 13. An apology to Red Sox fans. The only team ever in baseball to take a 3-0 series to a game seven I was playing the probabilities…. Shortcomings of first-order logic. Consider dental diagnosis

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## CS 416 Artificial Intelligence

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### CS 416Artificial Intelligence

Lecture 14

Uncertainty

Chapter 13

An apology to Red Sox fans
• The only team ever in baseball to take a 3-0 series to a game seven
• I was playing theprobabilities…
Shortcomings of first-order logic
• Consider dental diagnosis
• Not all patients with toothaches have cavities. There are other causes of toothaches
Shortcomings of first-order logic
• What’s wrong with this?
• An unlimited number of toothache causes
Shortcomings of first-order logic
• Alternatively, create a causal rule
• Again, not all cavities cause pain. Must expand
Shortcomings of first-order logic
• Both diagnostic and causal rules require countless qualifications
• Difficult to be exhaustive
• Too much work
• We don’t know all the qualifications
• Even correctly qualified rules may not be useful if the real-time application of the rules is missing data
Shortcomings of first-order logic
• As an alternative to exhaustive logic…
• Probability Theory
• Serves as a hedge against our laziness and ignorance
Degrees of belief
• I believe the glass is full with 50% chance
• Note this does not indicate the statement is half-true
• We are not talking about a glass half-full
• “The glass is full” is the only statement being considered
• My statement indicates I believe with 50% that the statement is true. There are no claims about what other beliefs I have regarding the glass.
• Fuzzy logic handles partial-truths
Decision Theory
• What is rational behavior in context of probability?
• Pick answer that satisfies goals with highest probability of actually working?
• Sometimes more risk is acceptable
• Must have a utility function that measures the many factors related to an agent’s happiness with an outcome
• An agent is rational if and only if it chooses the action that yields the highest expected utility, averaged over all the possible outcomes of the action
Building probability notation
• Propositions
• Like propositional logic. The things we believe
• Atomic Events
• A complete specification of the state of the world
• Prior Probability
• Probability something is true in absence of other data
• Conditional Probability
• Probability something is true given something else is known
Propositions
• Like propositional logic
• Random variables refer to parts of the world with unknown status
• Random variables have a well-defined domain
• Boolean
• Discrete (countable)
• Continuous
Atomic events
• A complete specification of the world
• All variables in the world are assigned values
• Only one atomic event can be true
• The set of all atomic events is exhaustive – at least one must be true
• Any atomic even entails the truth or falsehood of every proposition
Prior probability
• The degree of belief in the absence of other info
• P (Weather)
• P (Weather == sunny) = 0.7
• P (Weather == rainy) = 0.2
• P (Weather == cloudy) = 0.08
• P (Weather == snowy) = 0.02
• P (Weather) = <0.7, 0.2, 0.08, 0.02>
• Probability distribution for the random variable Weather
Prior probability - Discrete
• Joint probability distribution
• P (Weather, Natural Disaster) = an n x m table of probs
• n = instances of weather
• m = instances of natural disasters
• Full joint probability distribution
• Probabilities for all variables are established
• What about continuous variables where a table won’t suffice?
Prior probability - Continuous
• Probability density functions (PDFs)
• P (X = x) = Uniform [18, 26] (x)
• The probability that tomorrow’s temperature is 20.5 degrees Celsius is U [18, 26] (20.5) = 0.125
Conditional probability
• The probability of a given all we know is b
• P (a | b)
• Written as an unconditional probability
Axioms of probability
• All probabilities are between 0 and 1
• Necessarily true propositions have probability 1Necessarily false propositions have probability 0
• The probability of disjunction is:
Using axioms of probability
• The probability of a proposition is equal to the sum of the probabilities of the atomic events in which it holds:
An example
• Maginalization:
• Conditioning:
Normalization
• Two previous calculations had the same denominator
• P(cavity | toothache) = a P(cavity, toothache)
• = a [P(cavity, toothache, catch) + P(cavity, toothache, ~catch)]
• = a [<0.108, 0.016> + <0.012, 0.064>] = a<0.12, 0.08> = <0.6, 0.4>
• Generalized (X = cavity, e = toothache, y = catch)
• P (X, e, y) is a subset of the full joint distribution
Using the full joint distribution
• It does not scale well…
• n Boolean variables
• Table size O (2n)
• Process time O (2n)
Independence
• Independence of variables in a domain can dramatically reduce the amount of information necessary to specify the full joint distribution
• Adding weather (four states) to this table requires creating four versions of it (one for each weather state) = 8*4=32 cells
Independence
• P (toothache, catch, cavity, Weather=cloudy) = P(Weather=cloudy | toothache, catch, cavity) * P(toothache, catch, cavity)
• Because weather and dentistry are independent
• P (Weather=cloudy | toothache, catch, cavity) = P (Weather = cloudy)
• P (toothache, catch, cavity, Weather=cloudy) = P(Weather=cloudy) * P(toothache, catch, cavity)4-cell table 8-cell table
Bayes’ Rule
• Useful when you know three things and need to know the fourth
Example
• Meningitis
• Doctor knows meningitis causes stiff necks 50% of the time
• Doctor knows unconditional facts
• The probability of having meningitis is 1 / 50,000
• The probability of having a stiff neck is 1 / 20
• The probability of having meningitis given a stiff neck:
Power of Bayes’ rule
• Why not collect more diagnostic evidence?
• Statistically sample to learn P (m | s) = 1 / 5,000
• If P(m) changes… due to outbreak… Bayes’ computation adjusts automatically, but sampled P(m | s) is rigid
Conditional independence
• Consider the infeasibility of full joint distributions
• We must know P(toothache and catch) for all Cavity values
• Simplify using independence
• Toothache and catch are not independent
• Toothache and catch are independent given the presence or absence of a cavity
Conditional independence
• Toothache and catch are independent given the presence or absence of a cavity
• If you know you have a cavity, there’s no reason to believe the toothache and the dentist’s pick are related
Conditional independence
• In general, when a single cause influences multiple effects, all of which are conditionally independent (given the cause)
Naïve Bayes
• Even when “effect” variables are not conditionally independent, this model is sometimes used
• Sometimes called a Bayesian Classifier