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## CS 416 Artificial Intelligence

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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

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