Larry Holder School of EECS Washington State University 1 } - - PowerPoint PPT Presentation
Larry Holder School of EECS Washington State University 1 } Sometimes the truth or falsity of facts in the world is unknown } Sources of agents uncertainty Incompleteness of rules Incorrectness of rules Limited and ambiguous
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} Sometimes the truth or falsity of facts in the
} Sources of agent’s uncertainty
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} Fully observable vs. partially observable } Deterministic vs. stochastic } Episodic vs. sequential } Static vs. dynamic } Discrete vs. continuous } Single agent vs. multiagent
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} P(Pit1,3) = ? } P(Pit2,2) = ? } P(Pit3,1) = ?
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} Choose action A that maximizes expected
} I.e., maximizes Prob(A) * Utility(A) } Prob(A) = probability A will succeed } Utility(A) = value to agent of A’s outcomes
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} A probability model associates a numeric
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} The probability P(a) of a proposition ‘a’ is the
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} What is the probability of the proposition that a
} Consider the 36 possible outcomes of rolling two
} P(Total=8) = P(Die1=2∧Die2=6) +
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} An unconditional or prior probability is the
} A conditional or posterior probability is the
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} Fraction of worlds in which ‘a’ and ‘b’ are true
} P(Total=8 | Die1=2)
} Product rule: P(a ∧ b) = P(a | b) P(b)
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} Random variable: a variable in probability
} Domain: set of possible values for a random
} Probability distribution: probability of each
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Note: Boldfaced
} Conditional probability distribution
} Joint probability distribution
} Full joint probability distribution
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} P(¬a) = 1 – P(a) } P(a ∨ b) = P(a) + P(b) – P(a ∧ b)
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a b ¬a
} Approach #1: Just ask Spock…
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“Errand of Mercy” (1967)
} Given full joint probability distribution, we
} Example: Tooth World
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Toothache = true Toothache = false Catch = true Catch = false Catch = true Catch = false Cavity = true .108 .012 .072 .008 Cavity = false .016 .064 .144 .576
} Answer questions (perform probabilistic
} Examples
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toothache ¬toothache catch ¬catch catch ¬catch cavity .108 .012 .072 .008 ¬cavity .016 .064 .144 .576 Note: For Boolean random variables, write X=true as x, and X=false as ¬x.
} Marginalization is the process of finding the
} Conditioning:
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combinations of values z in {(catch, tooth), (catch, ¬tooth),
(¬catch, tooth), (¬catch, ¬tooth)}
} Conditional probabilities } Note that denominator is the same in both
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} Don’t need normalization constant to
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} General rule } Want to know the probability distribution over
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} Full joint probability distributions are typically
} If we know some variables are independent of
} If two variables X and Y are independent, then
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} For example, add Weather variable to Tooth
} Assuming your teeth don’t affect the weather
} Full joint distribution described by two tables of
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} Foundational rule of probabilistic reasoning } Turns a diagnostic question into a causal one } In general, given some evidence e:
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Thomas Bayes (1701-1761)
} Example: Diagnosing cancer } Given
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} Example: Diagnosing cancer } Variables
} Given
} P(cancer|pos)=? } P(cancer|pos) = P(cancer ∧ pos) / P(pos) = ?
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} Applying Bayes rule…
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} Could also compute P(pos) via normalization } P(cancer|pos) = αP(pos|cancer)P(cancer) = } P(healthy|pos) = αP(pos|healthy)P(healthy) = } α = } P(pos) = 1/α = } Many times, P(effect|cause) is easier to determine
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} Ci = car behind Door i } Oi = opens Door i } Pick Door 2 } Opens Door 3 } Should you change your guess? } P(C1 | O3) = ?
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“Numb3rs” S1E13, 2005.
} How to compute P(a | b ^ c ^ …)?
} Easy if we have full joint probability
} Or, using Bayes rule: a[P(b ^ c ^ … | a) P(a)]
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} If b, c, … are “caused” by ‘a’, but not by each
} I.e., b, c, … are independent given ‘a’ } Two propositions (effects) are conditionally
} Now we only need to know the probabilities of
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} Example (assuming toothache and catch are
} Using this in Bayes rule
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} In general, if random variables X and Y are
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} Want P(Cause | Effect1,Effect2,…,Effectn) } Assume Effect1,Effect2,…,Effectn are
} Applying Bayes Rule
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} Be careful
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} P(Pit1,3) = ? } P(Pit2,2) = ? } P(Pit3,1) = ?
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} “query” = Pit1,3 } “frontier” = {Pit2,2 , Pit3,1} } “other” = other 10 pit variables } “known” = ¬pit1,1 ^ ¬pit1,2 ^ ¬pit2,1 } “breeze” = ¬breeze1,1 ^ breeze1,2 ^ breeze2,1 } Note: “breeze” is conditionally independent of
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} And given independence of pits Pi,j } P(P1,3 | known, breeze) =
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P1,3=true P1,3=false
} So, P(P1,3=true) = P(P3,1=true) = 0.31 } P(P2,2=true) = 0.86 } Probabilistic agent “knows more” than logical
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} Probability theory allows us to reason about
} Bayes rule allows us to change diagnostic
} Conditional independence allows us to
} Probabilistic agent can outperform logical
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