probability. uncertainty let action a t = leave for airport t minutes before flight from logan...
TRANSCRIPT
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PROBABILITY
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Uncertainty
Let action At = leave for airport t minutes before flight from Logan Airport
Will At get me there on time?
Problems:1. Partial observability (road state, other drivers'
plans, etc.).2. Noisy sensors (traffic reports).3. Uncertainty in action outcomes (flat tire, etc.).4. Immense complexity of modeling and
predicting traffic.
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Uncertainty
Let action At = leave for airport t minutes before flight from Logan Airport
Will At get me there on time?
A purely logical approach either:1) risks falsehood: “A120 will get me there on
time,” or 2) leads to conclusions that are too weak for
decision making:“A120 will get me there on time if there's no
accident on I-90and it doesn't rain and my tires remain intact,
etc., etc.”
(A1440 might reasonably be said to get me there on time but I'd have to stay overnight in the airport …)
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Methods for handling uncertainty Default logic:
Assume my car does not have a flat tire Assume A120 works unless contradicted by evidence Issues: What assumptions are reasonable? How to
handle contradiction?
Probability Model agent's degree of belief Given the available evidence, A120 will get me there on
time with probability 0.4
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Probability
Probabilistic assertions summarize effects of laziness: failure to enumerate exceptions, qualifications,
etc. ignorance: lack of relevant facts, initial conditions, etc.
Subjective probability: Probabilities relate propositions to agent's own state
of knowledge e.g., P(A120 | no reported accidents) = 0.6
These are not assertions about the world, but represent belief about the whether the assertion is true.
Probabilities of propositions change with new evidence: e.g., P(A120 | no reported accidents, 5 a.m.) = 0.15
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Making decisions under uncertainty
Suppose I believe the following:P(A 135 gets me there on time | ...) = 0.04
P(A 180 gets me there on time | ...) = 0.70
P(A 240 gets me there on time | ...) = 0.95
P(A 1440 gets me there on time | ...) = 0.9999
Which action to choose? Depends on my preferences for missing flight vs.
airport cuisine, etc.
Utility theory is used to represent and infer preferences
Decision theory = utility theory + probability theory
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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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Syntax
Basic element: random variable
Boolean random variables e.g., Cavity (do I have a cavity?)
Discrete random variables e.g., Weather is one of
<sunny,rainy,cloudy,snow> Continuous random variables
represented by probability density functions (PDFs), valued [0,1]
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Syntax
Elementary propositions constructed by assignment of a value to a random variable e.g., Weather = sunny, Cavity = false (abbreviated as sunny or cavity)
Complex propositions formed from elementary propositions and standard logical connectives e.g., (Weather = sunny Cavity = false) e.g., (sunny cavity)
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Syntax
Atomic event: A complete specification of the state of the world. A complete assignment of values to variables.
E.g., if the world consists of only two Boolean variables Cavity and Toothache, then there are 4 distinct atomic events:
Cavity = false Toothache = falseCavity = false Toothache = trueCavity = true Toothache = falseCavity = true Toothache = true
Atomic events are mutually exclusive and exhaustive
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Axioms of probability (Kolmogorov’s Axioms)
For any propositions A, B1. 0 ≤ P(A) ≤ 12. P(true) = 1 and P(false) = 03. P(A B) = P(A) + P(B) - P(A B)
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Prove
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Prove
(Axiom 3 with )
(by logical equivalence)
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Priors
Prior or unconditional probabilities of propositions correspond to belief prior to arrival of any (new) evidence.
e.g., P(cavity) =P(Cavity=true)= 0.1 P(Weather=sunny) = 0.72P(cavity Weather=sunny) = 0.072
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Distributions
Probability distribution gives probabilities of all possible values of the random variable. Weather is one of <sunny,rain,cloudy,snow> P(Weather) = <0.72, 0.1, 0.08, 0.1> (Normalized, i.e., sums to 1. Also note the bold
font..)
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Joint Probability
Joint probability distribution for a set of random variables:
The entire table sums to 1.
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Inference by Enumeration
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Inference by Enumeration
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Conditional Probability
Conditional or posterior probability Degree of belief of something being true
given knowledge about situation
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Inference by Enumeration
=0.4
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Conditional Probability
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Absolute Independence
If A and B are independent
𝑃 ( h h𝑇𝑜𝑜𝑡 𝑎𝑐 𝑒 , h𝐶𝑎𝑡𝑐 ,𝐶𝑎𝑣𝑖𝑡𝑦 , h𝑊𝑒𝑎𝑡 𝑒𝑟 )=𝑃 ( h h𝑇𝑜𝑜𝑡 𝑎𝑐 𝑒 , h𝐶𝑎𝑡𝑐 ,𝐶𝑎𝑣𝑖𝑡𝑦 )𝑃 ( h𝑊𝑒𝑎𝑡 𝑒𝑟 )
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Absolute Independence
Absolute independence powerful but rare For n independent biased coins, O(2n) →O(n)
2*2*2*4=32 entries 2*2*2+4=12 entries
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Conditional independence
If I have a cavity, the probability that the probe catches in it doesn't depend on whether I have a toothache: P(catch | toothache, cavity) = P(catch | cavity)
The same independence holds if I haven't got a cavity: P(catch | toothache,cavity) = P(catch | cavity)
Catch is conditionally independent of Toothache given Cavity: P(Catch | Toothache,Cavity) = P(Catch | Cavity)
Equivalent statements:P(Toothache | Catch, Cavity) = P(Toothache | Cavity)P(Toothache, Catch | Cavity) = P(Toothache | Cavity) P(Catch | Cavity)
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Bayes' Rule
Product rule
Divide both sides by
Bayes' rule
or in distribution form
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Bayes' Rule
Much easier to calculate/estimate than the part on the left!
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Bayes Rule: Example
Let M be meningitis, and S be stiff neck, with
0.00080.1
0.00010.8
P(s)
P(m)m)|P(ss)|P(m
Note: probability of meningitis still very small!
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Bayes' Rule and conditional independence
[by Bayes’ Rule]
[by conditional independence]
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Naïve Bayes Model
A single cause influences several effects, all of which are conditionally independent
Simplifying assumption! Usually incorrect, but works well in practice.
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Summary
Probability is a formalism for uncertain knowledge Basic probability statements include prior probabilities and
conditional probabilities The full joint probability distribution specifies probability of
each complete assignment of values to variables. Usually too large to create and use in its explicit form.
Absolute independence between subsets of random variables allows the full joint distribution to be factored into smaller joint distributions. Absolute independent rarely occurs in practice.
Bayes’ rule allows unknown probabilities to be computed from known conditional probabilities. Applying Bayes’ rule with many variables runs into the same
scaling problem as above. Conditional independence, brought about by direct causal
relationships in the domain, may allow the full joint distribution to be factored into smaller, conditional distributions. Naïve Bayes model assumes conditional independence of all effect
variables.