CS 188: Artificial Intelligence Spring 2007

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CS 188: Artificial Intelligence Spring 2007. Lecture 11: Probability 2/20/2007. Srini Narayanan – ICSI and UC Berkeley. Announcements. HW1 graded Solutions to HW 2 posted Wednesday HW 3 due Thursday 11:59 PM. Today. Probability Random Variables Joint and Conditional Distributions
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CS 188: Artificial IntelligenceSpring 2007Lecture 11: Probability2/20/2007Srini Narayanan – ICSI and UC BerkeleyAnnouncements
  • HW1 graded
  • Solutions to HW 2 posted Wednesday
  • HW 3 due Thursday 11:59 PM
  • Today
  • Probability
  • Random Variables
  • Joint and Conditional Distributions
  • Bayes Rule
  • Independence
  • You’ll need all this stuff for the next few weeks, so make sure you go over it!
  • What is this?Uncertainty Uncertainty
  • Let action At = leave for airport t minutes before flight
  • Will At get me there on time?
  • Problems:
  • partial observability (road state, other drivers' plans, etc.)
  • noisy sensors (KCBS traffic reports)
  • uncertainty in action outcomes (flat tire, etc.)
  • immense complexity of modeling and predicting traffic
  • A purely logical approach either
  • Risks falsehood: “A25 will get me there on time” or
  • Leads to conclusions that are too weak for decision making:
  • “A25 will get me there on time if there's no accident on the bridge, and 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…
  • Probabilities
  • Probabilistic approach
  • Given the available evidence, A25 will get me there on time with probability 0.04
  • P(A25 | no reported accidents) = 0.04
  • Probabilities change with new evidence:
  • P(A25 | no reported accidents, 5 a.m.) = 0.15
  • P(A25 | no reported accidents, 5 a.m., raining) = 0.08
  • i.e., observing evidence causes beliefs to be updated
  • Probabilities Everywhere?
  • Not just for games of chance!
  • I’m snuffling: am I sick?
  • Email contains “FREE!”: is it spam?
  • Tooth hurts: have cavity?
  • Safe to cross street?
  • 60 min enough to get to the airport?
  • Robot rotated wheel three times, how far did it advance?
  • Why can a random variable have uncertainty?
  • Inherently random process (dice, etc)
  • Insufficient or weak evidence
  • Unmodeled variables
  • Ignorance of underlying processes
  • The world’s just noisy!
  • Probabilistic Models
  • CSP/Prop Logic:
  • Variables with domains
  • Constraints: map from assignments to true/false
  • Ideally: only certain variables directly interact
  • Probabilistic models:
  • (Random) variables with domains
  • Joint distributions: map from assignments (or outcomes) to positive numbers
  • Normalized: sum to 1.0
  • Ideally: only certain variables are directly correlated
  • Random Variables
  • A random variable is some aspect of the world about which we have uncertainty
  • R = Is it raining?
  • D = How long will it take to drive to work?
  • L = Where am I?
  • We denote random variables with capital letters
  • Like in a CSP, each random variable has a domain
  • R in {true, false}
  • D in [0, ]
  • L in possible locations
  • Distributions on Random Vars
  • A joint distribution over a set of random variables:
  • is a map from assignments (or outcomes, or atomic events) to reals:
  • Size of distribution if n variables with domain sizes d?
  • Must obey:
  • For all but the smallest distributions, impractical to write out
  • Examples
  • An event is a set E of assignments (or outcomes)
  • From a joint distribution, we can calculate the probability of any event
  • Probability that it’s warm AND sunny?
  • Probability that it’s warm?
  • Probability that it’s warm OR sunny?
  • Marginalization
  • Marginalization (or summing out) is projecting a joint distribution to a sub-distribution over subset of variables
  • Conditional Probabilities
  • A conditional probability is the probability of an event given another event (usually evidence)
  • Conditional Probabilities
  • Conditional or posterior probabilities:
  • E.g., P(cavity | toothache) = 0.8
  • Given that toothache is all I know…
  • Notation for conditional distributions:
  • P(cavity | toothache) = a single number
  • P(Cavity, Toothache) = 2x2 table summing to 1
  • P(Cavity | Toothache) = Two 2-element vectors, each summing to 1
  • If we know more:
  • P(cavity | toothache, catch) = 0.9
  • P(cavity | toothache, cavity) = 1
  • Note: the less specific belief remains valid after more evidence arrives, but is not always useful
  • New evidence may be irrelevant, allowing simplification:
  • P(cavity | toothache, traffic) = P(cavity | toothache) = 0.8
  • This kind of inference, guided by domain knowledge, is crucial
  • Conditioning
  • Conditional probabilities are the ratio of two probabilities:
  • Normalization Trick
  • A trick to get the whole conditional distribution at once:
  • Get the joint probabilities for each value of the query variable
  • Renormalize the resulting vector
  • NormalizeSelectThe Product Rule
  • Sometimes joint P(X,Y) is easy to get
  • Sometimes easier to get conditional P(X|Y)
  • Example: P(sun, dry)?
  • Lewis Carroll's Sack Problem
  • Sack contains a red or blue token, 50/50
  • We add a red token
  • If we draw a red token, what’s the
  • chance of drawing a second red token?
  • Variables:
  • F={r,b} is the original token
  • D={r,b} is the first token we draw
  • Query: P(F=r|D=r)
  • Lewis Carroll's Sack Problem
  • Now we have P(F,D)
  • Want P(F=r|D=r)
  • Bayes’ Rule
  • Two ways to factor a joint distribution over two variables:
  • Dividing, we get:
  • Why is this at all helpful?
  • Lets us invert a conditional distribution
  • Often the one conditional is tricky but the other simple
  • Foundation of many systems we’ll see later (e.g. ASR, MT)
  • In the running for most important AI equation!
  • That’s my rule!More Bayes’ Rule
  • Diagnostic probability from causal probability:
  • Example:
  • m is meningitis, s is stiff neck
  • Note: posterior probability of meningitis still very small
  • Note: you should still get stiff necks checked out! Why?
  • Inference by Enumeration
  • P(sun)?
  • P(sun | winter)?
  • P(sun | winter, warm)?
  • Inference by Enumeration
  • General case:
  • Evidence variables:
  • Query variables:
  • Hidden variables:
  • We want:
  • First, select the entries consistent with the evidence
  • Second, sum out H:
  • Finally, normalize the remaining entries to conditionalize
  • Obvious problems:
  • Worst-case time complexity O(dn)
  • Space complexity O(dn) to store the joint distribution
  • All variablesIndependence
  • Two variables are independent if:
  • This says that their joint distribution factors into a product two simpler distributions
  • Independence is a modeling assumption
  • Empirical joint distributions: at best “close” to independent
  • What could we assume for {Weather, Traffic, Cavity}?
  • How many parameters in the joint model?
  • How many parameters in the independent model?
  • Independence is like something from CSPs: what?
  • Example: Independence
  • N fair, independent coin flips:
  • Example: Independence?
  • Arbitrary joint distributions can be poorly modeled by independent factors
  • Conditional Independence
  • P(Toothache,Cavity,Catch) has 23 = 8 entries (7 independent entries)
  • 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 don’t have 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)
  • Conditional Independence
  • Unconditional (absolute) independence is very rare (why?)
  • Conditional independence is our most basic and robust form of knowledge about uncertain environments:
  • What about this domain:
  • Traffic
  • Umbrella
  • Raining
  • What about fire, smoke, alarm?
  • The Chain Rule II
  • Can always factor any joint distribution as an incremental product of conditional distributions
  • Why?
  • This actually claims nothing…
  • What are the sizes of the tables we supply?
  • The Chain Rule III
  • Trivial decomposition:
  • With conditional independence:
  • Conditional independence is our most basic and robust form of knowledge about uncertain environments
  • Graphical models (next class) will help us work with independence
  • The Chain Rule IV
  • Write out full joint distribution using chain rule:
  • P(Toothache, Catch, Cavity)
  • = P(Toothache | Catch, Cavity) P(Catch, Cavity) = P(Toothache | Catch, Cavity) P(Catch | Cavity) P(Cavity) = P(Toothache | Cavity) P(Catch | Cavity) P(Cavity)CavP(Cavity)
  • Graphical model notation:
  • Each variable is a node
  • The parents of a node are the other variables which the decomposed joint conditions on
  • MUCH more on this to come!
  • TCatP(Toothache | Cavity)P(Catch | Cavity)Combining Evidence P(cavity | toothache, catch) =  P(toothache, catch | cavity) P(cavity) =  P(toothache | cavity) P(catch | cavity) P(cavity)
  • This is an example of a naive Bayes model:
  • CE1E2En
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