Warm-up. Probability in the movies

Transcription

1 Warm-up Probability in the movies

2 Announcements Upcoming due dates Wednesday 10/14 11:59pm Homework 5 Friday 10/16 5:00 pm Project 2 Probability sessions Continuing for the next couple of weeks, including tomorrow Wednesdays, 9-10 am in 299 Cory and 6-7 pm in 531 Cory Contest Results on Thursday! Midterm Results Regrades due Monday, 11:59 pm Response to course survey

3 Course Feedback Response In general, fairly positive Project (Fri), Written HW (Mon), Midterm (Thur) Pushing homework due dates to Wednesdays Note: edx homeworks due Wednesday before project due Friday Written homework not clear about answer format We can fix that Office hours are crowded The homework deadline change should help some Will try to schedule homework parties (likely in evenings) We ll look for bigger rooms too (kinda hard sometimes) Please contact us if you feel you are not getting the help you need via standard mechanisms Piazza response rate mixed reviews We will certainly work to do better Keep up the great student responses

4 Course Feedback Response Grade clarity Added details to course policies tab on edx Stuart and Pat should wear brighter colors (multiple comments) Stuart and I are going shopping this weekend

5 Where we are in the course We re done with Part I Search and Planning! Part II: Probabilistic Reasoning Diagnosis Speech recognition Tracking objects Robot mapping Genetics Error correcting codes lots more! Part III: Machine Learning

6 CS 188: Artificial Intelligence Probability Instructors: Stuart Russell and Pat Virtue University of California, Berkeley

7 Today Probability Random Variables Joint and Marginal Distributions Conditional Distribution Product Rule, Chain Rule, Bayes Rule Inference All of these concepts were in CS 70, but now you ll be using them as basic elements in a practical technology for reasoning

8 Uncertainty Wheel of Fortune

9 Uncertainty My flight to Chicago is scheduled to leave SFO at 10:30 am Let action A t = order leave home t minutes before flight Will A t ensure that I catch the plane? Problems: partial observability (road state, other drivers plans, etc.) noisy sensors (radio traffic reports, Google maps) immense complexity of modelling and predicting traffic, security line, etc. lack of knowledge of world dynamics (will tire burst? Will that guy in the security line remember to take out his keys?)

10 Responses to uncertainty Ignore it map directly from percept stream (known) to actions Hopeless! Some sort of softening of logical rules (fudge factors) A CatchPlane CatchPlane 0.95 MajorTrafficJam Hence, chaining these together, A Oops Probability MajorTrafficJam Given the available evidence and the choice A 120, I will catch the plane with probability 0.92

11 Probability Probabilistic assertions summarize effects of ignorance: lack of relevant facts, initial conditions, etc. laziness: failure to list all exceptions, compute detailed predictions, etc. Subjective or Bayesian probability: Probabilities relate propositions to one s own state of knowledge E.g., P(CatchPlane A 120, sunny) = 0.92 Not claiming a probabilistic tendency in the actual situation (traffic is not like quantum mechanics) Probabilities of propositions change with new evidence: E.g., P(CatchPlane A 120, sunny, NoDelaysOnBridge) = 0.96

12 Decisions Suppose I believe P(CatchPlane A 60, all my evidence ) = 0.51 P(CatchPlane A 120, all my evidence ) = 0.97 P(CatchPlane A 1440, all my evidence ) = Which action should I choose? Depends on my preferences for, e.g., missing flight, airport food, etc. Utility theory is used to represent and infer preferences Decision theory = utility theory + probability theory Maximize expected utility : a* = argmax a s P(s a) U(s)

13 Inference in Ghostbusters A ghost is in the grid somewhere Sensor readings tell how close a square is to the ghost On the ghost: red 1 or 2 away: orange 3 or 4 away: yellow 5+ away: green Sensors are noisy, but we know P(Color(x,y) DistanceFromGhost(x,y)) P(red 3) P(orange 3) P(yellow 3) P(green 3)

14 Video of Demo Ghostbusters No probability

15 Basic laws of probability Begin with a set of possible worlds E.g., 6 possible rolls of a die, {1, 2, 3, 4, 5, 6} A probability model assigns a number P( ) to each world E.g., P(1) = P(2) = P(3) = P(5) = P(5) = P(6) = 1/6. These numbers must satisfy 0 P( ) 1 P( ) = 1 1/6 1/6 1/6 1/6 1/6 1/6

16 Basic laws contd. An event is any subset of E.g., roll < 4 is the set {1,2,3} E.g., roll is odd is the set {1,3,5} The probability of an event is the sum of probabilities over its worlds P(A) = A P( ) E.g., P(roll < 4) = P(1) + P(2) + P(3) = 1/2 De Finetti (1931): anyone who bets according to probabilities that violate these laws can be forced to lose money on every set of bets

17 Random Variables A random variable is some aspect of the world (formally a deterministic function of ) about which we (may) be uncertain R = Is it raining? Odd = Is the dice roll an odd number? T = Is it hot or cold? D = How long will it take to get to the airport? L Ghost = Where is the ghost? We denote random variables with capital letters Like variables in a CSP, random variables have domains Odd in {true, false} e.g. Odd(1)=true, Odd(6) = false often write the event Odd=true as odd, Odd=false as odd T in {hot, cold} D in [0, ) L in possible locations, maybe {(0,0), (0,1), }

18 Probability Distributions Associate a probability with each value; sums to 1 Temperature: Weather: P(T) T P hot 0.5 cold 0.5 P(W) W P sun 0.6 rain 0.1 fog 0.3 meteor 0.0

19 Probability Distributions Every probability model automatically fixes a distribution for every random variable P(T) P(W) T P W P hot 0.5 sun 0.6 cold 0.5 rain 0.1 fog 0.3 meteor 0.0 A distribution is a TABLE of probabilities of values A probability (lower case value) is a single number P(W=rain) = 0.1 Shorthand notation: P(hot) = P(T=hot) P(cold) = P(T=cold) P(rain) = P(W=rain) OK if all domain entries are unique Special Boolean notation: P(happy) = P(Happy=true) P( happy) = P(Happy=false)

20 Joint Distributions A joint distribution over a set of random variables: X 1, X 2,, X n specifies a real number for each assignment (or outcome): P(X 1 =x 1, X 2 =x 2,, X n =x n ) P(x 1, x 2,, x n ) Joint distribution obeys same rules as univariate: P(x 1, x 2,, x n ) 0 x 1, x2,, xn P(x 1, x 2,, x n ) =1 P(T,W) T W P hot sun 0.4 hot rain 0.1 cold sun 0.2 cold rain 0.3

21 Joint Distributions Conceptual transition from single events to discrete distributions P(a, b) P(A,B) P(A,B) A B P(A,B) a b 0.1 a b 0.2 a b 0.2 a b 0.5

22 Making possible worlds In many cases we begin with random variables and their domains construct possible worlds as assignments of values to all variables (cf CSPs) E.g., two dice rolls Roll 1 and Roll 2 How many possible worlds? What are their probabilities? Size of distribution for n variables with domain size d? d n For all but the smallest distributions, cannot write out by hand!

23 Probabilities of events Recall that the probability of an event is the sum of probabilities of its worlds So, given a joint distribution over all variables, can compute any event probability! Probability that it s hot AND sunny? Probability that it s hot? Probability that it s hot OR sunny? Typically, the events we care about are partial assignments, like P(T=hot) P(T,W) T W P hot sun 0.4 hot rain 0.1 cold sun 0.2 cold rain 0.3

24 Quiz: Events P(X=true, Y=true)? P(X=true)? P(X Y)? P(X,Y) X Y P true true 0.2 true false 0.3 false true 0.4 false false 0.1

25 Quick Break So you're telling me there's a chance

26 Marginal Distributions Marginal distributions are sub-tables which eliminate variables Marginalization (summing out): Combine collapsed rows by adding P(T) P(T,W) T W P hot sun 0.4 hot rain 0.1 cold sun 0.2 cold rain 0.3 P(X=x 1 ) = y P(X=x 1, Y=y ) T P hot 0.5 cold 0.5 P(W) W P sun 0.6 rain 0.4

27 Quiz: Marginal Distributions P(X) P(X,Y) X Y P true true 0.2 true false 0.3 P(x) = y P(x, y ) X true false P(Y) P false true 0.4 false false 0.1 P(y) = x P(x, y) Y true false P

28 Conditional Probabilities A simple relation between joint and conditional probabilities In fact, this is taken as the definition of a conditional probability P(a b) = P(a, b) P(b) P(a,b) P(T,W) T W P hot sun 0.4 hot rain 0.1 cold sun 0.2 cold rain 0.3 P(a) P(b) P(W=s T=c) = P(W=s,T=c) P(T=c) = P(W=s,T=c) + P(W=r,T=c) = = 0.5 = 0.2/0.5 = 0.4

29 Conditional Probabilities Conceptual transition from single events to discrete distributions P(a b) P(A b) P(A B) P(A b) A B P(A B) a b 0.3 a b 0.7 P(A B) A B P(A B) a b 0.3 a b 0.7 a b 0.4 a b 0.6

30 Quiz: Conditional Probabilities P(X=true Y=true)? P(X,Y) X Y P true true 0.2 true false 0.3 false true 0.4 false false 0.1 P(X=false Y=true)? P(Y X=true)?

31 P(W T ) Conditional Distributions Conditional distributions are probability distributions over some variables given fixed values of others Conditional Distributions P(W T=hot) W P sun 0.8 rain 0.2 P(W T=cold) W P sun 0.4 rain 0.6 Joint Distribution P(T,W) T W P hot sun 0.4 hot rain 0.1 cold sun 0.2 cold rain 0.3

32 Normalization Trick P(W=s T=c) = P(W=s,T=c) P(T=c) P(T,W) T W P hot sun 0.4 hot rain 0.1 cold sun 0.2 cold rain 0.3 P(W=r T=c) = P(W=s,T=c). P(W=s,T=c) + P(W=r,T=c) 0.2. = = = = P(W=r,T=c) P(T=c) P(W=r,T=c). P(W=s,T=c) + P(W=r,T=c) P(W T=c) W P sun 0.4 rain = =

33 Normalization Trick P(T,W) T W P hot sun 0.4 hot rain 0.1 cold sun 0.2 cold rain 0.3 P(W=s T=c) SELECT the joint probabilities matching the evidence = = P(W=s,T=c) P(T=c) P(W=s,T=c). P(W=s,T=c) + P(W=r,T=c) 0.2. = = P(c,W) T W P cold sun 0.2 cold rain 0.3 NORMALIZE the selection (make it sum to one) P(W T=c) W P sun 0.4 rain 0.6 P(W=r T=c) P(W=r,T=c) = P(T=c) P(W=r,T=c). = P(W=s,T=c) + P(W=r,T=c) 0.3. = =

34 Normalization Trick P(T,W) T W P hot sun 0.4 hot rain 0.1 cold sun 0.2 cold rain 0.3 SELECT the joint probabilities matching the evidence P(c,W) T W P cold sun 0.2 cold rain 0.3 NORMALIZE the selection (make it sum to one) P(W T=c) W P sun 0.4 rain 0.6 Why does this work? Sum of selection is P(evidence)! (P(T=cold), here)

35 Smaller set of remaining worlds Normalization

36 To Normalize (Dictionary) To bring or restore to a normal condition Procedure: All entries sum to ONE Step 1: Compute Z = sum over all entries [Note: AIMA uses = 1/Z] Step 2: Divide every entry by Z Example 1 Example 2 W P sun 0.2 Normalize rain 0.3 Z = 0.5 W P sun 0.4 rain 0.6 T W P hot sun 20 hot rain 5 cold sun 10 Normalize Z = 50 T W P hot sun 0.4 hot rain 0.1 cold sun 0.2 cold rain 15 cold rain 0.3

37 Probabilistic Inference Probabilistic inference: compute a desired probability from other known probabilities (e.g. conditional from joint) We generally compute conditional probabilities P(on time no reported accidents) = 0.90 These represent the agent s beliefs given the evidence Probabilities change with new evidence: P(on time no accidents, 5 a.m.) = 0.95 P(on time no accidents, 5 a.m., raining) = 0.80 Observing new evidence causes beliefs to be updated

38 Inference by Enumeration General case: Evidence variables: E 1,, E k = e 1,,e k Query* variable: Q Hidden variables: H 1,, H r X 1,, X n All variables We want: P(Q e 1,,e k ) * Works fine with multiple query variables, too Step 1: Select the entries consistent with the evidence Step 2: Sum out H to get joint of Query and evidence Step 3: Normalize Z = q P(q,e 1,,e k ) P(Q e 1,,e k ) = 1/Z P(Q,e 1,,e k ) P(Q,e 1,,e k ) = h 1,, hr P(Q,h 1,, h r, e 1,,e k ) X 1,, X n

39 Inference by Enumeration P(W)? S T W P summe r summe r summe r summe r hot sun 0.30 hot rain 0.05 cold sun 0.10 cold rain 0.05 winter hot sun 0.10 winter hot rain 0.05 winter cold sun 0.15 winter cold rain 0.20

40 Inference by Enumeration P(W)? P(W winter)? S T W P summe r summe r summe r summe r hot sun 0.30 hot rain 0.05 cold sun 0.10 cold rain 0.05 winter hot sun 0.10 winter hot rain 0.05 winter cold sun 0.15 winter cold rain 0.20

41 Inference by Enumeration P(W)? P(W winter)? P(W winter, hot)? S T W P summe r summe r summe r summe r hot sun 0.30 hot rain 0.05 cold sun 0.10 cold rain 0.05 winter hot sun 0.10 winter hot rain 0.05 winter cold sun 0.15 winter cold rain 0.20

42 Inference by Enumeration Obvious problems: Worst-case time complexity O(d n ) Space complexity O(d n ) to store the joint distribution

43 The Product Rule Sometimes have conditional distributions but want the joint P(a b) P(b) = P(a, b) P(a b) = P(a, b) P(b)

44 The Product Rule P(a b) P(b) = P(a, b) Example: P(D W) P(W) = P(D, W) P(D W) P(D, W) D W P wet sun 0.1 dry sun 0.9 wet rain 0.7 dry rain 0.3 P(W) W P sun 0.8 rain 0.2 D W P wet sun 0.08 dry sun 0.72 wet rain 0.14 dry rain 0.06

45 The Chain Rule More generally, can always write any joint distribution as an incremental product of conditional distributions P(x 1, x 2, x 3 ) = P(x 3 x 1, x 2 ) P(x 1, x 2 ) = P(x 3 x 1, x 2 ) P(x 2 x 1 ) P(x 1 ) P(x 1, x 2,, x n ) = i P(x i x 1,, x i-1 ) Why is this always true?

46 Bayes Rule

47 Bayes Rule Write the product rule both ways: P(a b) P(b) = P(a, b) = P(b a) P(a) That s my rule! Dividing left and right expressions, we get: P(a b) = Why is this at all helpful? P(b a) P(a) P(b) Lets us build one conditional from its reverse Often one conditional is tricky but the other one is simple Describes an update step from prior P(a) to posterior P(a b) Foundation of many systems we ll see later (e.g. ASR, MT) In the running for most important AI equation!

48 Bayes Rule 48

49 Inference with Bayes Rule Example: Diagnostic probability from causal probability: P(cause effect) = P(effect cause) P(cause) P(effect) Example: M: meningitis, S: stiff neck P(m) = P(s m) = 0.8 P(s) = 0.01 Example givens P(m s) = P(s m) P(m) P(s) = 0.8 x Note: posterior probability of meningitis still very small: (80x bigger why?) Note: you should still get stiff necks checked out! Why?

50 Next time Independence Conditional independence Bayes nets