Machine Learning. CS Spring 2015 a Bayesian Learning (I) Uncertainty

Transcription

1 Machine Learning CS Spring 2015 a Bayesian Learning (I) 1 Uncertainty Most real-world problems deal with uncertain information Diagnosis: Likely disease given observed symptoms Equipment repair: Likely component failure given sensor reading Cannot be represented by deterministic rules Headache => Fever Correct framework for representing uncertainty: Probability 2 1

2 Probability P(A) = Probability of event A = fraction of all possible worlds in which A is true. 3 Probability 4 2

3 Probability Immediately derived properties More generally: IF we know that exactly one of B 1, B 2..., B n are true (i.e., P(B 1 or B 2 or... B n ) = 1, and for all i, j unequal, P(B i and B j ) = 0) THEN we know: P(A) = P(A, B 1 ) + P(A, B 2 ) +... P(A, B n ) 5 Probability A random variable is a variable X that can take values x 1,..,x n with a probability P(X= x i ) attached to each i = 1,..,n 6 3

4 Example My mood can take one of two values: Happy, Sad. The weather can take one of three values: Rainy, Sunny Cloudy. Given P(Mood=Happy ^ Weather=Rainy ) = 0.2 P(Mood=Happy ^ Weather=Sunny ) = 0.1 P(Mood=Happy ^ Weather=Cloudy ) = 0.4 Can I compute P(Mood=Happy)? Can I compute P(Mood=Sad)? Can I compute P(Weather=Rainy)? 7 Conditional Probability P(A B) = Fraction of those worlds in which B is true for which A is also true. 8 4

5 Conditional Probability Example H = Headache P(H) = 1/2 F = Flu P(F) = 1/8 P(H F) = (Area of H and F region) (Area of F region) P(H F) = P(H,F)/P(F) P(H F) = 1/2 9 Conditional Probability Definition: Chain rule: Can you prove that P(A, B) <= P(A) for any events A and B? What can you say about P(A B) comparing to P(A)? 10 5

6 Conditional Probability Other useful relations: 11 Probabilistic Inference What is the probability that F is true given H is true? Given P(H) = 1/2 P(F) = 1/8 P(H F) =

7 Probabilistic Inference Correct reasoning: We know P(H), P(F), P(H F) and the two chain rules: Substituting the values: 13 Bayes Rule 14 7

8 Bayes Rule 15 Bayes Rule What if we do not know P(A)??? Use the relation: More general Bayes rule: 16 8

9 Bayes Rule Same rule for a non-binary random variable, except we need to sum over all the possible events 17 Generalizing Bayes Rule If we know that exactly one of A 1, A 2,..., A n are true, then: P(B) = P(B A 1 )P(A 1 ) + P(B A 2 )P(A 2 ) P(B A n ) P(A n ) and in general P(B X) = P(B A 1,X)P(A 1 X) P(B A n,x) P(A n X) So P( A k B, X ) = P( Ak X ) P( B Ak, X ) P( A X ) P( B A, X ) i i i 18 9

10 Medical Diagnosis A doctor knows that meningitis causes a stiff neck 50% of the time. The doctor knows that if a person is randomly selected from the US population, there s a 1/50,000 chance the person will have meningitis. The doctor knows that if a person is randomly selected from the US population, there s a 5% chance the person will have a stiff neck. You walk into the doctor complaining of the symptom of a stiff neck. What s the probability that the underlying cause is meningitis? 19 Joint Distribution Joint Distribution Table Given a set of variables A,B,C,. Generate a table with all the possible combinations of assignments to the variables in the rows For each row, list the corresponding joint probability For M binary variables size 2 M 20 10

11 Using the Joint Distribution Compute the probability of event E: 21 Inference Using the Joint Distribution Given that event E 1 occurs, what is the probability that E 2 occurs: 22 11

12 Inference Using the Joint Distribution 23 Inference General view: I have some evidence (Headache) how likely is a particular conclusion (Fever) 24 12

13 Generating the Joint Distribution Three possible ways of generating the joint distribution: 1. Human experts 2. Using known conditional probabilities (e.g., if we know P(C A,B), P(B A), and P(A), we know P(A,B,C) = P(C A,B)P(B A)P(A).) 3. Learning from data 25 Learning the Joint Distribution Suppose that we have recorded a lot of training data: The entry for P(A,B,~C) in the table is: 26 13

14 Learning the Joint Distribution Suppose that we have recorded a lot of training data: More generally, the entry for P(E) in the table is: 27 Real-Life Joint Distribution UCI Census Database P(Male Poor) = / =

15 So Far Basic probability concepts Bayes rule What are joint distributions Inference using joint distributions Learning joint distributions from data Problem: If we have M variables, we need 2 M entries in the joint distribution table An independence assumption leads to an efficient way to learn and to do inference Problem: estimate probabilities 29 Independence A and B are independent iff: In words: Knowing B does not affect how likely we think that A is true 30 15

16 Key Properties Symmetry: Joint distribution: Independence of complements: 31 Independence Suppose that A, B, C are independent Then any value of the joint distribution can be computed easily: In fact, we need only M numbers instead of 2 M for binary variables!! 32 16

17 Independence: General Case If X 1,..,X M are independent variables: Under the independence assumption, we can compute any value of the joint distribution We can answer any inference query How do we learn the distributions? Similar to earlier slides on joint distributions 33 Learning with the Independence Assumption Learning the distributions from data is simple and efficient In practice, the independence assumption may not be met but it is often a very useful approximation 34 17

18 So Far Basic probability concepts Bayes rule What are joint distributions Inference using joint distributions Learning joint distributions from data Independence assumption Problem: We now have the joint distribution. How can we use it to make decision Bayes Classifier 35 Note about Probability Estimation So far we have been using relative frequencies to approximate probability of an event C( u) f u = N We will discuss more probability estimation later 36 18

19 Three Prisoner Problem Three prisoners, A, B, and C, are locked in their cells. One of them will be executed the next day and others will be released. Only the governor knows which one will be executed. Prisoner A asked the governor to tell him which one of B and C will be released, and got the answer of B. Now what is the chance that A thinks he will be executed? 37 Monty Hall Problem Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice? 38 19