Uncertainty. Variables. assigns to each sentence numerical degree of belief between 0 and 1. uncertainty

Transcription

1 Bayes Classification n Uncertainty & robability n Baye's rule n Choosing Hypotheses- Maximum a posteriori n Maximum Likelihood - Baye's concept learning n Maximum Likelihood of real valued function n Bayes optimal Classifier n Joint distributions n Naive Bayes Classifier 1

2 Uncertainty n Our main tool is the probability theory, which assigns to each sentence numerical degree of belief between 0 and 1 n It provides a way of summarizing the uncertainty Variables n Boolean random variables: cavity might be true or false n Discrete random variables: weather might be sunny, rainy, cloudy, snow n Weather=sunny n Weather=rainy n Weather=cloudy n Weather=snow n Continuous random variables: the temperature has continuous values 2

3 Where do probabilities come from? n Frequents: n From experiments: form any finite sample, we can estimate the true fraction and also calculate how accurate our estimation is likely to be n Subjective: n Agent s believe n Objectivist: n True nature of the universe, that the probability up heads with probability 0.5 is a probability of the coin n Before the evidence is obtained; prior probability n a the prior probability that the proposition is true n cavity=0.1 n After the evidence is obtained; posterior probability n ab n The probability of a given that all we know is b n cavitytoothache=0.8 3

4 Axioms of robability Kolmogorov s axioms, first published in German 1933 n All probabilities are between 0 and 1. For any proposition a 0 a 1 n true=1, false=0 n The probability of disjunction is given by a b = a + b a b n roduct rule a b = a b b a b = b a a 4

5 Theorem of total probability If events A 1,..., A n are mutually exclusive with then Bayes s rule n Reverent Thomas Bayes He set down his findings on probability in "Essay Towards Solving a roblem in the Doctrine of Chances" 1763, published posthumously in the hilosophical Transactions of the Royal Society of London b a = a bb a 5

6 How can we arrive at Bayes s rule? n For any finite sample, we can estimate the true fraction and also calculate how accurate our estimation is likely to be. n By using samples, as in most physical measurements, we estimate the values. n This approach is called frequentist. n We approach the true value by counting the frequency of an event. Conditional probability n If Ω is the set of all possible events, n Ω = 1, then a Ω. n The cardinality determines the number of elements of a set, cardω is the number of elements of the set Ω, carda is the number of elements of the set a 6

7 Baye s rule 7

8 8 Diagnosis n What is the probability of meningitis in the patient with stiff neck? n A doctor knows that the disease meningitis causes the patient to have a stiff neck in 50% of the time -> sm n rior robabilities: That the patient has meningitis is 1/ > m That the patient has a stiff neck is 1/20 -> s m s = 0.5* = m s = s mm s Normalization x y y x x y x y y x x y = = 0.6, ,0.08, 1 = = + = α α α x y x y Y Y X X Y x y x y

9 Bayes Theorem n h = prior probability of hypothesis h n D = prior probability of training data D n hd = probability of h given D n Dh = probability of D given h Choosing Hypotheses n Generally want the most probable hypothesis given the training data n Maximum a posteriori hypothesis h MA : 9

10 n If assume h i =h j for all h i and h j, then can further simplify, and choose the n Maximum likelihood ML hypothesis 10

11 Example n Does patient have cancer or not? A patient takes a lab test and the result comes back positive. The test returns a correct positive result + in only 98% of the cases in which the disease is actually present, and a correct negative result - in only 97% of the cases in which the disease is not present Furthermore, of the entire population have this cancer Suppose a positive result + is returned... 11

12 Normalization cancer + = = cancer + = = n The result of Bayesian inference depends strongly on the prior probabilities, which must be available in order to apply the method Brute-Force Bayes Concept Learning n For each hypothesis h in H, calculate the posterior probability n Output the hypothesis h MA with the highest posterior probability 12

13 n Given no prior knowledge that one hypothesis is more likely than another, what values should we specify for h? n What choice shall we make for Dh? n Hypothesis generates data... n Choose h to be uniform distribution n for all h in H how often is h present? Frequency... n Dh=1 if h consistent with D n Dh=0 otherwise 13

14 D D = D = D h i h i h i H 1 1 H H h i VS H,D D = VS H,D H h i VS H,D n Version space VS H,D is the subset of consistent Hypotheses from H with the training examples in D n Hypothesis generates this data... if h is inconsistent with D h D = 1 1 H VS H,D H = 1 VS H,D if h is consistent with D 14

15 Hypothesis generates this data... Maximum Likelihood of real valued function.. see EM Clustering 15

16 n Maximize natural log of this instead... Bayes optimal Classifier A weighted majority classifier n What is the most probable classification of the new instance given the training data? n The most probable classification of the new instance is obtained by combining the prediction of all hypothesis, weighted by their posterior probabilities n If the classification of new example can take any value v j from some set V, then the probability v j D that the correct classification for the new instance is v j, is just remember, have to calculate h i D : 16

17 Bayes optimal classification: Gibbs Algorithm n Bayes optimal classifier provides best result, but can be expensive if many hypotheses we have to compute the posterior probability for every hypothesis n Gibbs algorithm: n Choose one hypothesis at random, according to h D n Use this to classify new instance n Surprisingly its expected error no worse than twice Bayes optimal Haussler et al

18 Joint distribution n A joint distribution for toothache, cavity, catch, dentist s probe catches in my tooth :- n We need to know the conditional probabilities of the conjunction of toothache and cavity n What can a dentist conclude if the probe catches in the aching tooth? toothache catch cavitycavity cavity toothache catch = toothache cavity n For n possible variables there are 2 n possible combinations Conditional Independence n Once we know that the patient has cavity we do not expect the probability of the probe catching to depend on the presence of toothache catch cavity toothache = catch cavity toothache cavity catch = toothache cavity n Independence between a and b a b = a b a = b 18

19 a b = a b toothache, catch, cavity, Weather = cloudy = = Weather = cloudy toothache, catch, cavity The decomposition of large probabilistic domains into weakly connected subsets via conditional independence is one of the most important developments in the recent history of AI This can work well, even the assumption is not true! A single cause directly influence a number of effects, all of which are conditionally independent cause, effect1, effect2,... effectn = cause effecti cause n i= 1 19

20 Naive Bayes Classifier n Along with decision trees, neural networks, nearest neighbor, one of the most practical learning methods n When to use: n Moderate or large training set available n Attributes that describe instances are conditionally independent given classification n Successful applications: n Diagnosis n Classifying text documents Naive Bayes Classifier n Assume target function f: X è V, where each instance x described by attributes a 1, a 2.. a n n Most probable value of fx is: 20

21 v NB n Naive Bayes assumption: n which gives Naive Bayes Algorithm n For each target value v j n ç estimate v j n For each attribute value a i of each attribute a n ç estimate a i v j 21

22 Training dataset Class: C1:buys_computer= yes C2:buys_computer= no Data sample: X = age<=30, Income=medium, Student=yes Credit_rating=Fair age income student credit_rating buys_computer <=30 high no fair no <=30 high no excellent no high no fair yes >40 medium no fair yes >40 low yes fair yes >40 low yes excellent no low yes excellent yes <=30 medium no fair no <=30 low yes fair yes >40 medium yes fair yes <=30 medium yes excellent yes medium no excellent yes high yes fair yes >40 medium no excellent no Naïve Bayesian Classifier: Example n Compute XC i for each class age= <30 buys_computer= yes = 2/9=0.222 age= <30 buys_computer= no = 3/5 =0.6 income= medium buys_computer= yes = 4/9 =0.444 income= medium buys_computer= no = 2/5 = 0.4 student= yes buys_computer= yes= 6/9 =0.667 student= yes buys_computer= no = 1/5=0.2 credit_rating= fair buys_computer= yes =6/9=0.667 credit_rating= fair buys_computer= no =2/5=0.4 buys_computer= yes =9/14 buys_computer= no =5/14 n X=age<=30,income =medium, student=yes,credit_rating=fair XC i : Xbuys_computer= yes = x x x =0.044 Xbuys_computer= no = 0.6 x 0.4 x 0.2 x 0.4 =0.019 XC i *C i : Xbuys_computer= yes * buys_computer= yes =0.028 Xbuys_computer= no * buys_computer= no =0.007 X belongs to class buys_computer=yes 22

23 n Conditional independence assumption is often violated n...but it works surprisingly well anyway Estimating robabilities n We have estimated probabilities by the fraction of times the event is observed to n c occur over the total number of opportunities n n It provides poor estimates when n c is very small n If none of the training instances with target value v j have attribute value a i? n n c is 0 23

24 n When n c is very small: n n is number of training examples for which v=v j n n c number of examples for which v=v j and a=a i n p is prior estimate n m is weight given to prior i.e. number of ``virtual'' examples Naïve Bayesian Classifier: Comments n Advantages : n Easy to implement n Good results obtained in most of the cases n Disadvantages n Assumption: class conditional independence, therefore loss of accuracy n ractically, dependencies exist among variables n E.g., hospitals: patients: rofile: age, family history etc Symptoms: fever, cough etc., Disease: lung cancer, diabetes etc n Dependencies among these cannot be modeled by Naïve Bayesian Classifier n How to deal with these dependencies? n Bayesian Belief Networks 24

25 n Uncertainty & robability n Baye's rule n Choosing Hypotheses- Maximum a posteriori n Maximum Likelihood - Baye's concept learning n Maximum Likelihood of real valued function n Bayes optimal Classifier n Joint distributions n Naive Bayes Classifier Bayesian Belief Networks Burglary B Earthquake E Alarm Burg. Earth. A t t.95 t f.94 f t.29 f f.001 JohnCalls A J t.90 f.05 MaryCalls A M t.7 f.01 25