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Naïve Bayes Classification Things We’d Like to Do • Spam Classification – Given an email, predict whether it is spam or not • Medical Diagnosis – Given a list of symptoms, predict whether a patient has disease X or not • Weather – Based on temperature, humidity, etc… predict if it will rain tomorrow • The relationship between attribute set and the class variable is non-deterministic. • Even if the attributes are same, the class label may differ in training set even and hence can not be predicted with certainty. • Reason: noisy data, certain other attributes are not included in the data. • Example: Task of predicting whether a person is at risk for heart disease based on the person’s diet and workout frequency. • So need an approach to model probabilistic relationship between attribute set and the class variable. Bayesian Classification • Problem statement: – Given features X1,X2,…,Xn – Predict a label Y Another Application • Digit Recognition Classifier 5 • X1,…,Xn {0,1} (Black vs. White pixels) • Y {5,6} (predict whether a digit is a 5 or a 6) Bayes Classifier • A probabilistic framework for solving classification problems P(A,C) P(C | A) • Conditional Probability: P(A) P(A,C) P(A | C) P(C) • Bayes theorem: P(A| C)P(C) P(C | A) P(A) Example of Bayes Theorem • Given: – A doctor knows that Cold causes fever 50% of the time – Prior probability of any patient having cold is 1/50,000 – Prior probability of any patient having fever is 1/20 • If a patient has fever, what’s the probability he/she has cold? P(F | C)P(C) 0.51/ 50000 P(C | F) 0.0002 P(F) 1/ 20 Bayesian Classifiers • Consider each attribute and class label as random variables • Given a record with attributes (A1, A2,…,An) – Goal is to predict class C – Specifically, we want to find the value of C that maximizes P(C| A1, A2,…,An ) • Can we estimate P(C| A1, A2,…,An ) directly from data? Bayesian Classifiers • Approach: – compute the posterior probability P(C | A1, A2, …, An) for all values of C using the Bayes theorem P(A1 A2 An | C)P(C) P(C | A1 A2 An ) P(A A A ) 1 2 n – Choose value of C that maximizes P(C | A1, A2, …, An) – Equivalent to choosing value of C that maximizes P(A1, A2, …, An|C) P(C) • How to estimate P(A1, A2, …, An | C )? Naïve Bayes Classifier • Assume independence among attributes Ai when class is given: – P(A1, A2, …, An |C) = P(A1| Cj) P(A2| Cj)… P(An| Cj) – Can estimate P(Ai| Cj) for all Ai and Cj. – New point is classified to Cj if P(Cj) P(Ai| Cj) is maximum. How to Estimate Probabilities from Data? categoricalcategorical continuousclass • Class: P(C) = Nc/N Tid Refund Marital Taxable – e.g., P(No) = 7/10, Evade Status Income P(Yes) = 3/10 1 Yes Single 125K No 2 No Married 100K No • For discrete attributes: 3 No Single 70K No 4 Yes Married 120K No P(Ai | Ck) = |Aik|/ Nk c 5 No Divorced 95K Yes 6 No Married 60K No – where |Aik| is number of 7 Yes Divorced 220K No instances having attribute Ai and belongs to class C 8 No Single 85K Yes k 9 No Married 75K No – Examples: 10 No Single 90K Yes P(Status=Married|No) = 4/7 10 P(Refund=Yes|Yes)=0 How to Estimate Probabilities from Data? • For continuous attributes: – Discretize the range into bins • one ordinal attribute per bin k • violates independence assumption – Two-way split: (A < v) or (A > v) • choose only one of the two splits as new attribute – Probability density estimation: • Assume attribute follows a normal distribution • Use data to estimate parameters of distribution (e.g., mean and standard deviation) • Once probability distribution is known, can use it to estimate the conditional probability P(Ai|c) How to Estimate Probabilities from Data? categoricalcategorical continuousclass Tid Refund Marital Taxable • Normal distribution: Status Income Evade ( A )2 i ij 1 2 2 1 Yes Single 125K No ij P(Ai | c j ) e 2 No Married 100K No 2 2 ij 3 No Single 70K No 4 Yes Married 120K No – One for each (Ai,ci) pair 5 No Divorced 95K Yes • 6 No Married 60K No For (Income, Class=No): 7 Yes Divorced 220K No – If Class=No 8 No Single 85K Yes • sample mean = 110 9 No Married 75K No • sample variance = 2975 10 No Single 90K Yes 10 (120110 )2 1 P(Income 120 | No) e 2 ( 2975 ) 0.0072 2 (54.54) Example of Naïve Bayes Classifier Given a Test Record: X (Refund No,Married,Income 120K) naive Bayes Classifier: P(Refund=Yes|No) = 3/7 P(X|Class=No) = P(Refund=No|Class=No) P(Refund=No|No) = 4/7 P(Married| Class=No) P(Refund=Yes|Yes) = 0 P(Income=120K| Class=No) P(Refund=No|Yes) = 1 = 4/7 4/7 0.0072 = 0.0024 P(Marital Status=Single|No) = 2/7 P(Marital Status=Divorced|No)=1/7 P(Marital Status=Married|No) = 4/7 P(X|Class=Yes) = P(Refund=No| Class=Yes) P(Marital Status=Single|Yes) = 2/7 P(Married| Class=Yes) P(Marital Status=Divorced|Yes)=1/7 P(Income=120K| Class=Yes) P(Marital Status=Married|Yes) = 0 = 1 0 1.2 10-9 = 0 For taxable income: If class=No: sample mean=110 Since P(X|No)P(No) > P(X|Yes)P(Yes) sample variance=2975 Therefore P(No|X) > P(Yes|X) If class=Yes: sample mean=90 sample variance=25 => Class = No Naïve Bayes Classifier • If one of the conditional probability is zero, then the entire expression becomes zero • Probability estimation:N Original : P(A | C) ic i N c c: number of classes N 1 Laplace : P(A | C) ic p: prior probability i N c c m: parameter Nic mp m - estimate : P(Ai | C) Nc m Example of Naïve Bayes Classifier Name Give Birth Can Fly Live in Water Have Legs Class human yes no no yes mammals A: attributes python no no no no non-mammals salmon no no yes no non-mammals M: mammals whale yes no yes no mammals frog no no sometimes yes non-mammals N: non-mammals komodo no no no yes non-mammals 6 6 2 2 bat yes yes no yes mammals P(A | M ) 0.06 pigeon no yes no yes non-mammals 7 7 7 7 cat yes no no yes mammals leopard shark yes no yes no non-mammals 1 10 3 4 turtle no no sometimes yes non-mammals P(A | N) 0.0042 penguin no no sometimes yes non-mammals 13 13 13 13 porcupine yes no no yes mammals eel no no yes no non-mammals 7 salamander no no sometimes yes non-mammals P(A | M )P(M ) 0.06 0.021 gila monster no no no yes non-mammals 20 platypus no no no yes mammals owl no yes no yes non-mammals 13 dolphin yes no yes no mammals P(A | N)P(N) 0.004 0.0027 eagle no yes no yes non-mammals 20 Give Birth Can Fly Live in Water Have Legs Class P(A|M)P(M) > yes no yes no ? P(A|N)P(N) => Mammals Naïve Bayes (Summary) • Robust to isolated noise points • Handle missing values by ignoring the instance during probability estimate calculations • Robust to irrelevant attributes • Independence assumption may not hold for some attributes – Use other techniques such as Bayesian Belief Networks (BBN) The Bayes Classifier • Use Bayes Rule! Likelihood Prior Normalization Constant • Why did this help? Well, we think that we might be able to specify how features are “generated” by the class label The Bayes Classifier • Let’s expand this for our digit recognition task: • To classify, we’ll simply compute these two probabilities and predict based on which one is greater Model Parameters • For the Bayes classifier, we need to “learn” two functions, the likelihood and the prior • How many parameters are required to specify the prior for our digit recognition example? Model Parameters • The problem with explicitly modeling P(X1,…,Xn|Y) is that there are usually way too many parameters: – We’ll run out of space – We’ll run out of time – And we’ll need tons of training data (which is usually not available) The Naïve Bayes Model • The Naïve Bayes Assumption: Assume that all features are independent given the class label Y • Equationally speaking: • (We will discuss the validity of this assumption later) Naïve Bayes Training • Now that we’ve decided to use a Naïve Bayes classifier, we need to train it with some data: MNIST Training Data Naïve Bayes Training • Training in Naïve Bayes is easy: – Estimate P(Y=v) as the fraction of records with Y=v – Estimate P(Xi=u|Y=v) as the fraction of records with Y=v for which Xi=u Naïve Bayes Training • In practice, some of these counts can be zero • Fix this by adding “virtual” counts: – This is called Smoothing Naïve Bayes Training • For binary digits, training amounts to averaging all of the training fives together and all of the training sixes together. Naïve Bayes Classification Another Example of the Naïve Bayes Classifier The weather data, with counts and probabilities outlook temperature humidity windy play yes no yes no yes no yes no yes no sunny 2 3 hot 2 2 high 3 4 false 6 2 9 5 overcast 4 0 mild 4 2 normal 6 1 true 3 3 rainy 3 2 cool 3 1 sunny 2/9 3/5 hot 2/9 2/5 high 3/9 4/5 false 6/9 2/5 9/14 5/14 overcast 4/9 0/5 mild 4/9 2/5 normal 6/9 1/5 true 3/9 3/5 rainy 3/9 2/5 cool 3/9 1/5 A new day outlook temperature humidity windy play sunny cool high true ? • Likelihood of yes 2 3 3 3 9 0.0053 9 9 9 9 14 • Likelihood of no 3 1 4 3 5 0.0206 5 5 5 5 14 • Therefore, the prediction is No The Naive Bayes Classifier for Data Sets with Numerical Attribute Values • One common practice to handle numerical attribute values is to assume normal distributions for numerical attributes.
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