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Università di Milano-Bicocca Laurea Magistrale in Informatica Corso di APPRENDIMENTO E APPROSSIMAZIONE Prof. Giancarlo Mauri Lezione 5 - Statistical Learning

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Outline Bayes theorem MAP, ML hypotheses Minimum Description Length principle Optimal Bayes classifier Naive Bayes classifier Expectation Maximization (EM) algorithm

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Two roles for Bayesian learning Provides practical learning algorithms Naive Bayes learning Bayesian belief network learning Combines prior knowledge (prior probabilities) with observed data Requires prior probabilities Provides useful conceptual framework Gold standard for evaluating other learning algorithms Additional insight into Occams razor

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Bayesian learning Advantages No hypothesis is eliminated, even if inconsistent with data Each hypothesis h is given a probability P(h) to be the correct one P(h) is incrementally modified after seen an example Hypotheses that make probabilistic predictions (eg, for medical diagnosis) are allowed Drawbacks Not easy to estimate prior probabilities Huge computational cost in the general case

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Bayesian learning View as updating of a probability distribution over the hypothesis space H H (random) hypothesis variable, values h 1, h 2, … Start with prior distribution P(H) jth observation gives the outcome d j of random variable D j - Training data D = {d 1, d 2, …, d N } Use Bayes theorem to compute posterior probability of each hypothesis

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Bayes theorem Allows to compute probabilities of hypotheses, given training sample D and prior probabilities P(D|h) P(h) P(h|D) = P(D) P(h) = prior probability of hypothesis h P(D) = probability of training set D P(h|D) = posterior probability of h given D P(D|h) = likelihood of D given h NB: increases with P(D|h) and P(h), decreases when P(D) increases

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Bayesian prediction Prediction based on weighted average of likelihood wrt hypotheses probabilities P(X|D) = i P(X|D,h i )P(h i |D) = i P(X|h i )P(h i |D) No need to pick one best-guess hypothesis!

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Example Un paziente fa un esame di laboratorio per un marcatore tumorale. Sappiamo che: Lincidenza della malattia su tutta la popolazione è dell8 per mille (probabilità a priori) Lesame dà un risultato positivo nel 98% dei casi in cui è presente la malattia (quindi 2% di falsi negativi) dà un risultato negativo nel 97% dei casi in cui non è presente la malattia (quindi 3% di falsi positivi)

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Example Abbiamo le seguenti probabilità a priori e condizionate: P(c) = 0,008 P( c) = 0,992 P(+|c) = 0,98 P(-|c) = 0,02 P(+| c) = 0,03 P(-| c) = 0,97 Se il test per il nostro paziente risulta positivo (D=+), quali sono le probabilità a posteriori che abbia o non abbia il cancro ? P(+|c)P(c) = 0.98x0.008 = = P(c|+) P(+| c)P( c) = 0.03x0.992 = = P( c|+)

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Example Dividendo per 0,0078+0,0298 per normalizzare a 1, otteniamo: P(c|+) = 0,21P( c|+) = 0,79 E un risultato controintuitivo, che si spiega col fatto che i falsi positivi, su una popolazione in stragrande maggioranza sana, diventano molto numerosi

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Example Suppose there are ve kinds of bags of candies: 10% are h 1 : 100% cherry candies 20% are h 2 : 75% cherry candies + 25% lime candies 40% are h 3 : 50% cherry candies + 50% lime candies 20% are h 4 : 25% cherry candies + 75% lime candies 10% are h 5 : 100% lime candies Then we observe candies drawn from some bag: What kind of bag is it? What flavour will the next candy be?

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Posterior probability of hypotheses Number of samples in d Posterior Probability of hypothesis

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Prediction probability

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Posterior probability of hypotheses The correct hypothesis in the limit will dominate the prediction, independently of the prior distribution, provided that the correct hypothesis is not given 0 probability The bayesian prediction is optimal, i.e. it will be correct more often than each other prediction

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MAP approximation Summing over the hypothesis space is often intractable (e.g., 18,446,744,073,709,551,616 Boolean functions of 6 attributes) Maximum a posteriori (MAP) learning: choose the most probable hypothesis wrt training data P(D|h) P(h) h MAP = arg max P(h|D) = arg max h H P(D) = arg max P(D|h) P(h) = arg max ( log P(D|h) + log P(h)) N.B. Log terms can be viewed as (negative of) bits to encode data given hypothesis + bits to encode hypothesis (basic idea of minimum description length (MDL) learning) For deterministic hypotheses, P(D|h) is 1 if consistent, 0 otherwise MAP = simplest consistent hypothesis

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Example Number of samples in d Posterior Probability of hypothesis After three samples, h 5 will be used for prediction with probability 1

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Brute force MAP learner 1. For each h in H, calculate the posterior probability P(D|h) P(h) P(h|D) = P(D) 2. Output the hypothesis h MAP with the highest posterior probability h MAP = arg max P(h|D) h H

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Relation to Concept Learning Consider our usual concept learning task instance space X, hypothesis space H, training examples D Consider the FIND-S learning algorithm (outputs most specific hypothesis from the version space VSH.D) What would Bayes rule produce as the MAP hypothesis? Does Find-S output a MAP hypothesis?

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Relation to Concept Learning Assume fixed set of instances x 1,…,x m Assume D is the set of classifications D = c(x 1 ),…,c(x m )> Let P(D|h) = 1 if h consistent with D = 0 otherwise Choose P(h) to be uniform distribution P(h) = 1/|H| for all h in H Then P(h|D) = 1/|VS H,D | if h consistent with D = 0 otherwise

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Evolution of posterior probabilities

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ML approximation Maximum likelihood (ML) hypothesis If prior probabilities are uniform (P(h i )=P(h j ) i,j), let choose h that maximizes likelihood of D h ML = arg max P(D|h) For large data sets, prior becomes irrelevant I.e., simply get the best fit to the data; identical to MAP for uniform prior (which is reasonable if all hypotheses are of the same complexity) ML is the standard (non-Bayesian) statistical learning method

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ML parameter learning Bag from a new manufacturer; fraction of cherry candies? Any is possible: continuum of hypotheses h is a parameter for this simple (binomial) family of models Suppose we unwrap N candies, c cherries and α = N-c limes These are i.i.d. (independent, identically distributed) observations, so P(D|h ) = N j=1 P(d j |h ) = c (1- ) α Maximize this w.r.t., which is easier for the log-likelihood: L(D|h ) = log P(D|h ) = N j=1 log P(d j |h ) = c log + α log(1- ) dL(D|h )P(h) c α c c = = 0 d c+ α N Seems sensible, but causes problems with 0 counts!

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xi Learning a Real Valued Function Consider any real-valued target function f Training examples x i,d i, where d i is noisy training value d i = f(x i ) + e i e i is random variable (noise) drawn indipendently for each x i according to some Gaussian distribution with mean=0 Then the maximum likelihood hypothesis h ML is the one that minimizes the sum of squared errors: h ML = arg min m i=1 (d i -h(x i )) 2 h H

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Learning a real valued function

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Minimum Description Length Principle

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Occams razor: prefer the shortest hypothesis MDL: prefer the hypothesis h that minimizes FORMULA Where L C (x) is the description length of x under econding C _______________________________________________________________ Example: H = decision trees, D = training data labels L C 1 (h) is # bits to describe tree h L C 2 (D|h) is # bits to describe D given h - Note L C 2 (D|h) = 0 if examples classified perfectly by h. Need only describe exceptions Hence h MDL trades off tree size for training errors

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Minimum Description Length Principle FORMULE Interesting fact from information theory: The optimal (shortest coding length) code for an event with probability p is - log 2 p bits. So interpret (1): - log 2 P(h) is length of h under optimal code - log 2 P(D|h) is length of D given h under optimal code Prefer the hypothesis that minimizes Length(h) + length(misclassifications)

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Most Probable Classification of New Instances So far weve sought the most probable hypothesis given the data D (i.e., h MAP ) Given new instance x, what is its most probable classification? h MAP (x) is not the most probable classification! Consider: Three possible hypotheses: FORMULA Given new instance x, h 1 (x) = +, h 2 (x) = -, h 3 (x) = - Whats the most probable classification of x?

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Bayes Optimal Classifier Bayes optimal classification: FORMULA Example: P(h 1 |D) =.4, P(-|h 1 ) = 0, P(+|h 1 ) = 1 P(h 2 |D) =.3, P(-|h 2 ) = 0, P(+|h 2 ) = 0 P(h 3 |D) =.3, P(-|h 3 ) = 0, P(+|h 3 ) = 0 Therefore FORMULE And FORMULA

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Gibbs Classifier Bayes optimal classifier provides best result, but can be expensive if many hypotheses. Gibbs algorithm: 1. Choose one hypothesis at random, according to P(h|D) 2. Use this to classify new instance Surprising fact: assume target concepts are drawn at random from H according to priors on H. Then: FORUMLA Suppose correct, uniform prior distribution over H, then - Pick any hypothesis from VS, with uniform probability - Its expected error no worse than twice Bayes optimal

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Naive Bayes Classifier Along with decision trees, neural networks, nearest nbr, one of the most practical learning methods. When to use - Moderate or large training set available - Attributes that describe instances are conditionally independent given classification Successful applications: - Diagnosis - Classifying text documents

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Naive Bayes Classifier Assume target function f : X V, where each instance x described by attributes. Most probable value of f(x) is: FORMULE Naive Bayes assumption: FORMULA Which gives Naive Bayes classifier: FORMULA

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Naive Bayes Algorithm Naive_Bayes_Learn (examples) For each target value v j P( v j ) estimate P( v j ) For each attribute value a i of each attribute a P( a i |vj) estimate P( a i | v j ) Classify_New_Instance(x) FORMULA

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Naive Bayes: Example Consider Playtennis again and new instance Wanto to compute: FORMULA P(y) P(sun|y) P(cool|y) P(high|y) P(strong|y) =.005 P(n) P(sun|n) P(cool|n) P(high|n) P(strong|n) =.021 v NB = n

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Naive Bayes: Subtleties Conditional independence assumption is often violated FORMULA - …but it works surprisingly well anyway. Note dont need estimated posteriors P(v j |x) to be correct; need only that FORMULA - See [Domingos&Piazzoni, 1996] for analysis - Naive Bayes posteriors often unrealistically close to 1 or 0

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Naive Bayes: Subtleties 2. What if none of the training instances with target value v j have attribute value a j ? Then FORMULE Typical solution is Bayesian estimate for P( a j | v j ) FORMULA Where: - n is number of training examples for which v = v j - n c number of examples for which v = v j and a = a i - p is prior estimate for P( a j | v j ) - m is weight given to prior (i.e. number of virtual examples)

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Learning to Classify Text Why? - Learn which new articles are of interest - Learn to classify web pages by topic Naive Bayes is among most effective algorithm What attributes shall we use to represent text documents?

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Learning to Classify Text Target concept Interesting? : Document {+, -} 1. Represent each document by vector of words - one attribute per word position in document 2. Learning: use training examples to estimate - P(+) - P(-) - P(doc|+) - P(doc|-) Naive Bayes conditional independence assumption FORMULA Where_ _ _is probably that word in position I is_ _ _given_ _ _ One more assumption FORMULA

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Learn_Naive_Bayes_Text (Examples, V) 1. Collect all words and other tokens that occur in Examples - Vocabulary all distinct words and other tokens in Examples 2. Calculate the required P(v j ) and P(w k |v j ) probability terms - For each traget value v j in V do - Docs j subset of Examples for which the target value is v j FORMULA - Text j a single document created by concatenating all members of docs j - N total number of words in Text j (counting duplicate words multiple times) - For each word w k in Vocabulary *n k number of times word w k occurs in Text j *FORMULA

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Classify_Naive_Bayes_Text (Doc) - positions all word positions in Doc that contain tokens found in Vocabulary - Return v NB, where FORMULA

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Classify_Naive_Bayes_Text (Doc) Given 1000 training documents from each group learn to classify new document according to which newsgroup it came from comp.graphics misc.forsale comps.os.ms-windows.misc rec.autos comp.sys.ibm.pc.hardware rec.motorcycles comp.sys.mac.hardware rec.sport.baseball comp.windows.x rec.sport.hockey alt. atheism sci.space soc.religion.christian sci.crypt talk.religion.misc sci.electronics talk.politics.mideast sci.med talk.politics.misc talk.politics.guns Naive Bayes: 89% classification accuracy

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Reti Bayesiane Il classificatore ottimale di Bayes non fa assunzioni di indipendenza tra variabili ed è computazionalmente pesante Il classificatore ingenuo di Bayes è efficiente grazie allipotesi molto restrittiva di indipendenza condizionale di tutti gli attributi dato il valore obiettivo v Le reti Bayesiane descrivono lindipendenza condizionale tra sottoinsiemi di variabili

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Reti Bayesiane DEF. X, Y, Z variabili casuali discrete. X è condizionalmente indipendente da Y dato Z se: x i, y j, z k P(X= x i |Y= y j,Z= z k )= P(X= x i |Z= z k ) Facilmente estendibile a insiemi di variabili: P(X 1,…, X l |Y 1,…,Y m,Z 1,…, Z n )=P(X 1,…, X l | Z 1,…, Z n )

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Reti Bayesiane - Rappresentazione Grafo aciclico orientato I nodi rappresentano le variabili casuali Gli archi rappresentano la struttura di dipendenza condizionale: ogni variabile è condizionalmente indipendente dai suoi non discendenti, dati i suoi genitori Tabella di probabilità condizionali locali per ogni nodo, con la distribuzione di probabilità della variabile associata dati i predecessori immediati Si può calcolare: P(y 1,…, y n ) = i P(y i |gen(Y i ))

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Reti Bayesiane - Rappresentazione Temporale Gita Fuoco di campo Fulmine Tuono Incendio T,G T, G T,G T, G F 0,4 0,1 0,8 0,2 F 0,6 0,9 0,2 0,8 (Variabili a valori booleani)

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