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Introduction Discrete Predictors Validation Summary B AYESIAN N ETWORK C LASSIFIERS Pedro Larra naga Computational Intelligence Group Artificial Intelligence Department Universidad Polit ecnica de Madrid Bayesian Networks: From Theory

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Introduction Discrete Predictors Validation Summary

BAYESIAN NETWORK CLASSIFIERS

Pedro Larra˜ naga

Computational Intelligence Group Artificial Intelligence Department Universidad Polit´ ecnica de Madrid

Bayesian Networks: From Theory to Practice International Black Sea University Autumn School on Machine Learning 3-11 October 2019, Tbilisi, Georgia Pedro Larra˜ naga Bayesian Network Classifiers 1 / 52

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Introduction Discrete Predictors Validation Summary

Outline

1 Introduction 2 Discrete Predictors 3 Validation of Supervised Classifiers 4 Summary

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Outline

1 Introduction 2 Discrete Predictors 3 Validation of Supervised Classifiers 4 Summary

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Introduction Discrete Predictors Validation Summary

Supervised classification

X1 . . . Xn C (x(1), c(1)) x(1)

1

. . . x(1)

n

c(1) (x(2), c(2)) x(2)

1

. . . x(2)

n

c(2) . . . . . . . . . (x(N), c(N)) x(N)

1

. . . x(N)

n

c(N) x(N+1) x(N+1)

1

. . . x(N+1)

n

???

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Introduction Discrete Predictors Validation Summary

Applications domains

Supervised pattern recognition Decision support systems for diagnosis and prognosis Loan decision Spam detection Prediction of sport results Hand writing character recognition Weather forecast Prediction of the secondary structure of proteins . . .

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Optical character recognition

Figure: Hand writing character recognition

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Weather forecast

Figure: Methereology

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Computational biology

Figure: Prediction of the secondary structure of proteins

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Introduction Discrete Predictors Validation Summary

Paradigms for supervised classification

Statistical and machine learning Bayesian networks (Pearl, 1988) Classification trees (Quinlan, 1986; Breiman et al. 1984) Classifier systems (Holland, 1975) Discriminant analysis (Fisher, 1936) k–NN classifiers (Covert and Hart, 1967; Dasarathy, 1991) Logistic regression (Hosmer and Lemeshov, 1989) Neural networks (McCulloch and Pitts, 1943) Rule induction (Clark and Nibblet, 1989; Cohen, 1995; Holte, 1993) Support vector machines (Cristianini and Shawe–Taylor, 2000)

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Introduction Discrete Predictors Validation Summary

Bayesian network based classifiers

Hierarchy of classifiers Na¨ ıve Bayes (NB) (Minsky, 1961) Semina¨ ıve Bayes (Pazzani, 1997) Tree augmented na¨ ıve Bayes (TAN) (Friedman et al., 1997) k-dependence Bayesian classifier (k-DB) (Sahami, 1996) Markov blanket (Sierra and Larraaga, 1998) Bayesian multinets (Kontkanen et al., 2000)

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Outline

1 Introduction 2 Discrete Predictors 3 Validation of Supervised Classifiers 4 Summary

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Introduction

Fundamentals Cost matrix: cost(r, s) with r predicted class and s true class r, s = 1, . . . r0 Minimization of the total cost error (Bayes rule) γ(x) = arg min

c r0

  • k=1

cost(c, k)P(c|x1, . . . , xn)

In the case of a 0/1 loss function: γ(x) = arg max

c

P(c|x1, . . . , xn)

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Generative versus discriminative classifiers

Generative classifiers P(c|x1, . . . , xn) obtained in an undirected way P(c|x1, . . . , xn) ∝ P(c, x1, . . . , xn) ∝ P(c)P(x1, . . . , xn|c) Parameters estimated from the joint log–likelihood L

  • (x(1), c(1)), . . . , (x(N), c(N))
  • =

N

  • j=1

log P(x(j), c(j))

Discriminant analysis Na¨ ıve Bayes

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Generative versus discriminative classifiers

Discriminative classifiers Discriminative classifiers P(c|x1, . . . , xn) directly Parameters are estimated from the conditional log–likelihood: L

  • (c(1)|x(1)), . . . , (c(N)|x(N))
  • =

N

  • j=1

log P(c(j)|x(j))

Logistic regression

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From the classical diagnosis problem to the na¨ ıve Bayes

Classical diagnosis problem. Multiple diseases X1 . . . Xn Y1 . . . Ym (x(1), y(1)) x(1)

1

. . . x(1)

n

y(1)

1

. . . y(1)

m

(x(2), y(2)) x(2)

1

. . . x(2)

n

y(2)

1

. . . y(2)

m

. . . . . . . . . (x(N), y(N)) x(N)

1

. . . x(N)

n

y(N)

1

. . . y(N)

m

Table: Classical diagnosis problem

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From the classical diagnosis problem to the na¨ ıve Bayes

Classical diagnosis problem. Multiple diseases

(y∗

1 , . . . , y∗ m) = arg

max

(y1,...,ym) P(Y1 = y1, . . . , Ym = ym|X1 = x1, . . . , Xn = xn)

P(Y1 = y1, . . . , Ym = ym|X1 = x1, . . . , Xn = xn) ∝ P(Y1 = y1, . . . , Ym = ym)P(X1 = x1, . . . , Xn = xn|Y1 = y1, . . . , Ym = ym)

Number of parameters: 2m − 1 + (2n − 1)2m m = 3, n = 10 number of parameters ≃ 8 · 103 m = 5, n = 20 number of parameters ≃ 33 · 106 m = 10, n = 50 number of parameters ≃ 11 · 1017

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From the classical diagnosis problem to the na¨ ıve Bayes

Single disease

c∗ = arg max

c

P(C = c|X1 = x1, . . . , Xn = xn) P(C = c|X1 = x1, . . . , Xn = xn) ∝ P(C = c)P(X1 = x1, . . . , Xn = xn|C = c)

Number of parameters: (r0 − 1) + r0(2n − 1) r0 = 3, n = 10 number of parameters ≃ 3 · 103 r0 = 5, n = 20 number of parameters ≃ 5 · 106 r0 = 10, n = 50 number of parameters ≃ 11 · 1015

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From the classical diagnosis problem to the na¨ ıve Bayes

Single disease and symptoms conditionally independent given the disease c∗ = arg max

c

P(C = c|X1 = x1, . . . , Xn = xn) = arg max

c

P(C = c)

n

  • i=1

P(Xi = xi|C = c) Number of parameters: r0 − 1 + r0n r0 = 3, n = 10, number of parameters = 32 r0 = 5, n = 20, number of parameters = 104 r0 = 10, n = 50, number of parameters = 509

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Na¨ ıve Bayes as a probabilistic graphical model

Na¨ ıve Bayes (Minsky, 1961) Predictor variables conditionally independent given C c∗ = arg maxc P(C = c) n

i=1 P(Xi = xi|C = c)

Figure: Structure of a na¨ ıve Bayes

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Na¨ ıve Bayes (Minsky, 1961)

Pattern recognition versus machine learning Long tradition in the pattern recognition community: Minsky (1961), van Woerkom and Brodman (1961), Warner et al. (1961), Bailey (1964), Boyle et al. (1966), Maron (1961), Duda and Hart (1973) Introduced in the machine learning field by Cestnik et al. (1987). Different names:

idiot Bayes: Ohmann et al. (1988) na¨ ıve Bayes: Kononenko (1990) simple Bayes: Gammerman and Thatcher (1991) independent Bayes: Todd and Stamper (1994)

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Na¨ ıve Bayes (Minsky, 1961)

Theoretical results Minsky (1961). The decision surfaces in a na¨ ıve Bayes classifier with binary predictor variables are hyperplanes Peot (1996). Generalization of the previous result for the case of nominal (no binary) predictor variables Duda and Hart (1973). For ordinal predictor variables, the decision surfaces are polynomials Domingos and Pazzani (1997). Although the estimation of p(c|x1, . . . , xn) is not well calibrated, na¨ ıve Bayes can

  • btain competitive accuracies

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Seminave Bayes (Pazzani, 1997)

Step 1. Initialize the set of variables to be used to the null set. Classify all the examples as being of the class with higher p(c) Step 2. Repeat in every iteration the best option between: (a) Consider each variable that is not in the model as a new

  • ne to be included in it.

The chosen variable should be added as conditionally independent of the variables in the model given the class (b) Join each variable not present in the model with a variable that is in the model Evaluate each possible option by means of the estimation

  • f the percentage of cases well classified

Until no improvement can be obtained Figure: Pseudocode of the forward sequential selection and joining (FSSJ) algorithm (Pazzani, 1997)

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Building process (FSSJ)

(a) (b) (c)

(a) The selective naive Bayes with X2 has yielded the best accuracy. (b) After building the models with these sets of predictor variables: {X2, X1}, {X2, X3}, {X2, X4}, {(X2, X1)}, {(X2, X3)} and {(X2, X4)}, the last option is selected according to its accuracy. (c) The winner model out of {X1, (X2, X4)}, {X3, (X2, X4)}, {(X1, X2, X4)}, and {(X3, X2, X4)}. The accuracy does not improve with {X1, X3, (X2, X4)}, {(X1, X3), (X2, X4)}, and {X3, (X1, X2, X4)}, and the process stops Pedro Larra˜ naga Bayesian Network Classifiers 23 / 52

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Tree augmented nave Bayes (TAN) (Friedman et al., 1997)

Mutual information between X e Y MI(X, Y) =

rX

  • i=1

rY

  • j=1

P(xi, yj) log P(xi, yj) P(xi)P(yj) measures the uncertainty reduction in one of the variables when the value of the other variable is known Mutual information between X and Y conditioned by C MI(X, Y|C) =

r0

  • k=1

P(ck)MI(X, Y|C = ck) =

rX

  • i=1

rY

  • j=1

r0

  • k=1

P(xi, yj, ck) log P(xi, yj|ck) P(xi|ck)P(yj|ck)

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TAN algorithm (Friedman et al. 1997)

Step 1. Calculate I(Xi, Xj | C) with i < j i, j = 1, . . . , n Step 2. Build an undirected complete graph, where the nodes correspond to the predictor variables: X1, . . . , Xn. Assign to the edge connecting variables Xi and Xj a weight given by I(Xi, Xj | C) Step 3. Assign the largest two branches to the tree to be constructed Step 4. Examine the next largest branch and add it to the tree unless it forms a loop. In the latter case discard it and examine the next largest branch Step 5. Repeat Step 4 until n − 1 branches have been added Step 6. Transform the undirected graph in a directed one, by choosing a random variable as the root Step 7. Build the TAN structure adding a node labelled as C, and later add one arc from C to each of the predictor variables Figure: Pseudocode of tree augmented na¨

ıve Bayes (TAN) algorithm (Friedman et al. 1997) Pedro Larra˜ naga Bayesian Network Classifiers 25 / 52

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TAN building process

I(X1, X3|C) > I(X2, X4|C) > I(X1, X2|C) > I(X3, X4|C) > I(X1, X4|C) > I(X3, X5|C) > I(X1, X5|C) > I(X2, X3|C) > I(X2, X5|C) > I(X4, X5|C) (a) (b) (c) (c) (d) (e) (g) (h) (a-c) Edges are added according to conditional mutual information quantities arranged in ascending order. (d-e) Edges X3 − X4 and X1 − X4 (dashed lines) cannot be added since they form a cycle. (f) Maximum weighted spanning tree. (g) The directed tree obtained by choosing X1 as the root node. (h) Final TAN structure Pedro Larra˜ naga Bayesian Network Classifiers 26 / 52

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k-DB algorithm (Sahami, 1996)

Scheme of k-DB Calculate I(Xi, C) and I(Xi, Xj|C) for each pair of variables At every iteration, add the Xmax variable not included in the model with the highest I(Xi, C) Set C and the k variables with the highest I(Xj, Xmax|C) as the parents of Xmax

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k-DB building process

Figure: Example for k-DB with k = 2. I(X3, C) > I(X1, C) > I(X4, C) > I(X5, C) > I(X2, C) I(X3, X4|C) > I(X2, X5|C) > I(X1, X3|C) > I(X1, X2|C) > I(X2, X4|C) > I(X2, X3|C) > I(X1, X4|C) > I(X4, X5|C) > I(X1, X5|C) > I(X3, X5|C)

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k-DB building process

Figure: Example for k-DB with k = 2. I(X3, C) > I(X1, C) > I(X4, C) > I(X5, C) > I(X2, C) I(X3, X4|C) > I(X2, X5|C) > I(X1, X3|C) > I(X1, X2|C) > I(X2, X4|C) > I(X2, X3|C) > I(X1, X4|C) > I(X4, X5|C) > I(X1, X5|C) > I(X3, X5|C)

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k-DB building process

Figure: Example for k-DB with k = 2. I(X3, C) > I(X1, C) > I(X4, C) > I(X5, C) > I(X2, C) I(X3, X4|C) > I(X2, X5|C) > I(X1, X3|C) > I(X1, X2|C) > I(X2, X4|C) > I(X2, X3|C) > I(X1, X4|C) > I(X4, X5|C) > I(X1, X5|C) > I(X3, X5|C) P(c|x1, x2, x3, x4, x5) ∝ P(c)P(x1|x3, c)P(x2|x1, x5, c)P(x3|c)P(x4|x1, x3, c)P(x5|x1, x4, c)

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Markov blanket (Sierra and Larraaga, 1998)

Parents, children and parents of the children The value of a variable only depends on the value of: its parents, its children, and the parent of the children This is called the Markov blanket (MB) of the variable P(C|MB(C), X \ MB(C)) = P(C|MB(C)) Figure: Markov blanket of variable A

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Markov blanket

Prediction with a Markov blanket Figure: Example of a Markov blanket for C. P(c|x1, . . . , x9) = P(c|x1, x2, x3, x4, x6) ∝ P(c, x1, x2, x3, x4, x6) = P(x6)P(x3)P(x4)P(c|x6)P(x1|x3, c)P(x2|c, x4)

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Bayesian multinet (Kontkanen et al. 2000)

Prediction with a Bayesian multinet Figure: Example of a Bayesian multinet for classification

P(C = 0|x, y, v, z) ∝ P(C = 0)P(x|C = 0)P(y|x, C = 0)P(v|x, C = 0)P(z|y, C = 0) P(C = 1|x, y, v, z) ∝ P(C = 1)P(x|C = 1)P(y|x, C = 1)P(v|x, C = 1)P(z|v, C = 1)

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Outline

1 Introduction 2 Discrete Predictors 3 Validation of Supervised Classifiers 4 Summary

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Measuring the performance of a supervised classifier

Comparison criteria Accuracy (Nadeau and Bengio, 2003) Brier score Area under the ROC curve (Zou, 2002) Complexity of the inductor Transparency of the model Simplicity of the model Comprehensibility of the model ...

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Measuring the performance of a supervised classifier

Confusion matrix C True class +

  • CM Predicted class

+ a b

  • c

d Figures of merit

Accuracy:

a+d a+b+c+d

Error rate:

c+b a+b+c+d

Rate of true positives (sensibility):

a a+c

Rate of true negatives (specificity):

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Estimating methods. No honest

ˆ pM = 1 N

N

  • i=1

δ(c(i) = c(i)

M )

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Estimating methods. Train and test

ˆ pM = 1 N − N1

N−N1

  • i=1

δ(c(N1+i) = c(N1+i)

M

)

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Estimating methods. Train and test several times

ˆ pM = 1 B

B

  • i=1

ˆ pi

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Estimating methods. k–fold cross validation

ˆ pM = 1 k

k

  • i=1

ˆ pi

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Estimating methods. 0, 632 bootstrapping

ˆ pa = 1 B

B

  • i=1

ˆ pi,a ˆ p0 = 1 B

B

  • i=1

ˆ pi,0 ˆ pM = ˆ p0.632Bo = (0.368ˆ pa + 0.632ˆ p0)

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About the different estimation methods

Train and test: when N is large Train and test several times: there is no control about which cases are used for training (or testing) the classifier k–fold cross validation: unbiased estimation of the probability of success of the classifier, but with a high variance 0.632 bootstrapping: asymptotically unbiased and with small variance

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Brier score

Calibrating a probabilistic classifier Calibration measure for a classifier that assigns a posterior probability for each value of the class Assuming that the true class value for x is C = 0, we are interested in distinguishing: p(CM = 0|x) = 0.51 and p(CM = 0|x) = 0.97 We would like classifiers that are almost sure when taking decisions These classifiers will have a lower value for the Brier measure

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Brier score

X1 . . . Xn C p(CM = 0|x) p(CM = 1|x) (x(1), c(1)) x(1)

1

. . . x(1)

n

1 0, 18 0, 82 (x(2), c(2)) x(2)

1

. . . x(2)

n

0, 51 0, 49 . . . . . . . . . . . . (x(N), c(N)) x(N)

1

. . . x(N)

n

1 0, 55 0.45 B = 1 N

N

  • i=1

2

  • c=1

[p(CM = c|x(i)) − δ(c(i), c(i)

M )]2

B = 1

N [(0, 18 − 0)2 + (0, 82 − 1)2 + (0.51 − 1)2 + (0.49 − 0)2 +

... + (0, 55 − 0)2 + (0.45 − 1)2]

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Cost sensitive classification

Total costs vs accuracy In general the costs of false positive and false negative are not the same Interesting to search for the classifier providing the smallest total cost This classifier can be different to the one with the highest accuracy (probability of correct classification)

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Cost sensitive classification

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ROC curve

(1- sensibility, specificity) In some situations it is difficult to estimate the cost matrix Receiver operating characteristic (ROC) curve based analysis

In 1970 ROC curve is used for the first time in a medical diagnosis problem In the 90s in machine learning CM = 1 ⇐ ⇒ p(C = 1|x) > t, where t denotes a threshold ROC curve: For each t in the interval [0, 1] we represent the corresponding bidimensional point (false positive ratet, true positive ratet)

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ROC curve

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ROC curve. Comparing two classifier with the area under the ROC curve (AUC)

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Outline

1 Introduction 2 Discrete Predictors 3 Validation of Supervised Classifiers 4 Summary

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Bayesian network based classifiers

Provides a posterior probability for each possible value of the class Competitive results (accuracy, Brier, ROC) with the state of the art in supervised classifiers Knowledge discovery from the structure of the Bayesian network Honest validation is mandatory

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BAYESIAN NETWORK CLASSIFIERS

Pedro Larra˜ naga

Computational Intelligence Group Artificial Intelligence Department Universidad Polit´ ecnica de Madrid

Bayesian Networks: From Theory to Practice International Black Sea University Autumn School on Machine Learning 3-11 October 2019, Tbilisi, Georgia Pedro Larra˜ naga Bayesian Network Classifiers 52 / 52