A. Darwiche Sensitivity Analysis in Bayesian Networks Adnan Darwiche Computer Science Department

A. Darwiche Bayesian network classifiers Given: –A Bayesian network N –A class variable C –A set of variables, attributes E = {E 1, …, E n } Each instantiation e is called an instance –A probability threshold p Define Bayesian network classifier F:

A. Darwiche Naïve Bayes Classifiers PPr(p) yes0.87 no0.13 PUPr(u|p) yes-ve0.27 no+ve0.107 PBPr(b|p) yes-ve0.36 no+ve0.106 PSPr(s|p) yes-ve0.10 no+ve0.01 UBSPr(P = yes | u, b, s) +ve ve -ve ve-ve+ve ve-ve ve+ve ve+ve-ve ve +ve ve  0.9? Yes No Yes No Yes Which sets of test results confirm pregnancy, with probability no less than 90%? Pregnant? (P) Urine test (U) Blood test (B) Scanning test (S)

A. Darwiche Reasoning about Bayesian network classifiers Given N and N’, do they induce the same classifier? If not, which, and how many, instances do they disagree on? Given N, what are the allowable changes to a CPT which will not change the current classifier? How many distinct classifiers can be induced by changing some CPT?

A. Darwiche Reasoning about Bayesian network classifiers We can answer these questions by enumerating all instances e explicitly However, this is often infeasible given the exponential number of instances Instead, we propose to build a tractable logical representation of the classifier F N This allows us to answer the above questions in time linear in the size of the representation

A. Darwiche From Numbers to Decisions + Probabilistic Inference 0.87yes 0.13no Pr(p)P 0.27-veyes no P ve Pr(u|p)U 0.36-veyes no P ve Pr(b|p)B 0.10-veyes no P 0.01+ve Pr(s|p)S Pregnant? (P) Urine test (U) Blood test (B) Scanning test (S) Decision Function Test results: U, B, S Yes, No

A. Darwiche U +ve -ve B S Yes +ve -ve No -ve +ve Situation: U=+ve, B=-ve, S=-ve 0.87yes 0.13no Pr(p)P 0.27-veyes no P ve Pr(u|p)U 0.36-veyes no P ve Pr(b|p)B 0.10-veyes no P 0.01+ve Pr(s|p)S Pregnant? (P) Urine test (U) Blood test (B) Scanning test (S) Ordered Decision Diagram + Probabilistic Inference From Numbers to Decisions

A. Darwiche X1 X2 X3 1 0 Binary Decision Diagram Test-once property

A. Darwiche Improving Reliability of Sensors Currently False negative 27.0% False positive 10.7% Pregnant? (P) Urine test (U) Blood test (B) Scanning test (S) Same decisions (in all situations) if new test is: False negative 10% False positive 5% Different decisions (in some situations) if new test: False negative 5% False positive 2.5% Can characterize these situations, compute their likelihood, analyze their properties Yes if > 90%

A. Darwiche Adding New Sensors Pregnant? (P) Urine test (U) Blood test (B) Scanning test (S) New test (N) Can characterize these situations, compute their likelihood, analyze their properties Same decisions (in all situations) if: False negative 40% False positive 20% Different (in some situations) decisions if: False negative 20% False positive 10% Yes if > 90%

A. Darwiche Naïve Bayes classifier Class variable C Attributes E PPr(p) yes0.87 no0.13 PUPr(u|p) yes-ve0.27 no+ve0.107 PBPr(b|p) yes-ve0.36 no+ve0.106 PSPr(s|p) yes-ve0.10 no+ve0.01

A. Darwiche Naïve Bayes classifier Prior log-oddsWeight of evidence e i

A. Darwiche Changing the prior log-odds in a naïve Bayes classifier If we change the CPT of C, thereby changing the prior log-odds from log O(c) to log O’(c), will we still have the same classifier?

A. Darwiche Changing the prior log-odds in a naïve Bayes classifier 

A. Darwiche Equivalence of NB classifiers

A. Darwiche Equivalence of NB classifiers Equivalent iff prior of P in F N’  [0.684, 0.970) Change prior of P

A. Darwiche Path 1Path 2 Sub-ODD D 1 Sub-ODD D 2

A. Darwiche Path 1Path 2 Sub-ODD D 1 = D 2

A. Darwiche Theoretical results of algorithm Space complexity: –Total number of nodes in the ODD  O(b n/2 ) Time complexity: –O(nb n/2 ) Improves greatly over brute-force approach: –Total number of instances = O(b n )

A. Darwiche Experimental results of algorithm Network# Attributes# Instances# Nodes bound# Nodes Tic-tac-toe Votes Spect22 4  Breast-cancer-w9 1  Hepatitis19 2  Kr-vs-kp36 1  Mushroom22 1  

A. Darwiche

A. Darwiche