Computer Automated Reasoning Science Group Conditional Independence in Testing Bayesian Networks Yujia Shen, Haiying Huang, Arthur Choi, Adnan Darwiche. Computer Science Department, UCLA Computer Science Department Conditional Independence in Testing Bayesian Networks
Fuse Knowledge with Expressiveness DEEP LEARNING • Neural networks are universal approximators. • They are data hungry. Pr ( Label | Feature ) Feature Sampled functions that are represented using a simple neural network. Computer Science Department Conditional Independence in Testing Bayesian Networks June 9, 2019 2
Fuse Knowledge with Expressiveness BAYESIAN NETWORKS • BNs utilize data efficiently using conditional independence assumptions. P ? (¯ z ) P ? ( z ) + + X * * * * θ ¯ θ z | ¯ θ ¯ θ z | x * * z | ¯ z | x x x + + * * Y Z λ ¯ θ ¯ λ x θ x * * * * x x θ ¯ θ y | ¯ θ ¯ θ y | x λ ¯ λ y y | ¯ y | x x y x Computer Science Department Conditional Independence in Testing Bayesian Networks June 9, 2019 3
Fuse Knowledge with Expressiveness EXPRESSIVENESS IN BAYESIAN NETWORKS • BNs utilize data efficiently using conditional independence assumptions. • Marginal queries are not universal approximators. Ground truth Best fit for BN Computer Science Department Conditional Independence in Testing Bayesian Networks June 9, 2019 4
Fuse Knowledge with Expressiveness TESTING BAYESIAN NETWORK • Testing Bayesian networks are universal approximators [Choi, Darwiche(2018)]. Ground truth Best fit for TBN Best fit for BN Universal Approximator Computer Science Department Conditional Independence in Testing Bayesian Networks June 9, 2019 5
<latexit sha1_base64="Fdp5m+1URdtiDP3EbzMyZDz15A=">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</latexit> <latexit sha1_base64="28t5eMjcvZCUmP7wae8louf6PqM=">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</latexit> Testing Bayesian Network A SET OF DISTRIBUTIONS • TBN represents a set of distributions. • Different evidence selects different distribution for inference. Pr x =0 . 2 ,y =0 . 2 ( z = 1 | x = 0 . 2 , y = 0 . 2) X Pr x =0 . 6 ,y =0 . 4 ( z = 1 | x = 0 . 6 , y = 0 . 4) Y Z Computer Science Department Conditional Independence in Testing Bayesian Networks June 9, 2019 6
<latexit sha1_base64="NA5fsBhU/a+XkaBYb6PV/vHj4zI=">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</latexit> <latexit sha1_base64="RGesx19ZqPhHqT2QjXdOhg4S52g=">ACGHicdVBLSwMxGMzWV62vVY9egkXwIHVTxba3ohePFewDtmvJpmkbmn2QZIW67M/w4l/x4kERr735b8y2XVDRgcAwM1/yZdyQM6ks69PILS2vrK7l1wsbm1vbO+buXksGkSC0SQIeiI6LJeXMp03FKedUFDsuZy23fFV6rfvqZAs8G/VJKSOh4c+GzClZ65mncnV1i6HrxFbJsiyE0ElKUOXC0qRWq5ZRNWmIu/ghSXpmMQvBLASzESpVECzR65rTbD0jkUV8RjqW0kRUqJ8ZCMcJpUuhGkoaYjPGQ2pr62KPSiWc7JfBIK304CIQ+voIz9ftEjD0pJ56rkx5WI/nbS8W/PDtSg6oTMz+MFPXJ/KFBxKEKYNoS7DNBieITARTO8KyQgLTJTusqBLyH4K/yetcgmdlco358X65aKOPDgAh+AYIFABdXANGqAJCHgEz+AVvBlPxovxbnzMozljMbMPfsCYfgEaLpyz</latexit> <latexit sha1_base64="s+7dbYZNjxeTAuOpN81VKRfJneU=">AB+3icbZDLTsJAFIZP8YZ4q7h0M5GYwIa0aKIbE6Ibl5gImEBDpsMUJkwvmZkaCvIqblxojFtfxJ1v41C6UPBPJvnyn3NyzvxuxJlUlvVt5NbWNza38tuFnd29/QPzsNiSYSwIbZKQh+LBxZJyFtCmYorTh0hQ7Luct3RzbzefqRCsjC4V0lEHR8PAuYxgpW2emaxIcrjp2RSQVcoxUmlZ5asqpUKrYKdQkyNXrmV7cfktingSIcS9mxrUg5UywUI5zOCt1Y0giTER7QjsYA+1Q60/T2GTrVTh95odAvUCh1f09MsS9l4ru608dqKJdrc/O/WidW3qUzZUEUKxqQxSIv5kiFaB4E6jNBieKJBkwE07ciMsQCE6XjKugQ7OUvr0KrVrXPqrW781L9OosjD8dwAmWw4QLqcAsNaAKBMTzDK7wZM+PFeDc+Fq05I5s5gj8yPn8AXiWTWw=</latexit> <latexit sha1_base64="D5jV7Pj2ESGU8qgxYfahYif6go0=">ACGXicbZC7TsMwFIYdrqXcAowsERUSA4qSNGrLVsHCWCR6kdpQOa7TWnUush2kEOU1WHgVFgYQYoSJt8FpM0DLkSx9+v9z7OPfjSjhwjC+lZXVtfWNzdJWeXtnd29fPTjs8DBmCLdRSEPWcyHlAS4LYiguBcxDH2X4q47vcr97j1mnITBrUgi7PhwHBCPICikNFSNdDC7pM/GrpMa+kWjZtm1c0M3jLpmTlYdbtqZy12lyYPWTZUK7mZl7YMZgEVUFRrqH4ORiGKfRwIRCHnfdOIhJNCJgiOCsPYo4jiKZwjPsSA+hj7qSzpTLtVCojzQuZPIHQZurviRT6nCe+Kzt9KCZ80cvF/7x+LyGk5IgigUO0PwhL6aCLU8Jm1EGEaCJhIgYkTuqEJZBAJGWZhmAufnkZOpZuVnXrxq40L4s4SuAYnIAzYI6aIJr0AJtgMAjeAav4E15Ul6Ud+Vj3rqiFDNH4E8pXz/My50M</latexit> Conditional Independence in TBN Suppose X is d-separated from Y given Z. In classical Bayesian networks, In testing Bayesian networks, Pr ( x | yz ) = Pr ( x | z ) Pr yz ( x | yz ) = Pr z ( x | z ) . . Pr yz is the joint distribution selected under evidence yz Pr z is the joint distribution selected under evidence z Computer Science Department Conditional Independence in Testing Bayesian Networks June 9, 2019 7
Fuse Knowledge with Expressiveness Testing Model Assumptions Universal Approximators Bayesian Bayesian Networks Neural Networks Networks Computer Science Department Conditional Independence in Testing Bayesian Networks June 9, 2019 8
Thank You Thank You Thank You Conditional Independence in Testing Bayesian Networks Computer Science Department Conditional Independence in Testing Bayesian Networks June 9, 2019 9
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