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8.1.5-Reasoning&D-Separation (1)

8.1.5-Reasoning&D-Separation (1) - Machine Learning...

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Unformatted text preview: Machine Learning Srihari 1 Reasoning Patterns and D- Separation Sargur Srihari [email protected] Machine Learning Srihari Topics • Reasoning Patterns – Causal and Evidential Reasoning • D-separation – Direct Connection – Indirect Connection • Soundness and Completeness • Algorithm for D-separation • I-Equivalence 2 Machine Learning Srihari Bayesian Network: Student Model Graph and CPDs Chain rule for Bayesian network Val(I)={i =low intelligence, i 1 = high intelligence} Val(D)={d =easy, d 1 =hard} Val(G)={g 1 =A, g 2 =B, g 3 =C} Val(S)={s =low, s 1 =high) Val(L)={l =weak, l 1 =strong} 3 Grade Letter SAT Intelligence Difficulty d 1 d 0.6 0.4 i 1 i 0.7 0.3 i i 1 s 1 s 0.95 0.2 0.05 0.8 g 1 g 2 g 2 l 1 l 0.1 0.4 0.99 0.9 0.6 0.01 i , d i , d 1 i , d i , d 1 g 2 g 3 g 1 0.3 0.05 0.9 0.5 0.4 0.25 0.08 0.3 0.3 0.7 0.02 0.2 P ( D , I , G , S , L ) = P ( D ) P ( I ) P ( G | D , I ) P ( S | I ) P ( L | G ) P ( i 1 , d , g 2 , s 1 , l ) = P ( i 1 ) P ( d ) P ( g 2 | i 1 , d ) P ( s 1 | i 1 ) P ( l | g 2 ) =0.3 ⋅ 0.6 ⋅ 0.08 ⋅ 0.8 ⋅ 0.4=0.004608 Machine Learning Srihari Reasoning Patterns Reasoning about a student George using the model • Causal Reasoning – George is interested in knowing as to how likely he is to get a strong letter (based on intelligence, difficulty)? • Evidential Reasoning – Recruiter is interested in knowing whether George is intelligent (based on letter, SAT) 4 Grade Letter SAT Intelligence Difficulty d 1 d 0.6 0.4 i 1 i 0.7 0.3 i i 1 s 1 s 0.95 0.2 0.05 0.8 g 1 g 2 g 2 l 1 l 0.1 0.4 0.99 0.9 0.6 0.01 i , d i , d 1 i , d i , d 1 g 2 g 3 g 1 0.3 0.05 0.9 0.5 0.4 0.25 0.08 0.3 0.3 0.7 0.02 0.2 Machine Learning Srihari Causal Reasoning Observe how probabilities change as evidence is obtained 1. How likely is George to get a strong letter (knowing nothing else)? • P(l 1 )=0.502 • Obtained by summing-out other variables in joint distribution 2 But George is not so intelligent ( i ) • P(l 1 |i )=0.389 3. Next we find out ECON101 is easy ( d ) • P(l 1 |i , d )=0.513 5 Query is Example of Causal Reasoning: Predicting downstream effects of factors such as intelligence Grade Letter SAT Intelligence Difficulty d 1 d 0.6 0.4 i 1 i 0.7 0.3 i i 1 s 1 s 0.95 0.2 0.05 0.8 g 1 g 2 g 2 l 1 l 0.1 0.4 0.99 0.9 0.6 0.01 i , d i , d 1 i , d i , d 1 g 2 g 3 g 1 0.3 0.05 0.9 0.5 0.4 0.25 0.08 0.3 0.3 0.7 0.02 0.2 P ( D , I , G , S , l 1 ) = P ( D ) P ( I ) P ( G | D , I ) P ( S | I ) P ( l 1 | G ) D , I , G , S ∑ Machine Learning Srihari Evidential Reasoning • Recruiter wants to hire intelligent student • A priori George is 30% likely to be intelligent • P(i 1 )=0.3 • Finds that George received grade C ( g 3 ) in ECON101 • P(i 1 |g 3 )=0.079 • Similarly probability class is difficult goes up from 0.4 to • P(d 1 |g 3 )=0.629 • If recruiter has lost grade but has letter • P(i 1 |l )=0.14 6 Grade Letter SAT Intelligence Difficulty d 1 d 0.6 0.4 i 1 i 0.7 0.3 i i 1 s 1 s 0.95 0.2 0.05 0.8 g 1 g 2 g...
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8.1.5-Reasoning&D-Separation (1) - Machine Learning...

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