10-ProbModel

# 10-ProbModel - Probabilistic Model CS273 Data and Knowledge...

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Probabilistic Model S273 ata and Knowledge Bases CS273 - Data and Knowledge Bases Xifeng Yan Computer Science niversity of California at Santa Barbara University of California at Santa Barbara

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Department of Computer Science Readings: 1. Probabilistic retrieval based on staged logistic regression, William S. Cooper, Fredric C. Gey, Daniel P. Dabney 2. A Probabilistic Model of Information Retrieval: Development and Status (1998) K. Sparck Jones, S. Walkerr , S. Robertson CS273: Data and Knowledge Bases | University of California at Santa Barbara 2
Department of Computer Science The Basic Question What is the probability that THIS document is relevant to THIS query? Formally… 3 random variables: query Q, document D, levance R 1} relevance R {0,1} Given a particular query q, a particular document d, p(R=1|Q=q,D=d)=? CS273: Data and Knowledge Bases | University of California at Santa Barbara 3 slides by courtesy of Zhai with modifications

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Department of Computer Science etour Detour . .. rief Review of Probability & Statistics Brief Review of Probability & Statistics CS273: Data and Knowledge Bases | University of California at Santa Barbara 4
Department of Computer Science Purpose of Prob. & Statistics Deductive vs. Plausible reasoning Incomplete knowledge -> uncertainty How do we quantify inference under uncertainty? Probability: models of random process/experiment (how the data is observed) Statistics: draw conclusions on the whole population based on samples CS273: Data and Knowledge Bases | University of California at Santa Barbara 5

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Department of Computer Science Basic Concepts in Probability Sample space: all possible outcomes, e.g., Tossing 2 fair coins, S ={HH, HT, TH, TT} Event: E S, E happens iff outcome in E, e.g., E={HH} (all heads) E={HH,TT} (same face) Probability of Event : 1 P(E) 0, s.t. P(S)=1 (outcome always in S) P(A B)=P(A)+P(B) if (A B)= CS273: Data and Knowledge Bases | University of California at Santa Barbara 6
Department of Computer Science Basic Concepts of Prob. (cont.) Conditional Probability :P(B|A)=P(A B)/P(A) P(A B) = P(A)P(B|A) =P(B)P(B|A) So, P(A|B)=P(B|A)P(A)/P(B) For independent events, P(A B) = P(A)P(B), so P(A|B)=P(A) Total probability: If A 1 , …, A n form a partition of S, then P(B)= P(B S)=P(B A 1 )+…+P(B A n ) (why?) o P(A )=P(B|A (A P(B) (Bayes’ Rule) So, P(A i |B) P(B|A i )P(A i )/P(B) (Bayes Rule) CS273: Data and Knowledge Bases | University of California at Santa Barbara 7

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Department of Computer Science Interpretation of Bayes’ Rule Hypothesis space: H={H 1 ,…,H n } Evidence: E ) ( ) | ( ) | ( H P H E P E H P i i i ) ( we want to pick the most likely hypothesis H*, we can drop p(E) If we want to pick the most likely hypothesis H, we can drop p(E) Posterior probability of H i Prior probability of H i ikelihood of data/e idence ) ( ) | ( ) | ( i i i H P H E P E H P CS273: Data and Knowledge Bases | University of California at Santa Barbara 8 Likelihood of data/evidence if H i is true
Department of Computer Science Random Variable X: S  (“measure” of outcome)

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10-ProbModel - Probabilistic Model CS273 Data and Knowledge...

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