ET-805 Bayesian Knowledge Tracing Ramkumar.Rajendran@iitb.ac.in
Activity - TPS Think individually and write down Bayes’ Theorem? (2 minutes) Pair with your neighbour and check your answer. If both are same write down why it is correct? Else, discuss why the formula that you wrote is correct (3 minutes) Share to the class 2
Activity - Class Response 3
Bayes’ Theorem Pass Study - Relating conditional probabilities - How probable is one event given that the other event occured - Example: Two events a) Passing the exam and b) Studying hard - What is the probability that a student passed an exam given that s/he studied for the exam? P (P/S) = P (P ∩ S)/ P(S) 4
Bayes’ Theorem Pass Study P (P/S) = P (P ∩ S)/ P(S) (from last slide) Similarly P (S/P) = P (P ∩ S)/ P(P) P (P ∩ S)= P (P/S). P(S) = P (S/P). P(P) = P (P/S) = P (S/P). P(P) / P(S) Bayes’ Theorem: P (A/B) = P(B/A). P(A)/ P (B) 5
Activity - Quiz A couple has two children, one of which is a boy. What is the probability that they have two boys? P(A) = Both kids are boys = ¼ P(B) = one of the kids is boy = ¾ P (A/B) = P(B/A). P(A)/P(B) = 1. ¼ / ¾ = ⅓ https://brilliant.org/wiki/bayes-theorem/ 6
Activity - Answer A couple has two children, one of which is a boy. What is the probability that they have two boys? Define two events: P(A) = Both children are boys = ¼ P (B) = one of their children is boy = ¾ P (A/B) = P(A).P (B/A)/ P(B) = ¼ . 1/ ¾ = ⅓ https://brilliant.org/wiki/bayes-theorem/ 7
Bayesian Network - A compact representation of joint probability distributions among set of random variables (nodes) and their conditional dependencies (arcs) - Directed Acyclic Graph (DAG) - Conditional Probability Table (CPT) 8
Confused! 9
Bayesian Network Slept after lunch Missed the Class 10
Bayesian Network Slept Had after Meeting lunch Missed the Class 11
Bayesian Network Slept Had after Meeting lunch (S) (M) Missed the Class (C) Watch Lecture Videos (V) 12
Bayesian Network S M Directed acyclic graph C - Static directions V - Cycles are not allowed Nodes are conditionally Independent - Example: Node S and V are dependent but if the value of C is known, then they will become independent - Similarly node S and M are independent. But if the value of C is known they will be dependent - The above relation is called conditional independency 13
Bayesian Network S M Conditional Probability Table C V Assume the values of nodes are boolean (yes or no) For Node S, How many probability values required to represent? One: Probability of Sleeping after lunch = P(S) For Node C, how many probability values are required to represent? Four: P(C/S,M) = Combination of S and M 14
Bayesian Network S M Conditional Probability Table C V P(X1, …,Xn) = ∏ P (Xi | Parents(Xi)) 15
Activity - TPS Bayesian Network - A compact representation of joint probability distributions among set of random variables (nodes) and their conditional dependencies (arcs) - Directed Acyclic Graph (DAG) - Conditional Probability Table (CPT) Think Why the bayesian network is compact? (2 minutes) Share 16
Bayesian Network S M - Boolean values C - Total 4 nodes, hence we will have to store V 2 (pow) 4 values to represent the fully connected network (2 4 - 1) - Number of probability values in Bayesian Network: 1 (S) + 1 (M) + 4 (C) + 2 (V) = 8 Compact representation of joint probability distributions over all the variables in the network 17
Sample Bayesian Network Conati, C. (2010). Bayesian student modeling. In Advances in intelligent tutoring systems (pp. 281-299). Springer, Berlin, Heidelberg. 18
Bayesian Networks as Learner Model Wayang Outpost - Time spent per problem, time spent per action, average incorrect action to model student’s attitude 19
Activity - One minute thinking In the previous example, which node we can update at every time instance based on student’s response? 20
Activity - One minute thinking In the previous example, which node we can update at every time instance based on student’s response? - The node concept indicates whether the student understood the concept or not. - Hence, we can update the conditional probability table (CPT) of Concept based on student’s response. - Updating CPT of Bayesian network is Dynamic Bayesian Network (DBN) 21
Bayesian Knowledge Tracing (BKT) We want to update a node called concept to indicate whether the student learned a skill or not. If we have a node for student’s response: C Node = Student’s Response to a question. Boolean value. R 1 = correct, 0 = incorrect Corbett, A. T., & Anderson, J. R. (1994). Knowledge tracing: Modeling the acquisition of procedural knowledge. User modeling and user-adapted interaction , 4 (4), 253-278. Baker, R. S., Corbett, A. T., & Aleven, V. (2008, June). More accurate student modeling through contextual estimation of slip and guess probabilities in bayesian knowledge tracing. In Intelligent Tutoring Systems (pp. 406-415). Springer, Berlin, Heidelberg. 22
Activity - TPS Update the P(Concept/response) = Increase if the response = 1, decrease otherwise. Think Write one drawback of the above method the update student’s skill on concept (2 minutes) Pair with your neighbour and copy their answer. Discuss and rank the drawback in the order of importance (3 minutes) Share 23
Activity - Class Response - Guess - By mistake he might selected wrong answer - Silly mistakes - Slip - 24
Bayesian Knowledge Tracing - Student might give correct answer by guess - Student might have understood the concept but answered wrongly by mistake (Slip) 25
Bayesian Knowledge Tracing p(Lo) Initial Learning - the probability of prior knowledge p(T) Acquisition - the probability a rule will make the transition from the unlearned to the learned state following an opportunity to apply the rule p(G) Guess - the probability a student will guess correctly if a rule is in the unlearned state p(S) Slip - the probability a student will slip (make a mistake) if a rule is in the learned state 26
BKT Initial Probability - Prior knowledge 27
Activity - TPS Think How do you measure the probability of learning P(L) if the response to the question is correct Share 28
Updating P(Learning) at State t If the answer is correct, it should be based on student’s prob(L) at state t-1 and not slip to the total probability of answering the question. 29
Activity - TPS Think How do you measure the probability of learning P(L) if the response to the question is incorrect Share P(L4/ observation = incorrect) = P (L3).P(S)/(P(L3).P(S) + 1 -P(L3)) 30
Updating P(Learning) at State t If the answer is incorrect: 31
Updating P(Skill Mastery) To update the P(Learning) for next state t+1, use the probability of learning into probability of transition 32
Activity - TPS Think list down one drawback of BKT (2 minutes) Pair and copy your neighbours answer. Arrange the drawbacks in the order of importance (3 minutes) Share 33
Activity - Class Response 34
BKT Explained Using Examples http://www.cs.williams.edu/~iris/res/bkt/ 35
Next Class Class after Next: Log Data Analytics ● What is Learning Analytics ● 36
Last Activity - Muddy Points List down - two important and - two least clear (muddy) points from today’s class - https://tinyurl.com/et8 05mp 37
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