Figure 12 A Bayesian network modeling granularity relationships.
Related Figures (20)
Fig. 2. A step-by-step modeling of the sneezing example. Prior probability distribution is an unconditional distribution that does not change, no matter the observations we make. Cat is one of the two root nodes, in our example network. We obtain the probability distribution for that node by answering a question “how likely is a couple (like Paul and Susan) to have a cat? ” The answer will be a number between 0 and 1 (or between 0 and 100%, which can later be transformed into a probability by dividing it by a 100). Let us take a guess that about 20% of couples own cats. Therefore, we arrive at the probability of 0.2 that Paul and Susan have a cat. Fig. 4. A Bayesian network equivalent to the one for the sneezing example from Fig. 2(e), but with all arcs reversed. Sneezing is very unlikely to happen in the absence of both the factors (0.01). Furthermore, Rhinitis has a slightly higher chance of causing Sneezing than does Cold (0.9 versus 0.85). When both factors are present, Sneezing is bound to occur (0.99). Two of the probabilities above are quite extreme. Despite that, they still leave some room for uncertainty. For example, John will be Sneezing, on average, in one out of a 100 situations when he has not caught Cold and was not having Rhinitis. Assigning probabilities of 0 o1 Fig. 5. The sneezing example model with parameters. Fig. 6. Possible configurations of three adjacent variables in a Bayesian network. BE EE A Ee ne ne ey SA eS SAE eS SRE Ie ROSAS we EMO eT UN CS Re ANS SEES ST Sen MU oe we Fel AER RT Ne AN eK CONES oes meee ee RLS ee. Neen ewe ge However, variables can be connected through other variables. The notion of conditional independence is at the center of cases like that. Fig. 6 shows the possible configurations of three adjacent variables that can be found in a BN. In the chain configuration (also called linear or serial), A and C are dependent if we do not know which state B is in. Once we know the state of B, A and C become independent. The state of Cis influenced only by the state of B, and no change in A is carried over to C. We say that C is independent of A given B. In the sneezing example, achain is formed by Allergies, Rhinitis, and Sneezing. If we do not know whether or not John is having Rhinitis, knowing whether or not he has Allergies influences our belief about him Sneezing. Ifhe has Allergies, probability of Sneezing is higher; if he has not, it is lower. However, once we know that John is not having Rhinitis whether or not he has Allergies becomes irrelevant from the point of view of Sneezing. Fig. 7 shows the evolution of the model’s parameters as new evidence is introduced. The first step is the computation of all a prio. probabilities. This step is commonly called the initialization process, and its results are known as the initial state. Fig. 7. Probability distributions associated with variables in the sneezing example at different stages of inference. When John starts Sneezing the probabilities are updated to account for this information. Note, that the probabilities of the positive states of all nodes in the network are higher. This is because the evidence available (“John is sneezing”) supports the positive state of all the variables. Fig. 9. Prerequisite relationship between two pieces of knowledge. Fig. 10. A Bayesian network backbone for prerequisites. Fig. 11. A Bayesian student model based on misconceptions. Fig. 13. A Bayesian network modeling granularity and prerequisite relationships simultaneously. Fig. 14. A Bayesian network modeling granularity and prerequisite relationships simultaneously - with intermediate variable introduced One way of making the machine learning process more controlled is to introduce structural constraints (enforce or prohibit links between certain variables) and learn the reminder of the structure along with parameters or learn exclusively the parameters. The accuracy of the resulting model can be subsequently investigated. Fig. 15. A Bayesian network modeling two ways of a learner’s knowledge acquisition. Fig. 16. A dynamic Bayesian network for student modeling. So far, we have discussed several simple BNs used to represent student’s features (mainly knowledge). The goal of this section is tc present a selection of more complex BNs that have been used to model a wide range of features and processes, such as: problem solving metacognitive skills, and emotional state and affect. 4.5.1. Problem solving A good example of modeling the problem solving process is the physics tutor ANDES (Conati et al.1997, 2002). The student model o} ANDES allows three kinds of assessment: a) plan recognition (discovering the most likely problem solving strategy that the student i: employing at a given point in time), b) prediction of student goals and actions, and c) long-time assessment of the student’s knowledge. A BN implemented in ANDES includes variables of the following types: Fig. 18. A BN supporting the Explanation Based Learning of Correctness (EBLC). Fig. 19. A Bayesian network for the Prime Climb game. By the moment, the nodes defined account for the skills and precedence relationships to be taken into account when teaching the nplex algorithm. More nodes are needed to collect evidence about a student: e Evidential problem nodes. As the simplex algorithm is a procedural domain, the evidential nodes are problems to be solved by the student. We therefore define propositional variables: Remember that the ability nodes have been defined as the set of skills necessary to solve certain kind of problems: in this way, there is a causal relationship between the ability and the problem. That is, only students who have the ability (i.e., the necessary set of skills) will be able to solve a particular problem. Fig. 20. A learning strategy for the simplex algorithm. Finally the Bayesian network that accounts for a student model for the Simplex algorithm is shown in Fig. 21. So now our BSM for the Simplex algorithm is finished. Recall that in this model there are only specific questions to test skills Sg to S14. Fig. 21. A Bayesian student model for the Simplex algorithm.
