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--- public:t-622-arti-13-1:lab_5_-_bayesian_networks [2013/03/12 09:15] – [Problem 2: The Pirate Treasure] stephan
+++ public:t-622-arti-13-1:lab_5_-_bayesian_networks [2024/04/29 13:33] (current) – external edit 127.0.0.1
@@ Line 9: / Line 9: @@
   * http://www.aispace.org/bayes/bayes.jnlp (using Java Web Start)
   * http://www.cs.cmu.edu/~javabayes/ (a slightly older Java applet)
+The following laws of probabilities might be helpful:
+  * <latex>$P(\neg A) = 1 - P(A)$</latex>
+  * Bayes Law: <latex>$P(A \land B) = P(A|B) * P(B)$</latex>
+  * Marginalization/Summing out: <latex>$P(A) = \sum_b P(A \land b) = P(A \land B) + P(A \land \neg B)$</latex>
+  * Independence 1: <latex>$P(A|B,C) = P(A|C)$</latex>, if <latex>$A$</latex> and <latex>$B$</latex> are independent given <latex>$C$</latex>.
+  * Independence 2: <latex>$P(A \land B|C) = P(A|C) * P(B|C)$</latex>, if <latex>$A$</latex> and <latex>$B$</latex> are independent given <latex>$C$</latex>. (This follows from Independence 1 and Bayes Law.)
+All those laws allow arbitrary additional conditions as long as all probabilities have the condition:
+  * Bayes Law: <latex>$P(A \land B|C) = P(A|B,C) * P(B|C)$</latex>
+  * Marginalization/Summing out: <latex>$P(A|C) = \sum_b P(A \land b|C) = P(A \land B|C) + P(A \land \neg B|C)$</latex>
 ==== Problem 1: Smelly Doors ====
@@ Line 18: / Line 29: @@
   * it can check whether the room smells bad or not and use that information as an indicator.
-  - Design a very simple Bayesian network usable by the NPC to decide when in front of a door, whether he should open it or not. Start by inventing reasonable probabilities for the relation between the contents of the room and its smell.
+  - Design a very simple Bayesian network (2 nodes) usable by the NPC to decide when in front of a door, whether he should open it or not. Start by inventing reasonable probabilities for the relation between the contents of the room and its smell.
   - How probable is it that the room contains a monster if the air smells bad?
   - Argue how, instead of using made-up probabilities, the NPC can learn as he opens doors and dynamically update the Bayesian network becoming smarter.
@@ Line 24: / Line 35: @@
 Hand in:
   - the Bayesian network (e.g., a screenshot and the probability tables for all nodes of the network)
+  - answer to question 2 above
   - a few sentences on how the NPC could learn the necessary probabilities