DA6. Belief Networks¶
Statement¶
Review the Unit 6 belief network available for download on the Reading Assignment page. The belief network extends the electrical domain to include an overhead projector.
Answer the following questions about how knowledge of the values of some variables would affect the probability of another variable:
- Can knowledge of the value of Projector_plugged_in affect your belief in the value of Sam_reading_book? Explain your answer.
- Can knowledge of Screen_lit_up affect your belief in Sam_reading_book? Explain your answer.
- Can knowledge of Projector_plugged_in affect your belief in Sam_reading_book given that you have observed a value for Screen_lit_up? Explain your answer.
- Which variables could have their probabilities changed if just Lamp_works was observed?
- Which variables could have their probabilities changed if just Power_in_prjector was observed?
Answer¶
Introduction¶
A belief network is a directed acyclic graph representing conditional dependence among a set of random variables. The random variables are the nodes. The arcs represent direct dependence The probabilistically of a variable depends on each of its parents, but is independent of its other predecessors (Poole & Mackworth, 2017).
1. Can knowledge of the value of Projector_plugged_in affect your belief in the value of Sam_reading_book?¶
No, the predecessors (parents) of Sam_reading_book are Room_Light_on, Light_Switch_on, and Power_in_Building. Projector_plugged_in has no parent, nor direct or indirect connection to Sam_reading_book. Therefore, Projector_plugged_in cannot affect the probability of Sam_reading_book.
Answer: No.
2. Can knowledge of Screen_lit_up affect your belief in Sam_reading_book?¶
The parent of Screen_lit_up are Mirror_Working, Projector_Lamp_On, Lamp_Works, Power_in_Projector, Projector_Switch_On, Power_in_Wire, Projector_plugged_in, and Power_in_Building.
The parent of Sam_reading_book are Room_Light_on, Light_Switch_on, and Power_in_Building.
Notice that Power_in_Building is a parent of both Screen_lit_up and Sam_reading_book. Therefore, If there is no Power_in_Building, then both Screen_lit_up and Sam_reading_book will have a probability of 0.
While there is no direct connection between Screen_lit_up and Sam_reading_book, if Screen_lit_up is 0 and Power_in_Building is 0, then Sam_reading_book will also have a probability of 0.
Answer: Yes.
3. Can knowledge of Projector_plugged_in affect your belief in Sam_reading_book given that you have observed a value for Screen_lit_up?¶
As we found in question 1, Projector_plugged_in has no direct or indirect connection to Sam_reading_book. However, Power_in_Wire is dependent (child of) on both Power_in_Building and Projector_plugged_in. Therefore, Screen_lit_up is 1, then Power_in_Wire must be 1, and Power_in_Building must be 1.
Therefore, Sam_reading_book will only depend on Room_Light_on and Light_Switch_on, which may give a different probability than when Screen_lit_up is 0.
Answer: Yes.
4. Which variables could have their probabilities changed if just Lamp_works was observed?¶
The children of Lamp_Works are Projector_Lamp_On, Screen_lit_up, and Ray_Says_Screen_is_Dark. If we observe Lamp_Works, then the probabilities of these children will change.
5. Which variables could have their probabilities changed if just Power_in_projector was observed?¶
The children of Power_in_Projector are Projector_Lamp_On, Screen_lit_up, and Ray_Says_Screen_is_Dark.
The parents of Power_in_Projector are Projector_Switch_On, Power_in_wire Projector_plugged_in, Power_in_Building, Room_Light_on, and Sam_reading_book.
If we observe Power_in_Projector, then the probabilities of its children, its parents, and all that are dependent on them will have their probabilities changed.
Answer: All except Mirror_Working, Light_Switch_on, and Lamp_Works.
Conclusion¶
| Question | Answer |
|---|---|
| Can knowledge of the value of Projector_plugged_in affect your belief in the value of Sam_reading_book? | No |
| Can knowledge of Screen_lit_up affect your belief in Sam_reading_book? | Yes |
| Can knowledge of Projector_plugged_in affect your belief in Sam_reading_book given that you have observed a value for Screen_lit_up? | Yes |
| Which variables could have their probabilities changed if just Lamp_works was observed? | Projector_Lamp_On, Screen_lit_up, and Ray_Says_Screen_is_Dark |
| Which variables could have their probabilities changed if just Power_in_projector was observed? | All except Mirror_Working, Light_Switch_on, and Lamp_Works |
References¶
- Poole, D. L., & Mackworth, A. K. (2017). Artificial Intelligence: Foundations of computational agents. Cambridge University Press. https://artint.info/2e/html/ArtInt2e.html Chapter 9: Reasoning under uncertainty.