1. The 3 prisoners dilemma Inferring more complex values
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The 3 prisoners dilemma
Inferring more complex values

2. The 3 prisoners dilemma - Story
There are currently 3 prisoners on death row - A, B and C.
Only one of them is to be executed and the other 2 murders will be released – Cuz’ logic.
Prisoner A makes a request to the guard – tell me which of the other 2 would be released.
To which he answers – Prisoner B.

3. The 3 prisoners dilemma – Story (2)
Prisoner A thinks to himself – before I asked – I had 33% chance of being executed.
Now – those chances have risen to 50%
What did I do wrong? I made certain not to ask for any information relevant to my own fate…

4. Definitions 𝐼 𝐵 is the proposition : “Prisoner B will be released”
𝐺 𝐴 is the proposition : “Prisoner A will be executed”
Since 𝐺 𝐴 ⇒ 𝐼 𝐵 (If A is executed, B is released) we get 𝑃 𝐼 𝐵 𝐺 𝐴 =1
𝑃 𝐺 𝐴 𝐼 𝐵 = 𝑃 𝐼 𝐵 𝐺 𝐴 𝑃( 𝐺 𝐴 ) 𝑃( 𝐼 𝐵 ) = = 1 2

5. When the bare facts won’t do
When facts are wrongly formulated, we might draw false conclusions
Prisoner A might think to himself – If the guard had named C instead of B then by sheer symmetry I would also have 50% of being executed. So I had 50% to begin with. Which is clearly false.
This happens because we omit the context – or rather, what was the question in the first place? – If the question was “Would B be executed?” this analysis would have been correct.

6. New context 𝐼 𝐵 ′ = “Guard said that B will be released”
𝑃 𝐺 𝐴 𝐼 𝐵 ′ = 𝑃 𝐼 𝐵 ′ 𝐺 𝐴 𝑃 𝐺 𝐴 𝑃 𝐼 𝐵 ′ = 1 2 ⋅ = 1 3

7. The thousand prisoners problem
Same setting – but now 999 are to be freed, 1 is to be executed.
Prisoner A finds a list – containing 998 names of to be freed prisoners.
Sadly for prisoner A, his name is not among them.

8. Implications of said list
If all the prisoners in the list were chosen at random – his chances of survival are abysmal.
If the query for that list would be – print me 998 right handed, innocent prisoners, while he’s the only left handed prisoner in Jail – he still has 999/1000 chance of survival.
Even though we don’t change the information we were given – the likelihood of A being executed changes.

9. Intuition
If a prisoner would be given the query (right handed only) before he found the list – the results would not surprise him, since he knows he would not be among them
If he found the list before the query – even though the math is exactly the same, psychologically – he would be freaking out.
This intuition basically translates into a prior. How so?

10. What else can Bayesian networks compute?

11. Boolean functions of variables
Let’s say we are interested in the value of ( 𝑥 2 ∨ 𝑥 3 )∧ 𝑥 6
We can do so by easily creating new OR and AND nodes.
Let 𝑄 ′ =𝑂𝑅( 𝑥 2 , 𝑥 3 )
𝑄=𝐴𝑁𝐷( 𝑄 ′ , 𝑥 6 )
Finally, we use the same inference tools.
𝑥 1
𝑥 2
𝑥 3
𝑥 5
𝑥 4 𝑄′
𝑥 6
Figure 4.35 𝑄

12. Fin