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The 3 prisoners dilemma
Inferring more complex values
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.
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…
Definitions
• 𝐼𝐵 is the proposition : “Prisoner B will be released”
• 𝐺𝐴 is the proposition : “Prisoner A will be executed”
• Since 𝐺𝐴 ⇒ 𝐼𝐵 (If A is executed, B is released) we get 𝑃 𝐼𝐵 𝐺𝐴 = 1
𝑃 𝐺𝐴 𝐼𝐵
1
𝑃 𝐼𝐵 𝐺𝐴 𝑃(𝐺𝐴 ) 3 1
=
= =
2 2
𝑃(𝐼𝐵 )
3
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.
New context
• 𝐼𝐵′ = “Guard said that B will be released”
𝑃
𝐺𝐴 𝐼𝐵′
=
𝑃
𝐼𝐵′
𝐺𝐴 𝑃 𝐺𝐴
𝑃 𝐼𝐵′
1 1
⋅
1
2
3
=
=
1
3
2
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.
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.
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?
What else can Bayesian networks
compute?
Boolean functions of variables
• Let’s say we are interested in the value of (𝑥2 ∨ 𝑥3 ) ∧ 𝑥6
• We can do so by easily creating new OR and AND nodes.
𝑥1
• Let 𝑄 ′ = 𝑂𝑅(𝑥2 , 𝑥3 )
𝑥2
𝑥3
𝑥5
• 𝑄 = 𝐴𝑁𝐷(𝑄 ′ , 𝑥6 )
𝑥4
𝑄′
• Finally, we use the same inference tools.
𝑥6
Figure 4.35
𝑄
Fin