Session 36, Monday, April

Warm up/Review

You are working with a bit string of length 8.  You are interested in two events...

A = First two bits are '11'

B = There are at least two consecutive 0s

P(A)=64/256

P(B)=201/256

P(Neither A nor B)=34/256

P(A^B) = ?

P(AVB) = ?

P(A|B) =  ?

P(B|A) = ?

 

( 43/256 , 222/256 , 43/201 , 43/64)

 

Inference by enumeration

Consider the following

  Summer Winter (¬Summer)
  Sunny ¬Sunny Sunny ¬Sunny
Warm 0.576 0.144 0.072 0.008
Cold (¬Warm) 0.016 0.064 0.012 0.108

 

P(Summer) = ???

P(Warm V Summer) = ???

P(¬Warm|Summer) = ???  (Concept of normalization involved here)

 

Independence

A and B are independent iff

P(A|B) = P(A) or P(B|A) = P(B).

Consider the joint distribution table for P(Summer,Sunny,Warm,Cavity)  This would be a table with 16 cells.  Something like

 

  Summer Winter (¬Summer)
  Sunny ¬Sunny Sunny ¬Sunny
Warm 0.4608 (0.1152) 0.1152 (0.0288) 0.0576 (.0144) 0.0064 (0.0016)
Cold (¬Warm) 0.0128 (0.0032) 0.0512 (0.0128) 0.0096 (0.0024) 0.0864 (0.0216)

 

 Presumably there is no relationship between Cavity and Summer.  However, we can test this by showing that

P(Sunny|Cavity) = P(Sunny)