Statistics

1 Let F� 
be the function satisfying (�A�B�|��X�)�=�F�(�(�A�|��X�)�,�(�B�|��A�X�)�)� 
for all A� 
, B�
, and X�
, as in Rule II(b). Letting x�=�(�A�|��X�)� 
, y �= �(�B�|��A�X�)� 
, and z�=�(�C�|��A�B�X�)� 
, show that 
 F�(�x�,�F�(�y�,�z�)�)�=�F�(�F�(�x�,�y�)�,�z�) �

 [Hint: both sides are equal to (�A�B�C�|��X�)� 
]

2) Give an example of a background assumption X and two propositions A� 
 and B �
such that X�A�
 is logically equivalent to X � B �
but A �
 is not logically equivalent to B �
. Does it follow that P [ A | X ] = P [ B | X ] ? 3 ) Give an argument using examples for why the plausibility ( A B | X ) should only depend on the plausibilities ( A | X ) and ( B | A X ) . For your support.

Common Sense Reasoning: Rule II(b) 2 ● Similarly, we assume a

functional relationship: �(AB|X) = F( (A|X) , (B|AX) ) �

3 ● Note: We are always

conditioning on X! �

4 ● Why not just use (A|X) and (B|X)? We need interaction effect. Example: Chance of brown left eye, chance of blue right eye; chance of both? �

5 ● Assume F is nondecreasing in each variable: (A|X) ≤ (A’|X’) and (B|AX) ≤ (B’|A’X’) implies (AB|X) ≤ (A’B’|X’) �