It has helped me appreciate how product rules (or additivity, if we apply a log transform) arises in many contexts. One thing I hadn’t appreciated when studying Cox theorem is that you do not need to respect “commutativity” to get a product rule (though obviously this restricts how you can group information). This was made very clear to me in example 3.
One thing that confused me in the first reading was that I misunderstood you as referring to the third requirement as associativity of F. Rereading this is not the case; you just say that the third requirement implies that F is associative. But I wish you had spelled out the implication, ie saying thatF(F(R(A),R(B|A)),R(C|A,B))=F(R(A),F(R(B|A),R(C|A,B))).
I really like this article.
It has helped me appreciate how product rules (or additivity, if we apply a log transform) arises in many contexts. One thing I hadn’t appreciated when studying Cox theorem is that you do not need to respect “commutativity” to get a product rule (though obviously this restricts how you can group information). This was made very clear to me in example 3.
One thing that confused me in the first reading was that I misunderstood you as referring to the third requirement as associativity of F. Rereading this is not the case; you just say that the third requirement implies that F is associative. But I wish you had spelled out the implication, ie saying thatF(F(R(A),R(B|A)),R(C|A,B))=F(R(A),F(R(B|A),R(C|A,B))).