Exact conditioning laboratory

Conditioning can preserve, ignore, or reverse an increasing event.

Every point below is one of the 16 subsets of X={a,b,c,d}. The blue event is always F={S:{a,b}⊆S}; switch the amber conditioning event B and move p to recompute every probability exactly.

Conditioning event B

Boolean lattice 2X

All outcomes, not a simulation

F B F∩B
Probability-weighted Boolean lattice on four elements Sixteen subsets arranged by rank. Node area represents exact product-measure probability; outlines and patterns show event membership.

Node area tracks μp({S})=p|S|(1−p)4−|S|.

What the theorem needs

One exact surplus carries the bound through conditioning.

Beyond the Boolean cube

Embed, estimate, condition back.

01 · embedU ⊆ P

Place the ordered problem inside 2X through an embedding f.

02 · estimatef(U)=F∩f(P)

Apply the Boolean product-measure estimate to the increasing event.

03 · condition backYp=f−1(Xp | Xp∈f(P))

Transfer the conclusion to the original finite poset.

Exact outcome table
Subset S|S|μp({S})In F?In B?