All simulators
Factorisation & Independence
Chains, forks and colliders on binary variables: the joint table from the factorisation and the independences it implies.
Day 47: Factorisation & the Markov PropertyP(z, x, y) = P(z) P(x | z) P(y | x)
| z | x | y | P(z, x, y) |
|---|---|---|---|
| 1 | 0 | 0 | 0.108 |
| 1 | 0 | 1 | 0.012 |
| 1 | 1 | 0 | 0.112 |
| 1 | 1 | 1 | 0.168 |
| 0 | 0 | 0 | 0.432 |
| 0 | 0 | 1 | 0.048 |
| 0 | 1 | 0 | 0.048 |
| 0 | 1 | 1 | 0.072 |
Z → Y overall
0.250
P(Y|Z=1) − P(Y|Z=0)
within X = 0
0.000
within X = 1
0.000
parameters, factorised
1 + 2 + 2 = 5
parameters, unrestricted
2³ − 1 = 7
Three binary variables with X in the middle. The joint table is built from the factorisation. Compare how Z predicts Y overall and within each value of X: chains and forks make the within-X differences zero; a collider makes the overall difference zero instead. (In the collider, Y is a fair coin and the sliders set how Z and Y raise X.)