When asking how much an intervention changes an outcome, a coefficient gives the change associated with one unit of a predictor or parent variable. In the linear causal model y = β0 + βx x + ny, βx is the coefficient for x: increasing x by one unit increases expected y by βx, while ny represents other causal factors and noise. section 6.4.4
Read the coefficient together with the rest of the model:
- β0 is the intercept, defining the baseline level before the parent variable contributes. section 6.4.4
- βx is a slope and, under the stated structural causal assumptions, represents the average causal effect of X on Y. section 6.4.4 section 6.1.1
- ny captures variation from exogenous causes; its normal scale parameter describes uncertainty around the expected response, not the coefficient itself. section 6.4.4
Femur length and height
The model y = 25 + 3x + ny uses x for femur length in centimeters and y for height in centimeters. The coefficient 3 asserts a proportional causal relationship: one additional centimeter of femur length corresponds to a three-centimeter increase in expected height. The residual term accounts for other causes of height. section 6.1.1
Listing 6.11 Pyro example of a linear Gaussian model
from pyro import sample
from pyro.distributions import Normal
def linear_gaussian():
n_x = sample("N_x", Normal(9., 3.))
n_y = sample("N_y", Normal(9., 3.))
x = 10. + n_x #1
y = 2. * x + n_y #2
return x, y
The Pyro example makes the coefficient operational: y = 2x + ny, with 2 multiplying the parent variable x. section 6.4.4
