Conditional expectation estimates an outcome after conditioning on observed evidence, such as E(U(Y)|X=x). It answers which option looks best among cases sharing that observed value, not necessarily which option would cause the best outcome. section 12.2.4
Distinguish the two decision quantities:
- E(U(Y)|X=x) conditions on cases where X equals x and can reflect confounding or selection. section 12.2.4
- E(U(Y_X=x)) evaluates an intervention that sets X to x and represents the causal decision target. section 12.2.4
- Optimizing the conditional expectation can agree with the causal answer, but parameter changes can make the two recommendations diverge. section 12.2.4
Investment choice
After changing one success-probability parameter, conditioning favors debt: E(U(Y)|X=debt)=57000 versus 53000 for equity. Intervention reverses the choice: E(U(Y_X=debt))=39000 versus 44000 for equity. A level-2 causal query therefore requires the interventional quantity. section 12.2.4
Listing 12.5 Calculate E(U(Y)|X=x) and E(U(YX=x))
def get_expectation(marginal): #1
u_values = marginal.state_names["U"] #1
probs = marginal.values #1
expectation = sum([x * p for x, p in zip(u_values, probs)]) #1
return expectation #1
infer = VariableElimination(model) #2
marginal_u_given_debt = infer.query( #2
variables=['U'], evidence={'X': 'debt'}) #2
marginal_u_given_equity = infer.query( #2
variables=['U'], evidence={'X': 'equity'}) #2
e_u_given_x_debt = get_expectation(marginal_u_given_debt) #2
e_u_given_x_equity = get_expectation(marginal_u_given_equity) #2
print("E(U(Y)|X=debt)=", e_u_given_x_debt) #2
print("E(U(Y)|X=equity)=", e_u_given_x_equity) #2
int_model_x_debt = do(model, {"X": "debt"}) #3
infer_debt = VariableElimination(int_model_x_debt) #3
marginal_u_given_debt = infer_debt.query(variables=['U']) #3
expectation_u_given_debt = get_expectation(marginal_u_given_debt) #3
print("E(U(Y_{X=debt}))=", expectation_u_given_debt) #3
int_model_x_equity = do(model, {"X": "equity"}) #3
infer_equity = VariableElimination(int_model_x_equity) #3
marginal_u_given_equity = infer_equity.query(variables=['U']) #3
expectation_u_given_equity = get_expectation(marginal_u_given_equity) #3
print("E(U(Y_{X=equity}))=", expectation_u_given_equity) #3
Computes conditional and interventional expectations with pgmpy inference, a utility-marginal helper, and do-based model variants. section 12.2.4
