Why Marginally Optimal Actions Can Fail Conditionally

Why a Marginally Best Action May Be Wrong

A decision made before seeing any predictors must use the outcome’s marginal distribution. But if predictors are available when the action is chosen, the outcome distribution can differ across predictor values. A single action can therefore be optimal on average and still be inferior for some predictor values. The key distinction is between choosing one constant action and choosing a rule that responds to the observed predictors.

This distinction applies to classification and numerical prediction alike. In classification, the marginally most likely class need not be the best class for every predictor group. Under squared loss, the marginal mean minimizes risk among constant predictions, while the conditional mean can vary with the predictors. The question is not whether a marginal action is optimal among constants; it is whether it remains optimal when predictor-dependent actions are allowed.

Marginal Risk and Conditional Risk

Let \(Y\) be an outcome, \(Z\) the observed predictor, \(\mathcal A\) an action set, and \(L(a,Y)\) the loss from taking action \(a\) when the outcome is \(Y\). A constant action is chosen without using \(Z\). A decision rule \(g\) chooses the action \(g(Z)\) after observing \(Z\). Suppose the losses are integrable for the actions and rules being compared.

A marginally optimal action minimizes \(R(a)\) over constant actions. A conditional minimizer chooses, for each predictor value, an action minimizing \(r(a,Z)\). When the action set is finite, a measurable conditional minimizing rule can be obtained by ordering the actions and breaking ties in favor of the first minimizer. This avoids any ambiguity in defining the rule.

The equality \(R(a)=E[r(a,Z)]\) follows from the defining property of conditional expectation. It says that marginal risk averages conditional risk across predictor values. It does not say that an action minimizing this average must minimize each term being averaged.

When Conditional Minimization Improves Risk

The comparison can be made precise for a finite action set. Let \(a_*\) be any marginally optimal constant action. At each predictor value, compare its conditional risk with the smallest available conditional risk. The gap is nonnegative; its average is exactly the risk improvement from using a conditional minimizing rule.

Proof. Since \(\mathcal A\) is finite, the conditional risks have a pointwise minimum, and the stated tie-breaking rule gives a measurable minimizer. Define \(\Delta=r(a_*,Z)-\min_{a\in\mathcal A}r(a,Z)\). By the definition of a minimum, \(\Delta\ge 0\) almost surely. Conditional minimization gives \(r(g(Z),Z)=\min_{a\in\mathcal A}r(a,Z)\) almost surely. Taking expectations and using the conditional-expectation identity,

A nonnegative random variable has strictly positive expectation exactly when it is positive with positive probability: if \(P(\Delta>0)>0\), then \(P(\Delta>1/k)>0\) for some positive integer \(k\), so \(E[\Delta]\ge k^{-1}P(\Delta>1/k)>0\). Conversely, if \(\Delta=0\) almost surely, then \(E[\Delta]=0\). This proves both the inequality and its strictness criterion. \(\square\)

The theorem does not assert that the conditional rule is better at every predictor value. It says that it is never worse when it can choose among the same actions, and that the overall improvement is strict precisely when the marginal action fails to be conditionally optimal on a set of positive probability.

Squared Loss: The Marginal Mean Versus the Conditional Mean

For a numerical outcome and squared loss \(L(a,Y)=(Y-a)^2\), the best constant action is the marginal mean, provided \(E[Y^2]<\infty\). When predictors are observed, the corresponding conditional action is the conditional mean. The following derivation shows why these are different decisions and quantifies the possible gain.

Derivation. Since \(Y\) is square-integrable, \(m\) is square-integrable. Write \(Y-g(Z)=(Y-m)+(m-g(Z))\), square, and condition on \(Z\):

The cross term is zero because \(E[Y-m\mid Z]=0\), and \(m-g(Z)\) is determined by \(Z\). Taking expectations proves the identity. Its first term does not depend on \(g\), and its second term is nonnegative and equals zero for \(g=m\); hence \(m\) minimizes risk. For the constant rule \(c=E[Y]=E[m]\), the risk difference is \(E[(m-c)^2]=\operatorname{Var}(m)\). It is strictly positive exactly when the conditional mean is not almost surely constant. \(\square\)

When the Marginal Choice Does Not Fail

Predictors do not automatically improve a decision. If a marginally optimal action is also conditionally optimal almost surely, a conditional rule cannot reduce risk strictly. In the finite-action theorem, this is exactly the case \(\Delta=0\) almost surely. Under squared loss, it occurs when the conditional mean equals the marginal mean almost surely. The predictor may still contain information about other features of the outcome; for this loss, what matters for the optimal action is the conditional mean.

This distinction matters when interpreting an analysis: a marginal decision answers what to do without predictor information, while a conditional decision answers what to do after observing it. Comparing the two is meaningful only when the timing and information available at decision time are clear.

Common Mistake

A reliable check is to ask two separate questions: what action minimizes risk when predictors are unavailable, and what action minimizes conditional risk when they are observed? The answers can coincide, but that must follow from the conditional risks rather than from the marginal optimum alone.