Deterministic and Randomized Admissibility

Why the Rule Class Matters

Admissibility is a comparison between a procedure and the other procedures allowed in the comparison. If procedures must choose an action deterministically, a rule can be admissible even when a randomized procedure has lower risk at every parameter value. The distinction is not a change in the loss or the statistical model: it is a change in the set of competitors.

In a finite decision problem, randomization has a precise geometric effect. It allows risk vectors that are mixtures of deterministic risk vectors. A mixture may improve on a deterministic rule even when none of its component rules improves on that rule by itself. Conversely, not every randomized rule is useful: its risk may be dominated by a deterministic procedure.

Two Classes of Procedures

Let the parameter space \(\Theta\), observation space \(\mathcal X\), and action space \(\mathcal A\) all be finite. Under parameter \(\theta\), the observation has probability mass function \(p_\theta(x)\), and taking action \(a\) incurs loss \(L(\theta,a)\). Assume the losses are finite. These conditions ensure the risk sums below are well-defined.

The risk of a deterministic procedure is \(R(\theta,\delta)=\sum_{x\in\mathcal X}p_\theta(x)L(\theta,\delta(x))\). For a randomized procedure it is \(R(\theta,q)=\sum_{x\in\mathcal X}p_\theta(x)\sum_{a\in\mathcal A}q(a\mid x)L(\theta,a)\). Both risks average over the observation; the second also averages over the procedure’s random choice of action.

The risk vector lists a procedure’s risk at each parameter value. Pointwise risk comparisons apply to these vectors: domination means no larger risk in every coordinate and smaller risk in at least one. The key question is whether the risk vectors attainable by randomized procedures include a dominating mixture that the deterministic class does not contain.

Randomized Risks Are Mixtures

A randomized procedure can be understood as a probability distribution over deterministic procedures. In this representation, one first draws a complete deterministic rule, then applies it to the observation. The next result makes the equivalence exact for finite problems.

Proof. Let \(\mathcal G\) be the finite set of all deterministic functions \(g:\mathcal X\to\mathcal A\). Given a randomized procedure \(q\), assign to each \(g\in\mathcal G\) the weight \(\lambda_g=\prod_{x\in\mathcal X}q(g(x)\mid x)\). These weights are nonnegative, and \(\sum_{g\in\mathcal G}\lambda_g=\prod_{x\in\mathcal X}\sum_{a\in\mathcal A}q(a\mid x)=1\). For any fixed observation \(x\) and action \(a\), their marginal satisfies \(\sum_{g:g(x)=a}\lambda_g=q(a\mid x)\): the factor at \(x\) is \(q(a\mid x)\), and the factors for the other observations each sum to one.

Thus the mixture’s expected loss conditional on \(X=x\) is \(\sum_{g\in\mathcal G}\lambda_g L(\theta,g(x))=\sum_{a\in\mathcal A}q(a\mid x)L(\theta,a)\). Multiplying by \(p_\theta(x)\), summing over \(x\), and then over the finite set of rules gives \(R(\theta,q)=\sum_{g\in\mathcal G}\lambda_g R(\theta,g)\) for every \(\theta\). This proves that each randomized risk vector is in the convex hull of the deterministic risk vectors. Conversely, a probability mixture of deterministic procedures is a randomized procedure: draw a rule using the mixture weights and, at each observation, take that rule’s action. Its risk is the corresponding weighted average. Hence the two sets of risk vectors are equal. \(\square\)

An immediate consequence is a practical test. A deterministic rule is admissible among randomized procedures precisely when no point in the convex hull of deterministic risk vectors has risks no larger at all parameter values and strictly smaller at at least one. For deterministic admissibility, it is enough to check the deterministic risk vectors themselves. Those two checks need not give the same answer.

When Randomization Creates Domination

The example illustrates why testing only the component rules can miss a randomized improvement. The relevant comparison is with the whole convex hull of deterministic risk vectors, not merely with its original vertices.

Randomized Admissibility and Useful Comparisons

This is the complementary possibility: randomization can add admissible trade-off points without making every randomized procedure a dominating improvement. The mixture result identifies the attainable risks, but admissibility still requires checking the pointwise risk inequalities.

Randomization Does Not Always Help

A Reliable Admissibility Check

For a finite problem, the theorem gives a direct audit. First list the deterministic procedures and their risk vectors. Then, for a deterministic candidate, check whether any deterministic vector dominates it to assess deterministic admissibility. To assess randomized admissibility, check whether any convex combination of those vectors dominates it. The latter search allows mixtures that are not themselves risk vectors of any single deterministic rule.

Common Mistake

A common mistake is to argue, “No deterministic rule dominates this candidate, so it is admissible.” That conclusion establishes admissibility only in the deterministic class. In the first worked example, neither \(b\) nor \(c\) dominates \(a\), but the equal mixture has risk \((0.75,0.75)\), strictly below \((1,1)\) in both coordinates. The correction is to name the class being considered and, for randomized admissibility, check mixtures as well as individual rules.