Correction

Abstract

This is to provide corrections to Theorems 1 and 3 in Martin and Liu (2013). The latter correction also casts further light on the role of nested predictive random sets.

This article refers to:

Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference

Martin, R., and Liu, C. (2013), “Inferential Models: A Framework for Prior-Free Posterior Probabilistic Inference,” Journal of the American Statistical Association, 108, 301–313

CORRECTION OF THEOREM 1

In the main text, for validity of the predictive random set , the support was assumed only to be nested, that is, for any , either S⊆S′ or S′⊆S. However, some additional technical conditions are required for the proof to go through.

Fix a topology on the auxiliary variable space , and let the σ-algebra defined there contain all the open sets. In addition to being nested, we shall assume that contains both and , and that all of its contents are closed subsets of . These additional requirements result in no real loss of generality. Indeed, those predictive random sets in Corollary 1 of the main text already satisfy these. These extra conditions also make the statement and proof of the theorem more transparent.

Theorem 1′.

Let be a nested collection of closed and, hence, -measurable subsets of that contains and . Define a predictive random set , with distribution , supported on , such that

Then is valid in the sense of Definition 1 in the main text.

Proof.

Set . For any α ∈ (0, 1), let S _α be the smallest such that . In particular, . Since each S is closed, so is S _α; it is also measurable by our assumptions about the richness of the σ-algebra on . The key observation is that Q(u) > 1 − α iff u ∈ S^c _α. Therefore, by continuity of from above, we get

where the limit is over all S decreasing to S _α. By construction, each such S satisfies . So, finally, we get and, since α is arbitrary, the claimed validity is proved.

CORRECTION/EXTENSION OF THEOREM 3

Theorem 3 in the main text says that nested predictive random sets are more efficient than those which are not nested. However, the nested predictive random set constructed in that theorem is not necessarily valid. Since validity is a key to the inferential model (IM) analysis, it would be desirable if the new nested predictive random set was also valid. We accomplish this in Theorem 3′. First, we need the following lemma.

Lemma.

On a space equipped with probability , let be a valid predictive random set for . Choose a collection of -measurable subsets of , and set . Then

for any subset of such that is -measurable.

Proof.

First, note that if , then . Therefore, if , then . This argument implies

Since is valid, we have

Combining this with the inequality in the previous display, we get

which implies .

A measurability question was overlooked in the main text. In particular, the sets in Equation (1) below (also defined in the proof of Theorem 3 in the main text) are not automatically measurable. To confirm this, we shall add one more modification; note that this is not needed if the sampling model is discrete. To start, for the given topology on , keep the same assumptions about the corresponding σ-algebra as above. Now, recall the a-events defined in the proof of Proposition 1 in the main text. Here, we shall replace with its closure. This does not affect any properties of the resulting belief function when is nonatomic. In all the examples we have considered, can be taken as continuous; this is a particularly convenient choice, in light of Corollary 1 in the main text.

Theorem 3′.

Suppose that either is a discrete space, or that the assumptions in the previous paragraph hold. Fix A⊆Θ and assume condition (2.10) in the main text. Given any valid predictive random set , there exists a nested and valid predictive random set such that for each .

Proof.

For the given A and , set . Define a collection of subsets of as

where is the new closed a-event. If necessary, add and to to satisfy the requirement in Theorem 1′. This collection will serve as the support for the new predictive random set . First, we can see that is nested: if b(y) ⩾ b(x), then S_y ⊇S_x . Second, since the new a-events are closed, each S′ _x in (1) is closed and, hence, -measurable. Third, define the measure for to satisfy

According to Theorem 1′, the new is valid. Moreover, by the lemma and the definition of S′ _x , we have

Finally, we have a comparison of the belief functions corresponding to and :

the first inequality follows from monotonicity of and the fact that for each x, and the second inequality follows from (2).

Acknowledgments

The authors thank Liang Hong for some helpful suggestions. This work is partially supported by the U.S. National Science Foundation, grants DMS–1007678, DMS–1208841, and DMS–1208833.