Most likely balls in Banach spaces: Existence and nonexistence

We establish a general criterion for the existence of convex sets of fixed shape as, for example, balls of a given radius, of maximal probability on Banach spaces. We also provide counterexamples, showing that their existence may fail even in some common situations.


Introduction
In this note we address the natural question if for a (Borel) probability measure µ on a separable Banach space X and for a given radius there is always a "ball of maximum likelihood", i.e., if the maximum of µ(B) is attained among all balls B ⊂ X of radius r > 0. For small radii, such a maximizer-if existent-can be viewed as an approximation to and a regularization of a "mode" referring to a "point of maximum likelihood" for the given measure µ as r becomes small.More generally, instead of balls of a fixed radius our main existence theorem will also apply to the system of all translates of a fixed convex set C.
This issue has received considerable attention recently, notably in the area of Bayesian inverse problems (cf.[8]) where the problem arises to devise maximum a posteriori (MAP) estimators, in particular for Gaussian priors on infinite dimensional (separable) Banach spaces, [2,3,7,1,4,5].Indeed, seminal results in that area made implicit use of the existence of balls of maximum likelihood (cp. the discussion in [4]), while very recently it has been noted that the question of their existence can be circumvented by considering the asymptotics of almost maximizers in order to obtain MAP estimators [4,5].Nevertheless, besides being a question of intrinsic interest, the problem remains relevant as the quest for the position of a (small) ball of positive radius with maximal probability amounts to solving a regularized point optimization problem.As such it typically enjoys improved stability properties and might even be favorable from a modeling perspective.In particular this will be the case in situations when, possibly due to data uncertainties in the presence of noise, it is preferable to estimate parameter regions rather than single points.
The problem is addressed for general metric spaces in some detail in [6, Sect.4.2].There the authors provide a collection of technical sufficient conditions for the existence of maximum likelihood balls ("radius-r modes" in their terminology).They also give counterexamples for some particular measures on specific spaces and certain ranges of r.The farthest reaching existence result for measures on Banach spaces known to date seems to be [6,Thm. 4.9], which proves that on the sequence spaces ℓ p , 1 < p < ∞, maximum likelihood balls do exist for measures that do not charge any sphere in ℓ p .In particular this applies to "radius-r maximum likelihood a posteriori estimators for Gaussian priors" on ℓ p , 1 < p < ∞, which are absolutely continuous with respect to a non-degenerate Gaussian.It appears that no counterexample on a Banach space is known to date.
The purpose of this note is twofold.First, we establish a general existence theorem for maximum likelihood convex shapes (and, in particular, balls of any radius) on Banach spaces.In particular this will apply to every separable and reflexive space (and ℓ 1 ), thus closing a gap in the seminal contribution [2].Second, by way of various examples we also show that existence my fail in some natural situations.In fact, we will provide a couple of counterexamples on c 0 and the Wiener space, which might even be absolutely continuous with respect to a non-degenerate Gaussian measure, and which do not allow for balls of maximal probability for any value of radius r.

A general existence result
Throughout we assume that X is a separable (real) Banach space.By B r and B • r we denote the closed and, respectively, open ball of radius r in X centered at 0. For x ∈ X, C ⊂ X we write x + C =: C(x).Suppose µ is a Borel probability measure on X. Theorem 2.1.Suppose X is the separable dual of a Banach space.Let C ⊂ X be a bounded weak*-closed convex set C ⊂ X with non-empty interior.Then there exists for all x ∈ X.
Remark 2.2. 1.In particular, this applies to every separable reflexive Banach space X.
2. An admissible choice for C is C = B r for any r > 0.
Proof.Without loss of generality we assume 0 is an interior point of C. We consider a maximizing sequence of translates C(x n ), i.e., Since X is separable and C has non-empty interior, it is easy to see that m 0 > 0. Clearly, the sequence ( As µ is tight (on the polish space X), also the family (µ n ) of restrictions µ n = µ ¬ C(x n ) is tight.Thus Prohorov's theorem implies µ n w −→ µ 0 weakly (in duality with C b (X)) for a (not relabeled) subsequence and a finite measure µ 0 .It follows that By assumption, X has a separable predual X * , so by Alaoglu's theorem we may pass to a further subsequence (not relabeled) such that x n * ⇀ x 0 weakly* in X for some x 0 ∈ X.We will now prove that µ 0 is supported on C(x 0 ).
To this end, we fix any z / ∈ C(x 0 ).Since C is weak*-closed, with the help of the Hahn-Banach theorem for the dual pairing (X, X * ) we can choose an element x * ∈ X * and then an ε > 0 such that sup x * , y : On the other hand, the Portmanteau theorem implies As a consequence we conclude that, in case µ 0 (B Passing to yet another subsequence (not relabeled) we get y n * ⇀ y for some y ∈ C. It follows that x n + y n * ⇀ x 0 + y ∈ C(x 0 ) ∩ B ε (z), which contradicts (1).So we must have µ 0 (B • ε (z)) = 0.This proves that supp µ 0 ⊂ C(x 0 ).
It remains to observe that µ 0 ≤ µ, which follows from the outer regularity of the Borel measures µ 0 and µ and from the fact that for any open subset U ⊂ X the Portmanteau theorem gives Summarizing we find that which, by definition of m 0 , proves µ(C(x 0 )) = m 0 .

Examples of non-existence
We discuss a number of concrete cases, where balls of maximum likelihood do not exist.There is a common underlying idea in all of them which would easily allow to generate further examples along these lines.
Our first two examples are on the Banach space X = c 0 of (real) null-sequences equipped with the sup-norm, which is separable and even has a separable dual (namely, ℓ 1 ), but is not a dual space itself.
For any ball B r (x), x = (x 1 , x 2 , . ..) one has Choosing k 0 such that x k 0 < r and setting This shows that x → µ(B r (x)) does not have a maximizer.(Its supremum is m 0 = k∈N 1 − e −2kr , which can be seen by maximizing each factor separately and considering the maximizing sequence (x In order to give an example where µ is absolutely continuous with respect to a Gaussian measure on X = c 0 , we first let µ 0 = k∈N N (0, k −2 ), where N (0, σ 2 ) denotes the Gaussian on R with variance mean 0 and σ 2 and cumulative distribution function X → Φ(x/σ).An easy application of the Borel-Cantelli lemma shows µ 0 (X) = 1.We consider the (closed) set A ⊂ X given by and note that µ 0 (A) (the probability that the coordinate process does not pass the moving boundary We then define µ by conditioning on A, i.e., we set µ = 1 µ(A) 1l A µ 0 .A similar reasoning as above shows that balls of maximal probability do not exist (and the supremum is explicitly given as We now give some examples on the Wiener space X = {ω ∈ C[0, 1] : ω(0) = 0} equipped, as usual, with the sup-norm in order to show that non-existence of maximum likelihood balls is encountered in common situations in a continuous time setting.They follow the similar basic idea of the previous two examples.
In what follows we let µ 0 be the Wiener measure on X so that the coordinate process (ω(t)) t∈[0,1] is Brownian motion.