###### Proof.

[section]section pageheadfoot pagenumber [0.525ex]0.6pt [0.525ex]0.6pt
[0.525ex]0.6pt [0.525ex]0.6pt *[0.525ex]0.6pt [0.525ex]0.6pt

###### An order-theoretic perspective on modes
and maximum a posteriori estimation in
Bayesian inverse problems

###### Abstract

Abstract. It is often desirable to summarise a probability measure on a space
$X$ in terms of a mode, or MAP estimator, i.e. a point of maximum probability.
Such points can be rigorously defined using masses of metric balls in the
small-radius limit. However, the theory is not entirely straightforward: the
literature contains multiple notions of mode and various examples of
pathological measures that have no mode in any sense. Since the masses of
balls induce natural orderings on the points of $X$, this article aims to shed
light on some of the problems in non-parametric MAP estimation by taking an
order-theoretic perspective, which appears to be a new one in the inverse
problems community. This point of view opens up attractive proof strategies
based upon the Cantor and Kuratowski intersection theorems; it also reveals
that many of the pathologies arise from the distinction between greatest and
maximal elements of an order, and from the existence of incomparable elements
of $X$, which we show can be dense in $X$, even for an absolutely continuous
measure on $X=\mathbb{R}$.

Keywords. Bayesian inverse problems $\bullet$ local behaviour of measures
$\bullet$ maximum a posteriori estimation $\bullet$ modes of probability
measures $\bullet$ orders on metric spaces

2020 MSC. 06F99 $\bullet$ 28A75 $\bullet$ 28C15 $\bullet$ 60B05 $\bullet$
62F10 $\bullet$ 62R20

WarwickMathematics Institute and School of Engineering, University of Warwick,
Coventry, CV4 7AL, United Kingdom
(, ) TuringAlan Turing Institute, 96 Euston Road, London, NW1 2DB, United
Kingdom

##  1 Introduction

In diverse applications such as statistical inference and the analysis of
transition paths of random dynamical systems it is desirable to summarise a
complicated probability measure $\mu$ on a space $X$ by a single distinguished
point $x^{\star}\in X$ that is, in some sense, a “point of maximum
probability” under $\mu$ — i.e. a _mode_ or, in the Bayesian context, a
_maximum a posteriori_ (_MAP_) _estimator_. Many optimisation-based approaches
to inverse problems (e.g. Tikhonov regularisation of the misfit) aim to
calculate or approximate modes, at least heuristically understood. Over the
last decade, it has become common to define modes in terms of masses of metric
balls in the limit as the ball radius tends to zero, since this makes sense
even when $X$ is a very general — possibly infinite-dimensional — space, as is
often the case for modern inference problems (Stuart, 2010).

However, this “small balls” theory of modes is not entirely straightforward.
There are various definitions — e.g. the strong mode of Dashti et al. (2013),
the generalised strong mode of Clason et al. (2019), the weak mode of Helin
and Burger (2015) — with various subtle distinctions among them. Even the
existence theory for modes is not entirely straightforward: there are already
examples in the literature, and this article will supply further examples, of
relatively simple probability measures that have no mode. It can even be the
case that the average of two disjointly supported unimodal probability
measures may have no mode.

The purpose of this article is to formulate the notion of a mode in an order-
theoretic manner and thereby to clarify some of these pathologies in the
theory of modes. We claim that this is a natural step to take in view of the
heuristic understanding of modes as “most probable points”.

With an order-theoretic point of view, many of the difficulties can be seen to
arise from the distinction between greatest and maximal elements of a
preordered set $(X,\preccurlyeq)$ when the preorder $\preccurlyeq$ is not
total, i.e. when there exist _incomparable_ $x,x^{\prime}\in X$ for which
neither $x\preccurlyeq x^{\prime}$ nor $x\succcurlyeq x^{\prime}$ holds.
Simply put, a greatest element must dominate every other element of $X$,
whereas a maximal element need only dominate those with which it is
comparable; for a total preorder, maximal and greatest elements coincide.
Motivated by the needs of inverse problems theory, current notions of modes
correspond to greatest elements. However, many preorders lack maximal
elements, and even those that have maximal elements may lack greatest
elements; this is exactly the situation of the examples discussed in Examples
5.7 and 5.11. Thus, one might argue that current notions of mode are order-
theoretically “too strong”, and perhaps maximal elements should be considered
as modes, but possibly these are “too weak” for the needs of applications
communities. We hope that the present article will stimulate discussion on
this point.

#### Outline of the paper.

The rest of this paper is structured as follows:

Section 2 sets out basic notation for the rest of the paper, including a brief
recap of necessary concepts from functional analysis, measure theory, and
order theory.

Section 3 gives an overview of related work in this area, in particular the
“small balls” approach to defining MAP estimators for non-parametric
statistical inverse problems.

Section 4 introduces and analyses the total preorder $\preccurlyeq_{r}$ on $X$
induced by the $\mu$-measures of metric balls of fixed radius $r>0$. Because
the preorder $\preccurlyeq_{r}$ is total, its maximal elements are also
greatest, and can be seen as approximate “radius-$r$ modes” for $\mu$. We are
able to provide several criteria for the existence of such radius-$r$ modes
$x_{r}^{\star}$ (Theorem 4.6) as well as examples of measures that admit none
(Examples 4.7 and 4.8). As a prelude to the next section, we also consider the
convergence of $x_{r}^{\star}$ as $r\to 0$ (Theorems 4.11 and 4.12).

In Section 5 we attempt to take the limit as $r\to 0$ of the preorders
$\preccurlyeq_{r}$ to define a preorder $\preccurlyeq_{0}$ whose greatest
elements will be weak modes of $\mu$. However, because the preorder
$\preccurlyeq_{0}$ is not total, the distinction between greatest and maximal
elements becomes important. Incomparable maximal elements are particularly
troubling because their maximality means that one would like to think of them
as candidate modes, yet their incomparability means that one cannot actually
say which is “most probable” and hence a bona fide mode, as in Example 5.7. We
show that antichains (collections of mutually incomparable elements) can be
topologically dense in $X$ even when $\mu$ is absolutely continuous with
respect to Lebesgue measure on $X\subseteq\mathbb{R}$ (Theorem 5.11). We also
show that measures with a continuous Lebesgue density may have incomparable
elements, but never incomparable _maximal_ elements (Example 5.9, Proposition
5.14).

Section 6 gives some closing remarks, while technical supporting results can
be found in Appendix A, and Appendix B discusses some alternatives to the
limiting preorder $\preccurlyeq_{0}$ of Section 5 and illustrates their
shortcomings.

##  2 Problem setting and notation

###  2.1 Spaces of interest

Throughout, unless noted otherwise, $X$ will be a metric space with metric
$d$; we write $\mathscr{B}(X)$ for its Borel $\sigma$-algebra, i.e. the one
generated by the closed balls $B_{r}(x)\coloneqq\\{x^{\prime}\in X\mid
d(x,x^{\prime})\leqslant r\\}$, $x\in X$, $r\geqslant 0$; we also write
$\mathring{B}_{r}(x)\coloneqq\\{x^{\prime}\in X\mid d(x,x^{\prime})<r\\}$ for
the corresponding open ball. We will often assume that $X$ is complete and
separable, and occasionally we will specialise to the case of $X$ being a
separable Banach or Hilbert space.

###  2.2 Measures of non-compactness and intersection theorems

Our approach in Section 4 will make much use of measures of non-compactness
and intersection theorems; see Malkowsky and Rakočević (2019, Sections
7.5–7.8) for a thorough treatment of these concepts and their properties.

Briefly, given $A\subseteq X$, its _separation_ (or _Istrăţescu_) _measure of
non-compactness_ is

$\gamma(A)\coloneqq\inf\left\\{r\geqslant 0\,\middle|\,\text{there is no
$(x_{n})_{n\in\mathbb{N}}\subseteq A$ with
}\inf_{\begin{subarray}{c}m,n\in\mathbb{N}\\\ m\neq
n\end{subarray}}d(x_{m},x_{n})\geqslant r\right\\}.$ (2.1)

This is an increasing function with respect to inclusion of sets, is finite
precisely when $A$ is bounded, and is zero precisely when $A$ is pre-compact.
The function $\gamma$ is bi-Lipschitz equivalent with several other measures
of non-compactness such as the set (or Kuratowski) measure of non-compactness
and the ball (or Hausdorff) measure of non-compactness.

######  Theorem 2.1 (Generalised intersection theorem).

Let $(A_{n})_{n\in\mathbb{N}}$ be a decreasingly nested sequence of non-empty,
closed subsets of a topological space $X$ and let
$A\coloneqq\bigcap_{n\in\mathbb{N}}A_{n}$.

  1. (a)

(Cantor) If each $A_{n}$ is compact, then $A$ is non-empty and compact.

  2. (b)

(Cantor) If $X$ is a complete metric space and
$\mathop{\textup{diam}}\nolimits(A_{n})\to 0$ as $n\to\infty$, then $A$ is a
singleton.

  3. (c)

(Kuratowski) If $X$ is a complete metric space and $\gamma(A_{n})\to 0$ as
$n\to\infty$, then $A$ is non-empty and compact.

###  2.3 Measure-theoretic concepts

Given a metric space $X$, $\mathscr{P}(X)$ denotes the set of all probability
measures on $\mathscr{B}(X)$. Absolute continuity of $\mu$ with respect to
$\nu$ is denoted $\mu\ll\nu$. The _topological support_ of
$\mu\in\mathscr{P}(X)$ is

$\operatorname*{supp}(\mu)\coloneqq\\{x\in X\mid\text{for all $r>0$,
}\mu(B_{r}(x))>0\\},$ (2.2)

which is always a closed subset of $X$, and is non-empty when $X$ is separable
(or, equivalently, second countable or Lindelöf) (Aliprantis and Border, 2006,
Theorem 12.14).

The $n$-dimensional Lebesgue measure on $\mathbb{R}^{n}$ will be denoted
$\lambda^{n}$.

The quantity $\mu(B_{r}(x))$ will play a major role in this work, especially
when thought of as a function of $r>0$ for various choices of $x\in X$; we
shall call the map $r\mapsto\mu(B_{r}(x))$ the _radial cumulative distribution
function_ (RCDF) and some of its key properties are given in Lemma A.1 and
Corollary A.4.

###  2.4 Order-theoretic concepts

We summarise here some basic terms from order theory; for a comprehensive
introduction to order theory, see e.g. Davey and Priestley (2002).

In the course of this work, the set $X$ will be equipped with various
_preorders_ $\preccurlyeq$, i.e. relations satisfying both

  1. (a)

_reflexivity_ : for all $x\in X$, $x\preccurlyeq x$; and

  2. (b)

_transitivity_ : for all $x,y,z\in X$, if both $x\preccurlyeq y$ and
$y\preccurlyeq z$, then $x\preccurlyeq z$.

For any such preorder, we will write $x\asymp x^{\prime}$ if both
$x\preccurlyeq x^{\prime}$ and $x\succcurlyeq x^{\prime}$ hold true, in which
case $x$ and $x^{\prime}$ are called _equivalent_ 111A preorder $\preccurlyeq$
is called a _partial order_ if it is _antisymmetric_ , i.e. if $x\asymp
x^{\prime}\implies x=x^{\prime}$, but almost none of the preorders that we
consider will actually be partial orders. in the preorder; we write $x\prec
x^{\prime}$ if $x\preccurlyeq x^{\prime}$ but $x\not\succcurlyeq x^{\prime}$.

If at least one of $x\preccurlyeq x^{\prime}$ and $x\succcurlyeq x^{\prime}$
holds true, then we call $x$ and $x^{\prime}$ _comparable_ ; if neither holds,
then we call them _incomparable_ and write $x\mathrel{\|}x^{\prime}$. A
preorder $\preccurlyeq$ is _total_ or _linear_ if there are no incomparable
elements. A subset of $X$ on which $\preccurlyeq$ is total is called a _chain_
, and a subset for which every two distinct elements are incomparable is
called an _antichain_.

We highlight and contrast two notions of a “biggest” element for a preorder:

######  Definition 2.2.

Let $X$ be a set equipped with a preorder $\preccurlyeq$.

  1. (a)

$g\in X$ is a _greatest element_ if, for every $x\in X$, $g\succcurlyeq x$.

  2. (b)

$m\in X$ is a _maximal element_ if, whenever $x\in X$ is such that
$m\preccurlyeq x$, it follows that $m\succcurlyeq x$ (and hence $m\asymp x$).
Equivalently, $m$ is maximal if there is no $x\in X$ with $x\succ m$.

  3. (c)

$u\in X$ is an _upper bound_ for $A\subseteq X$ if, for all $x\in A$,
$u\succcurlyeq x$.

Note in particular that a greatest element is also a maximal element, but it
must additionally be comparable to (and dominate) every element of $X$. On the
other hand, a maximal element is only required to dominate those elements of
$X$ with which it is comparable, and those elements could constitute a rather
small subset of $X$.

The most famous statement about the existence of maximal elements is _Zorn’s
lemma_ : under the Axiom of Choice, if $(X,\preccurlyeq)$ is a preordered
space in which every chain $Y\subseteq X$ has an upper bound, then $X$ has at
least one maximal element. However, Zorn’s lemma says nothing about the
existence of greatest elements.

We write $\mathop{\uparrow}\nolimits Y\coloneqq\\{x\in X\mid x\succcurlyeq
y\text{ for some }y\in Y\\}$ for the _upward closure_ of $Y\subseteq X$, and
further write, for $y\in X$, $\mathop{\uparrow}\nolimits
y\coloneqq\mathop{\uparrow}\nolimits\\{y\\}=\\{x\in X\mid x\succcurlyeq y\\}$,
so that $\mathop{\uparrow}\nolimits Y=\bigcup_{y\in
Y}\mathop{\uparrow}\nolimits y$.

Finally, since many of the preorders we consider will be parametrised by
radius $r\geqslant 0$, we will write $\preccurlyeq_{r}$ for the preorder,
$\mathrel{\|}_{r}$ for the induced relation of incomparability,
$\mathop{\uparrow}\nolimits_{r}Y$ for the upward closure of $Y$ with respect
to $\preccurlyeq_{r}$, etc.

##  3 Overview of related work

Modes, loosely understood as points of maximum probability, arise in many
areas of pure and applied mathematics. Two application domains where modes are
particularly prominent are the analysis of the transition paths of random
dynamical systems and the Bayesian approach to inverse problems.

The random dynamical systems setting is exemplified by mathematical models of
chemical reactions using diffusion processes. One is typically interested in
the (rare) transitions of the process from one energy well or metastable state
to another, and in particular one wishes to understand the transition paths
that a diffusion process is most likely to take. This amounts to a study of
the modes of the law $\mu$ of the diffusion process on the associated path
space $X$; e.g. for a molecule consisting of $n$ atoms in three-dimensional
space, $X=C([0,T];\mathbb{R}^{3n})$. The modes of $\mu$ are understood as
_minimum-action paths_ , and the behaviour of $\mu$ near the mode is
quantified using Freidlin–Wentzell theory or large deviations theory (Dembo
and Zeitouni, 1998; E et al., 2004; Freidlin and Wentzell, 1998).

In the Bayesian approach to inverse problems (Kaipio and Somersalo, 2005;
Stuart, 2010), the reconstruction of an $X$-valued parameter of interest from
observed $Y$-valued data is expressed in the form of a probability measure
$\mu\in\mathscr{P}(X)$, the _posterior distribution_. In many modern inverse
problems, particularly those coupled to partial differential equations, the
space $X$ is an infinite-dimensional function space or a high-dimensional
discretisation of such a space, e.g. via a system of finite elements.

The posterior measure $\mu$ arises from three ingredients: a _prior measure_
$\mu_{0}\in\mathscr{P}(X)$, which encodes (subjective) beliefs about the
parameter that are held in advance of knowing the observation mechanism or the
specific data that are observed; a _likelihood model_ , i.e. a family of
probability measures $L(\hbox
to5.71527pt{\hss$\cdot$\hss}|x)\in\mathscr{P}(Y)$, one for each $x\in X$,
which models how observed data would be expected to arise if the parameter
value $x$ were the truth; and a specific observed instance of the data, a
point $y\in Y$. Strictly speaking, the posterior measure $\mu$ is defined as
the disintegration (conditional distribution) of the joint measure
$\nu(\mathrm{d}x,\mathrm{d}y)\coloneqq
L(\mathrm{d}y|x)\mu_{0}(\mathrm{d}x)\in\mathscr{P}(X\times Y)$ along the
$y$-fibre (Chang and Pollard, 1997). For simplicity, however, we often
concentrate on the case that $\mu$ has a density with respect to $\mu_{0}$
given by Bayes’ formula,

$\mu(\mathrm{d}x)=\frac{\exp(-\Phi(x;y))\,\mu_{0}(\mathrm{d}x)}{\int_{X}\exp(-\Phi(x^{\prime};y))\,\mu_{0}(\mathrm{d}x^{\prime})},$
(3.1)

where $\Phi\colon X\times Y\to\mathbb{R}$ is called the _potential_. In simple
settings with $\dim Y<\infty$, the Lebesgue probability density of $L(\hbox
to5.71527pt{\hss$\cdot$\hss}|x)$ is proportional to $\exp(-\Phi(x;\hbox
to5.71527pt{\hss$\cdot$\hss}))$ and $\Phi$ can be interpreted as a non-
negative misfit functional. The case of infinite-dimensional data, $\dim
Y=\infty$, is considerably more subtle and does not generally admit a density
for $\mu$ with respect to $\mu_{0}$ as in (3.1); see e.g. Stuart (2010, Remark
3.8) and Lasanen (2012, Remark 9).

Since the full posterior distribution $\mu$ can be a rather intractable
object, it is often desirable to have access to a convenient point summary:
the two principal such point estimators are the _conditional mean estimator_
(i.e. the mean of $\mu$) and a _maximum a posteriori estimator_ (i.e. a mode,
or point of maximum probability, for $\mu$), and here we focus on this second
approach. Heuristically, at least when $X=\mathbb{R}^{d}$, a MAP estimator is
just an essential maximiser of the Lebesgue density of $\mu$, i.e. a minimiser
of the sum of $\Phi(\hbox to5.71527pt{\hss$\cdot$\hss};y)$ and the negative
logarithm of the Lebesgue density of $\mu_{0}$. However, this definition is
not effective if we have no access to Lebesgue densities; in particular, it
makes no sense when $\dim X=\infty$ (e.g. Sudakov, 1959).

To handle the general infinite-dimensional case, various definitions of modes
/ MAP estimators have been advanced over recent years, and we summarise them
here.222The definitions of Dashti et al. (2013), Helin and Burger (2015), and
Clason et al. (2019) were all stated in the case of a separable Banach space
$X$, but they generalise easily to the metric setting, as given here. Also,
their definitions used open rather than closed balls. One approach (Dürr and
Bach, 1978) is to understand a mode of the path measure $\mu$ of a diffusion
process as a minimiser of the _Onsager–Machlup (OM) functional_ $I_{\mu}$ of
$\mu$, which is defined by the relation

$\lim_{r\to
0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(x^{\prime}))}=\frac{\exp(-I_{\mu}(x))}{\exp(-I_{\mu}(x^{\prime}))}\quad\text{for
$x,x^{\prime}\in X$.}$ (3.2)

In some sense, $I_{\mu}$ is a formal negative log-density for $\mu$, but it is
in general only a partially defined extended-real-valued function. For
example, the OM functional of a Gaussian measure on a Hilbert space is finite
only on the Cameron–Martin space. The rigorous interpretation of modes as
minimisers of $I_{\mu}$ requires considerable care, especially since in some
cases it is not even possible to assign $+\infty$ as an exceptional value for
$I_{\mu}$: the ratio in (3.2) may oscillate and fail to converge as $r\to 0$.

A _strong mode_ of $\mu$ was defined by Dashti et al. (2013) to be any
$x^{\star}\in X$ such that

$\displaystyle\lim_{r\to 0}\frac{\mu(B_{r}(x^{\star}))}{M_{r}}=1,$ (3.3)
$\displaystyle M_{r}\coloneqq\sup_{x\in X}\mu(B_{r}(x)).$ (3.4)

(By Corollary A.2, separability of $X$ ensures that
$\operatorname*{supp}(\mu)\neq\varnothing$ and $M_{r}>0$.) Any strong mode
must lie in $\operatorname*{supp}(\mu)$, and the ratio in (3.3) is at most 1
for every choice of $x^{\star}\in X$, so

$\text{$x^{\star}$ is a strong mode}\iff\liminf_{r\to
0}\frac{\mu(B_{r}(x^{\star}))}{M_{r}}\geqslant 1\iff\limsup_{r\to
0}\frac{M_{r}}{\mu(B_{r}(x^{\star}))}\leqslant 1.$ (3.5)

However, Clason et al. (2019) observed that even elementary absolutely
continuous measures on $\mathbb{R}$ such as
$\mu(E)\coloneqq\int_{E\cap[-1,1]}\lvert x\rvert\,\mathrm{d}x$ do not have
strong modes, even though the Lebesgue density of $\mu$ is clearly maximised
at $\pm 1$. Therefore, they call $x^{\star}\in X$ a _generalised strong mode_
if, for every positive null sequence $(r_{n})_{n\in\mathbb{N}}$, there exists
a sequence $(x_{n})_{n\in\mathbb{N}}$ converging to $x^{\star}$ such that

$\lim_{n\to\infty}\frac{\mu(B_{r_{n}}(x_{n}))}{M_{r_{n}}}=1.$ (3.6)

Motivated by (3.5), $x^{\star}\in\operatorname*{supp}(\mu)\subseteq X$ is
called a _weak mode_ (Helin and Burger, 2015) if333In fact, Helin and Burger
(2015) used “$\lim$” in place of “$\limsup$” in (3.7), implicitly assuming the
existence of the limit. However, as Ayanbayev et al. (2022a) observe, this
yields an unsatisfying definition because it excludes the case in which the
ratio oscillates, while remaining bounded away from unity, from being a weak
mode. The desirable implication “strong mode $\implies$ weak mode” fails for
the original “$\lim$” version of the definition, but holds for the “$\limsup$”
version.

$\limsup_{r\to 0}\frac{\mu(B_{r}(x^{\prime}))}{\mu(B_{r}(x^{\star}))}\leqslant
1\text{ for all }x^{\prime}\in X.$ (3.7)

As a point of terminology, Helin and Burger (2015) were primarily interested
in the restricted case that $x^{\prime}\in x^{\star}+E$, where $x^{\star}\in
E$ and $E$ is a topologically dense linear subspace of a Banach space $X$, and
Lie and Sullivan (2018) later called this case an _$E$ -weak mode_.
Conversely, Ayanbayev et al. (2022a) call $x^{\star}$ satisfying (3.7) a
_global weak mode_. Since we are only going to consider global weak modes, we
can simply call them _weak modes_ without any ambiguity.

Under the assumption that the OM functional $I_{\mu}$ of $\mu$ is real-valued
on $\varnothing\neq E\subseteq X$ and

$\text{for some $x\in E$ and all $x^{\prime}\in X\setminus E$,}\quad\lim_{r\to
0}\frac{\mu(B_{r}(x^{\prime}))}{\mu(B_{r}(x))}=0,$ (3.8)

which Ayanbayev et al. (2022a, Definition 3.1) call _property $M(\mu,E)$_,
$I_{\mu}$ can be regarded as having the value $+\infty$ on $X\setminus E$ and
weak modes are precisely minimisers of this extended version of $I_{\mu}$.
This enabled Ayanbayev et al. (2022a, b) to establish a stability and
convergence theory for weak modes in terms of the $\Gamma$-convergence and
equicoercivity of the associated OM functionals.444Frustratingly, while there
are some situations in which strong modes can be characterised as minimisers
of Onsager–Machlup functionals (Agapiou et al., 2018; Dashti et al., 2013),
there are also situations in which this correspondence breaks down, even when
property $M(\mu,E)$ holds (Ayanbayev et al., 2022a, Example B.5). Furthermore,
as we show below in Lemma 5.3, weak modes are exactly the greatest elements of
a natural preorder $\preccurlyeq_{0}$ on $X$, namely the one induced by the
limiting ratios of masses of balls in the small-radius limit (Definition 5.1).

There are also local versions of the strong and weak modes (Agapiou et al.,
2018), in which $x^{\star}$ is only compared to points in a sufficiently small
ball $B_{\delta}(x^{\star})$, $\delta>0$, analogous to local maximisers of the
Lebesgue probability density function / local minimisers of the OM functional.

For $\mu$ of the form (3.1) with $\mu_{0}$ Gaussian and $X$ Hilbert, Dashti et
al. (2013) proved that $\mu$ has a strong mode by studying maximisers of the
radius-$r$ ball mass $x\mapsto\mu(B_{r}(x))$ for fixed $r>0$ — which we call
radius-$r$ modes in Section 4 — and arguing that a sequence of such maximisers
must converge to a strong mode. The arguments of Dashti et al. (2013) assume
the existence of radius-$r$ modes without proof; in Section 4.2, we prove
results on the existence of radius-$r$ modes in various settings but also
provide examples that have no such radius-$r$ modes. Despite the contributions
of Dashti et al. (2013), Kretschmann (2019), and Klebanov and Wacker (2022)
among others — and our own offerings — a surprising amount is still unknown
about the existence of radius-$r$ modes, let alone weak and strong modes, even
for “nicely” reweighted Gaussian measures on Banach spaces.

##  4 The positive-radius preorder

###  4.1 Definition and basic properties

A probability measure on a metric space $X$ induces a family of preorders on
$X$, one for each positive radius, in a very straightforward way:

######  Definition 4.1 (Positive-radius preorder).

Let $X$ be a metric space and let $\mu\in\mathscr{P}(X)$. For each $r>0$,
define a relation $\preccurlyeq_{r}$ on $X$ by

$\displaystyle x\preccurlyeq_{r}x^{\prime}$
$\displaystyle\iff\mu(B_{r}(x))\leqslant\mu(B_{r}(x^{\prime})).$ (4.1)

It is almost trivial to verify that $\preccurlyeq_{r}$ satisfies the axioms
for a preorder. We will write $x\asymp_{r}x^{\prime}$ if both
$x\preccurlyeq_{r}x^{\prime}$ and $x\succcurlyeq_{r}x^{\prime}$ hold, and
$x\mathrel{\|}_{r}x^{\prime}$ if neither $x\preccurlyeq_{r}x^{\prime}$ nor
$x\succcurlyeq_{r}x^{\prime}$ hold. In fact, though, incomparability never
arises for this preorder: totality of the usual order $\leqslant$ on
$\mathbb{R}$ implies totality of $\preccurlyeq_{r}$ on $X$. Totality implies
that the maximal and greatest elements of $X$ with respect to
$\preccurlyeq_{r}$ coincide (Lemma 4.3), which simplifies the discussion
considerably.

Upward closures with respect to $\preccurlyeq_{r}$ are notably well behaved.
In particular, Lemma 4.2(b) says that the relation $\preccurlyeq_{r}$ is
_upper semicontinuous_ (Aliprantis and Border, 2006, p.44).

######  Lemma 4.2 (Closedness, boundedness, and non-compactness of upward
closures).

Let $X$ be a metric space, let $\mu\in\mathscr{P}(X)$, and fix $r>0$.

  1. (a)

For each $t\geqslant 0$, $\\{x^{\prime}\in
X\mid\mu(B_{r}(x^{\prime}))\geqslant t\\}$ is closed.

  2. (b)

For each $x\in X$, $\mathop{\uparrow}\nolimits_{r}x\coloneqq\\{x^{\prime}\in
X\mid x^{\prime}\succcurlyeq_{r}x\\}$ is closed.

  3. (c)

For each $t>0$, $\\{x^{\prime}\in X\mid\mu(B_{r}(x^{\prime}))\geqslant t\\}$
is bounded, with separation measure of non-compactness
$\gamma(\\{x^{\prime}\in X\mid\mu(B_{r}(x^{\prime}))\geqslant t\\})\leqslant
2r$.

  4. (d)

For each $x\in X$ with $\mu(B_{r}(x))>0$, $\mathop{\uparrow}\nolimits_{r}x$ is
bounded with $\gamma(\mathop{\uparrow}\nolimits_{r}x)\leqslant 2r$.

###### Proof.

Claim (a) is immediate from the upper semicontinuity of the map
$x\mapsto\mu(B_{r}(x))$ (Lemma A.1(a)), and (b) is a special case of claim
(a).

Now fix $t>0$ and suppose for a contradiction that $(x_{n})_{n\in\mathbb{N}}$
is an unbounded sequence in $\\{x^{\prime}\in
X\mid\mu(B_{r}(x^{\prime}))\geqslant t\\}$. By passing to a subsequence if
necessary, we may assume that $d(x_{n},x_{n^{\prime}})>2r$ for all distinct
$n,n^{\prime}\in\mathbb{N}$. We thus obtain the contradiction that

$1=\mu(X)\geqslant\mu\left(\biguplus_{n\in\mathbb{N}}B_{r}(x_{n})\right)=\sum_{n\in\mathbb{N}}\mu(B_{r}(x_{n}))\geqslant\sum_{n\in\mathbb{N}}t=\infty.$

This shows that $\\{x^{\prime}\in X\mid\mu(B_{r}(x^{\prime}))\geqslant t\\}$
must be bounded and also that it admits no infinite subset with separation
$2r$, thus establishing (c), of which (d) is a special case. ∎

###  4.2 Existence and absence of greatest elements

Our first aim is to establish existence of greatest elements for
$\preccurlyeq_{r}$, which we also call _radius- $r$ modes_. Such points can be
seen as approximate modes555The intuition that radius-$r$ modes are
approximate modes must be treated sceptically. For example, consider
$\mu\in\mathscr{P}(X)$ with bimodal continuous Lebesgue density
$\rho(x)\propto\max\\{0,1-4(x-1)^{2}\\}+\max\\{0,1-4(x+1)^{2}\\}$, for which a
radius-$1$ mode is located at $0$, which is neither a maximiser of $\rho$ nor
even in $\operatorname*{supp}(\mu)$. with respect to the positive radius /
spatial resolution $r$; only in the next section will we attempt to take the
limit as $r\searrow 0$.

Lemma 4.3 now gives several equivalent conditions for a point to be a
radius-$r$ mode. The intersection criterion (d) will prove especially helpful
in what follows, in the sense that we establish existence of radius-$r$ modes
by showing that intersections of this type are non-empty.

######  Lemma 4.3 (Characterisation of radius-$r$ modes).

Let $X$ be any metric space, let $\mu\in\mathscr{P}(X)$, and let $r>0$. As in
(3.4), let $M_{r}\coloneqq\sup_{x\in X}\mu(B_{r}(x))$. Then the following are
equivalent and if one (and hence any) holds, then $x_{r}^{\star}\in X$ is
called a _radius- $r$ mode_:

  1. (a)

$x_{r}^{\star}$ is a $\preccurlyeq_{r}$-maximal element;

  2. (b)

$x_{r}^{\star}$ is a $\preccurlyeq_{r}$-greatest element;

  3. (c)

$x_{r}^{\star}\in\bigcap_{x\in X}\mathop{\uparrow}\nolimits_{r}x$;

  4. (d)

$x_{r}^{\star}\in\bigcap_{n\in\mathbb{N}}\mathop{\uparrow}\nolimits_{r}x_{n}$
for some sequence $(x_{n})_{n\in\mathbb{N}}\subseteq X$ with
$\mu(B_{r}(x_{n}))\nearrow M_{r}$ as $n\to\infty$;

  5. (e)

$\mu(B_{r}(x_{r}^{\star}))=M_{r}$.

###### Proof.

((a)$\iff$(b)) This equivalence holds because $\preccurlyeq_{r}$ is a total
preorder.

((b)$\iff$(c)) This equivalence is simply a restatement of the definition of
being greatest.

((c)$\implies$(d)) This implication is obvious, since
$\bigcap_{n\in\mathbb{N}}\mathop{\uparrow}\nolimits_{r}x_{n}\supseteq\bigcap_{x\in
X}\mathop{\uparrow}\nolimits_{r}x$.

((d)$\implies$(e)) Let $(x_{n})_{n\in\mathbb{N}}$ be a sequence in $X$ with
$\mu(B_{r}(x_{n}))\nearrow M_{r}$ and let
$x_{r}^{\star}\in\bigcap_{n\in\mathbb{N}}\mathop{\uparrow}\nolimits_{r}x_{n}$.
Then, for each $n$, $\mu(B_{r}(x_{r}^{\star}))\geqslant\mu(B_{r}(x_{n}))$, and
taking the limit as $n\to\infty$ shows that
$\mu(B_{r}(x_{r}^{\star}))\geqslant M_{r}$. The definition of $M_{r}$ implies
that $\mu(B_{r}(x_{r}^{\star}))\leqslant M_{r}$, and so
$\mu(B_{r}(x_{r}^{\star}))=M_{r}$.

((e)$\implies$(b)) Suppose that $x_{r}^{\star}$ has
$\mu(B_{r}(x_{r}^{\star}))=M_{r}$. Then, for any $x\in X$,
$\mu(B_{r}(x))\leqslant M_{r}$, i.e. $x\preccurlyeq_{r}x_{r}^{\star}$. Thus,
$x_{r}^{\star}$ is $\preccurlyeq_{r}$-greatest. ∎

A very simple existence result for radius-$r$ modes is the following:

######  Proposition 4.4 (Existence of radius-$r$ modes in compact spaces).

Let $X$ be a compact metric space, let $\mu\in\mathscr{P}(X)$, and let $r>0$.
Then $\preccurlyeq_{r}$ has at least one radius-$r$ mode $x_{r}^{\star}\in X$.

###### Proof.

This is a special case of Theorem 4.6(a), and also follows from Aliprantis and
Border (2006, Theorem 2.44), but a self-contained proof is given by observing
that the map $\mu(B_{r}(\hbox to5.71527pt{\hss$\cdot$\hss}))\colon X\to[0,1]$
is upper semicontinuous (Lemma A.1(a)) and hence has at least one global
maximiser $x_{r}^{\star}$ in the compact space $X$. ∎

We now adopt a very different approach to establishing the existence of
radius-$r$ modes, one based on applying intersection theorems to upward
closures with respect to $\preccurlyeq_{r}$. We begin with a very general
lemma; when Lemma 4.5 is used in practice, $\mathcal{T}$ will often be the
metric topology, but another useful case is the weak topology of a Banach
space.

######  Lemma 4.5.

Let $X$ be a separable metric space, $\mu\in\mathscr{P}(X)$, and $r>0$.
Suppose that $\mathcal{T}$ is a topology on $X$ such that, for some sequence
$(x_{n})_{n\in\mathbb{N}}\subset X$ with $\mu(B_{r}(x_{n}))\nearrow M_{r}>0$,
$\mathop{\uparrow}\nolimits_{r}x_{n}$ is $\mathcal{T}$-closed and
$\mathcal{T}$-compact for all sufficiently large $n$. Then the set
$\mathfrak{M}_{r}$ of radius-$r$ modes for $\mu$ is non-empty,
$\mathcal{T}$-compact, and
$\mathfrak{M}_{r}=\bigcap_{n\in\mathbb{N}}\mathop{\uparrow}\nolimits_{r}x_{n}$.

###### Proof.

Separability of $X$ implies that $M_{r}>0$. Let $(x_{n})_{n\in\mathbb{N}}$ be
such that $\mu(B_{r}(x_{n}))\nearrow M_{r}$ as $n\to\infty$. The sets
$\mathop{\uparrow}\nolimits_{r}x_{n}$ are non-empty; since the sequence
$\bigl{(}\mu(B_{r}(x_{n}))\bigr{)}_{n\in\mathbb{N}}$ is increasing,
$\mathop{\uparrow}\nolimits_{r}x_{n+1}\subseteq\mathop{\uparrow}\nolimits_{r}x_{n}$
for each $n$, i.e. they are decreasingly nested; by hypothesis, for
sufficiently large $n$, they are also $\mathcal{T}$-closed and
$\mathcal{T}$-compact. Therefore, by Cantor’s intersection theorem (Theorem
2.1(a)), their intersection is non-empty and $\mathcal{T}$-compact. This
intersection is precisely the set $\mathfrak{M}_{r}$ of radius-$r$ modes, as
already shown by Lemma 4.3. ∎

######  Theorem 4.6 (Existence of radius-$r$ modes).

Let $X$ be a separable metric space, $\mu\in\mathscr{P}(X)$, and $r>0$. Let
$\mathfrak{M}_{r}$ denote the set of radius-$r$ modes for $\mu$.

  1. (a)

Suppose that $X$ has the Heine–Borel property, i.e. that every closed and
bounded subset of $X$ is compact. Then $\mathfrak{M}_{r}$ is non-empty and
compact.

  2. (b)

Suppose that $X$ is complete and that $\mu$ is a doubling measure, i.e. there
exists a constant $C>0$ such that $\mu(B_{2r}(x))\leqslant C\mu(B_{r}(x))$ for
all $x\in X$ and $r>0$. Then $\mathfrak{M}_{r}$ is non-empty and compact.

  3. (c)

Suppose that $X$ is complete and there exists a point $o\in X$ and a function
$f\colon(0,\infty)^{2}\to(0,\infty)$ such that

$\text{for all $x\in B_{R}(o)$,}\quad\mu(B_{\delta}(x))\geqslant
f(\delta,R)>0.$ (4.2)

Then $\mathfrak{M}_{r}$ is non-empty and compact.

  4. (d)

Suppose that $X$ is complete and that there exists $(x_{n})_{n\in\mathbb{N}}$
with $\mu(B_{r}(x_{n}))\nearrow M_{r}$ and
$\gamma(\mathop{\uparrow}\nolimits_{r}x_{n})\to 0$ as $n\to\infty$. Then
$\mathfrak{M}_{r}$ is non-empty and compact.

  5. (e)

Suppose that $X$ is complete and that there exists $(x_{n})_{n\in\mathbb{N}}$
with $\mu(B_{r}(x_{n}))\nearrow M_{r}$ and
$\mathop{\textup{diam}}\nolimits(\mathop{\uparrow}\nolimits_{r}x_{n})\to 0$ as
$n\to\infty$. Then $\mathfrak{M}_{r}$ is a singleton.

  6. (f)

Suppose that $X$ is a Banach space and that there exists
$(x_{n})_{n\in\mathbb{N}}$ with $\mu(B_{r}(x_{n}))\nearrow M_{r}$, and that
$\mathop{\uparrow}\nolimits_{r}x_{n}$ is weakly compact for all sufficiently
large $n$. Then $\mathfrak{M}_{r}$ is non-empty and weakly compact.

  7. (g)

Suppose that $X$ is a reflexive Banach space and that there exists
$(x_{n})_{n\in\mathbb{N}}$ with $\mu(B_{r}(x_{n}))\nearrow M_{r}$ and that
$\mathop{\uparrow}\nolimits_{r}x_{n}$ is convex for all sufficiently large
$n$. Then $\mathfrak{M}_{r}$ is non-empty, weakly compact, and convex.

###### Proof.

  1. (a)

Lemma 4.2 ensures that every upward closure $\mathop{\uparrow}\nolimits_{r}x$,
$x\in X$, is closed and bounded in the metric topology on $X$. The claim now
follows from Lemma 4.5.

  2. (b)

By Björn and Björn (2011, Proposition 3.1), any complete metric space with a
doubling measure has the Heine–Borel property. The claim now follows from (a).

  3. (c)

Let $R>0$ and $\delta>0$ be arbitrary. The lower bound (4.2) implies that
there cannot be an infinite set of pairwise-disjoint balls
$B_{\delta}(x_{n})$, $n\in\mathbb{N}$, with centres $x_{n}\in B_{R}(o)$ since,
if there were, then we would obtain the contradiction

$1\geqslant\mu(B_{R+\delta}(o))\geqslant\mu\left(\biguplus_{n\in\mathbb{N}}B_{\delta}(x_{n})\right)=\sum_{n\in\mathbb{N}}\mu(B_{\delta}(x_{n}))\geqslant\sum_{n\in\mathbb{N}}f(\delta,R)=\infty.$

Since $\delta>0$ was arbitrary, $\gamma(B_{R}(o))=0$, i.e. $B_{R}(o)$ is
compact. Now, given any closed and bounded set $A\subseteq X$, choose $R>0$
large enough that $A\subseteq B_{R}(o)$ to see that $A$ must be compact.
Therefore, $X$ has the Heine–Borel property. The claim now follows from (a).

  4. (d)

The claim follows from Kuratowski’s intersection theorem (Theorem 2.1(c)).

  5. (e)

As already observed, by Lemma 4.2, each upward closure
$\mathop{\uparrow}\nolimits_{r}x_{n}$ is both closed and bounded in the metric
topology and they are decreasingly nested. Since $X$ is complete, Cantor’s
intersection theorem (Theorem 2.1(b)) yields that
$\mathfrak{M}_{r}=\bigcap_{n\in\mathbb{N}}\mathop{\uparrow}\nolimits_{r}x_{n}=\\{x_{r}^{\star}\\}$
for some $x_{r}^{\star}\in X$.

  6. (f)

This is simply Lemma 4.5 in the special case that $\mathcal{T}$ is the weak
topology of the separable Banach space $X$.

  7. (g)

Each closed, bounded, and convex subset of the separable, reflexive Banach
space $X$ is necessarily weakly compact, and so the claim follows from (f). ∎

Theorem 4.6 is by no means universally applicable, and indeed there are
measures that have no radius-$r$ modes, as the next two examples show.

######  Example 4.7 (An atomic measure with no radius-$r$ mode for $1\leqslant
r<2$).

Let $X=\mathbb{N}$ be equipped with the following variant of the discrete
metric:

$\Delta(k,\ell)\coloneqq\begin{cases}0,&\text{if $k=\ell$,}\\\ 2,&\text{if
$\min\\{k,\ell\\}$ is odd and $\max\\{k,\ell\\}=\min\\{k,\ell\\}+1$,}\\\
1,&\text{otherwise.}\end{cases}$ (4.3)

In the space $(X,\Delta)$, distinct points are a unit distance apart, with the
exception of each odd number and its successor, which are doubly spaced. Equip
this space with the measure
$\mu\coloneqq\sum_{k\in\mathbb{N}}2^{-k}\delta_{k}\in\mathscr{P}(X)$, where
$\delta_{k}$ is the unit Dirac measure centred at $k$. For arbitrary
$k\in\mathbb{N}$,

$\displaystyle\mu(B_{1}(2k-1))$
$\displaystyle=\mu(X\setminus\\{2k\\})=1-2^{-2k},$ (4.4)
$\displaystyle\mu(B_{1}(2k))$
$\displaystyle=\mu(X\setminus\\{2k-1\\})=1-2^{-(2k-1)}.$ (4.5)

Both (4.4) and (4.5) show that $M_{1}=1$; (4.4) shows that no odd number is a
radius-$1$ mode; (4.5) shows that no even number is a radius-$1$ mode. Thus,
$\mu$ has no radius-$1$ mode at all.

Similar arguments also show that $\mu$ has no radius-$r$ mode for $1\leqslant
r<2$; for $r\geqslant 2$, every point of $X$ is a radius-$r$ mode; for
$0<r<1$, the point $1\in X$ is the unique radius-$r$ mode.

It is interesting to relate this example to Theorem 4.6. In this setting, for
each $k\in X$,
$\mathop{\uparrow}\nolimits_{1}k\supseteq\\{k,k+2,k+4,\dots\\}$. This set is
non-compact with $\gamma(\mathop{\uparrow}\nolimits_{1}k)\geqslant 1$, since
it contains an infinite $1$-separated sequence. Thus, neither Theorem 4.6(a)
nor (d) can apply. Also, although the space $(X,\Delta)$ is complete,666Just
as in the case of the discrete metric, in this space, the properties of being
a Cauchy sequence, being a convergent sequence, and being eventually constant
all coincide. Theorem 4.6(e) does not apply because
$\mathop{\textup{diam}}\nolimits(\mathop{\uparrow}\nolimits_{1}k)\geqslant 1$.

######  Example 4.8 (A non-atomic measure with no radius-$r$ mode for any
$0<r<\nicefrac{{1}}{{8}}$.).

Building on the ideas of Example 4.7, consider the space

$\displaystyle X$
$\displaystyle\coloneqq\left\\{(\xi,k,m)\in\mathbb{R}\times\mathbb{N}^{2}\,\middle|\,\lvert\xi\rvert\leqslant
2^{-k-m-1}\right\\},$ (4.6)

equipped with the metric $d$ and probability measure $\mu$ given by

$\displaystyle d\bigl{(}(\xi,k,m),(\eta,\ell,n)\bigr{)}$
$\displaystyle\coloneqq\begin{cases}2,&\text{if $m\neq n$,}\\\
2^{-m}\Delta(k,\ell),&\text{if $m=n$ and $k\neq\ell$,}\\\
\lvert\xi-\eta\rvert,&\text{if $m=n$ and $k=\ell$,}\end{cases}$ (4.7)
$\displaystyle\mu\left(\biguplus_{k,m\in\mathbb{N}}\bigl{(}E_{k,m}\times\\{k\\}\times\\{m\\}\bigr{)}\right)$
$\displaystyle\coloneqq\frac{1}{Z}\sum_{k,m\in\mathbb{N}}\sigma^{-m}\lambda^{1}\bigl{(}E_{k,m}\cap[-2^{-k-m-1},2^{-k-m-1}]\bigr{)},$
(4.8)

for $E_{k,m}\in\mathscr{B}(\mathbb{R})$, where $\lambda^{1}$ is one-
dimensional Lebesgue measure, $\nicefrac{{1}}{{2}}<\sigma<1$ is a scaling
parameter, and the normalisation constant is
$Z\coloneqq\sum_{m\in\mathbb{N}}(2\sigma)^{-m}\in(1,\infty)$.

Now let $0<r<\nicefrac{{1}}{{8}}$ be arbitrary and let $n\in\mathbb{N}$ be
uniquely determined by $2^{-n}\leqslant r<2^{-n+1}$. We now determine $M_{r}$
and whether or not it can be attained by the masses of balls $B_{r}(x)$, where
$x=(\xi,k,m)$ has $m=n$, $m<n$, or $m>n$ respectively. Note that, since $r<2$,
the first case of (4.7) implies that
$B_{r}(x)\subseteq\mathbb{R}\times\mathbb{N}\times\\{m\\}$.

  1. (i)

First suppose that $m=n$. For odd $k\in\mathbb{N}$,

$\mu(B_{r}(x))=\mu(\mathbb{R}\times(\mathbb{N}\setminus\\{k+1\\})\times\\{m\\})=\frac{(2\sigma)^{-m}}{Z}(1-2^{-(k+1)}).$

Taking the limit as $k\to\infty$ shows that
$M_{r}\geqslant\frac{(2\sigma)^{-n}}{Z}$ but that no such ball realises this
supremal mass. The case of even $k$ is similar, just as in Example 4.7.

  2. (ii)

For $m>n$, since $B_{r}(x)\subseteq\mathbb{R}\times\mathbb{N}\times\\{m\\}$,
it follows that $x$ is not a radius-$r$ mode because

$\mu(B_{r}(x))\leqslant\frac{(2\sigma)^{-m}}{Z}<\frac{(2\sigma)^{-n}}{Z}\leqslant
M_{r}.$

  3. (iii)

If $m<n$, then $r<2^{-n}<2^{-m}$, and so the second case of (4.7) ensures that
$B_{r}(x)\subseteq\mathbb{R}\times\\{k\\}\times\\{m\\}$. The mass of such a
ball is maximised by the case $\xi=0$, $k=1$, $m=n-1$, in which case the ball
(which is a single line segment) has mass

$\displaystyle\mu(B_{r}(x))$
$\displaystyle=\frac{\sigma^{-m}}{Z}\lambda^{1}\bigl{(}[-r,r]\cap[-2^{-k-m-1},2^{-k-m-1}]\bigr{)}$
$\displaystyle\leqslant\sigma\frac{\sigma^{-n}}{Z}\lambda^{1}\bigl{(}[-2^{-n},2^{-n}]\cap[-2^{-n-1},2^{-n-1}]\bigr{)}$
$\displaystyle=\sigma\frac{(2\sigma)^{-n}}{Z}<\frac{(2\sigma)^{-n}}{Z},$

where the last inequality follows from the fact that $\sigma<1$.

Hence, $M_{r}=\frac{(2\sigma)^{-n}}{Z}$ but $\mu(B_{r}(x))<M_{r}$ for all
$x\in X$, i.e. $\mu$ has no radius-$r$ mode.

Thus, while Theorem 4.6 on the existence of radius-$r$ modes covers a variety
of well-behaved spaces and measures, Examples 4.7 and 4.8 show that existence
cannot be guaranteed for general spaces and measures.

Before moving on, we mention one interesting intermediate case, motivated by
applications to inverse problems, namely countable product measures on
weighted $\ell^{p}$ spaces (and isometric linear images of such spaces). This
is a broad class that includes Gaussian, Besov (Dashti et al., 2012; Lassas et
al., 2009), and Cauchy measures (Sullivan, 2017). It turns out that
reweightings of such measures always have radius-$r$ modes, i.e. Bayesian
posteriors with such measures as priors always have radius-$r$ MAP estimators.
We defer the precise statements to Theorems A.8 and A.9 in Section A.2 because
they do not have a particularly order-theoretic flavour.

Also, while we do prove the existence of radius-$r$ modes, we do not consider
taking limits as $r\to 0$ to obtain true MAP estimators for such posteriors.
The main difficulty here lies in proving that such a family
$(x_{r}^{\star})_{r>0}$ is bounded, so that a weakly convergent subsequence
can be extracted; this is not true in general, so one must argue using
properties of the prior and likelihood (e.g. when the prior is Gaussian or
Besov).

Indeed, the whole question of taking limits of radius-$r$ modes is a sensitive
one, and is the topic of the next section.

###  4.3 Convergence of greatest and near-greatest elements

If radius-$r$ modes $x_{r}^{\star}$ do exist for each $r>0$, it is then
natural to ask whether sequences of radius-$r$ modes can approximate true
modes, e.g. strong or weak modes. This approach is used by Dashti et al.
(2013, Theorem 3.5) to obtain strong modes for Bayesian posteriors arising
from Gaussian priors.

However, we have seen that existence of radius-$r$ modes can be difficult to
prove, and in some cases no radius-$r$ modes exist (Examples 4.7 and 4.8).
Taking limits of radius-$r$ modes is also problematic for more general
measures: the limit need not be a strong or weak mode, and not every mode can
be represented as the limit of radius-$r$ modes. Thus, one cannot hope to use
the approach of taking limits of radius-$r$ modes to find true modes if there
is no correspondence between modes and limits of radius-$r$ modes.
Nevertheless, we show that some of the difficulties can be overcome using
_asymptotic maximising families_ (AMFs) as proposed by Klebanov and Wacker
(2022).

To illustrate the problem and motivate the introduction of AMFs, we first give
an example of a measure with a bounded and continuous Lebesgue density
possessing a mode that cannot be represented as the limit of radius-$r$ modes.
The problem here is that balls around the points ${\pm 1}$ have asymptotically
equivalent mass, but each ball around ${+1}$ has slightly more mass than the
corresponding ball around ${-1}$; as a result, ${+1}$ “hides” the other mode
${-1}$.

(a) Unnormalised density (4.9) of the measure in Example 4.9. The point $+1$
is the unique radius-$r$ mode for all sufficiently small $r$.

(b) The ratio $\nicefrac{{\mu(B_{r}(+1))}}{{\mu(B_{r}(-1))}}$ converges to $1$
as $r\to 0$, but it is strictly greater than $1$ for any $r>0$.

Figure 4.1: Not every strong mode is the limit of a sequence of radius-$r$
modes: in Example 4.9, $-1$ is a strong mode but $+1$ is the unique radius-$r$
mode for all small $r$.

######  Example 4.9.

Define $\mu\in\mathscr{P}(\mathbb{R})$ by the Lebesgue density as shown in
Figure 4.1, given by

$\rho(x)\propto\max\\{1-(x-1)^{2},0\\}+\max\\{1-(x+1)^{2}-(x+1)^{4},0\\}.$
(4.9)

When $r$ is sufficiently small, $\mu(B_{r}(+1))=2r-\frac{2}{3}r^{3}$ and
$\mu(B_{r}(-1))=2r-\frac{2}{3}r^{3}-\frac{2}{5}r^{5}$, so there is a unique
radius-$r$ mode at $+1$. However, $+1$ and $-1$ are both strong modes: $+1$ is
a radius-$r$ mode for all sufficiently small $r$, so it is a strong mode
(Theorem 4.11) and $-1$ is a strong mode because

$\lim_{r\to 0}\frac{\mu(B_{r}(-1))}{M_{r}}=\lim_{r\to
0}\frac{\mu(B_{r}(+1))}{M_{r}}\lim_{r\to
0}\frac{\mu(B_{r}(-1))}{\mu(B_{r}(+1))}=1.$

Instead of representing modes as limits of radius-$r$ modes — which might not
be possible — one may consider families of points that are nearly greatest,
which always exist, and try to take limits of such families.

######  Definition 4.10.

Let $X$ be a metric space and let $\mu\in\mathscr{P}(X)$. A net
$(x_{r})_{r>0}\subseteq X$ is an _asymptotic maximising family_ (AMF) if there
exists a positive function $\varepsilon$ with $\lim_{r\to 0}\varepsilon(r)=0$
and

$\frac{\mu(B_{r}(x_{r}))}{M_{r}}\geqslant 1-\varepsilon(r)\quad\text{for all
$r>0$.}$ (4.10)

Note that every measure admits an AMF satisfying (4.10), even if the function
$\varepsilon$ is specified in advance, which is sometimes advantageous.

The following results shed light on the subtleties involved in taking limits
of radius-$r$ modes, or, more generally, AMFs.

######  Theorem 4.11.

Let $X$ be a separable metric space and let $\mu\in\mathscr{P}(X)$.

  1. (a)

If $(x^{\star})_{r>0}$ is an AMF with $x^{\star}\in X$ fixed, then $x^{\star}$
is a strong mode.

  2. (b)

If $x^{\star}$ is a radius-$r$ mode for all small enough $r>0$, then
$x^{\star}$ is a strong mode.

  3. (c)

If the AMF $(x_{r}^{\star})_{r>0}$ converges to $x^{\star}$, then $x^{\star}$
is a generalised strong mode.

###### Proof.

  1. (a)

As $\mu(B_{r}(x^{\star}))\geqslant(1-\varepsilon(r))M_{r}$, it is immediate
that $x^{\star}$ is a strong mode, because

$1\geqslant\lim_{r\to 0}\frac{\mu(B_{r}(x^{\star}))}{M_{r}}\geqslant\lim_{r\to
0}(1-\varepsilon(r))=1.$

  2. (b)

This is immediate from (a) as $(x^{\star})_{r>0}$ forms an AMF.

  3. (c)

This is precisely Clason et al. (2019, Lemma 2.4). ∎

The claim in (c) — which requires that the net $(x_{r}^{\star})_{r>0}$
converges to $x^{\star}$ along every subsequence — cannot be made stronger
without additional hypotheses: the limit $x^{\star}$ need not be a strong or
weak mode (as can be seen in Example 5.4(a), for which $x^{\star}=1$ is the
limit of an AMF which is neither a strong mode nor a weak mode). Furthermore,
one cannot weaken the hypotheses of (c) further: the points $\pm 1$ in Example
5.7 are limit points of an AMF but they are not even generalised modes.

The general question of classifying measures $\mu$ for which limits of
radius-$r$ modes are strong modes is still open, although Dashti et al. (2013)
show that reweightings of Gaussian measures on Hilbert spaces enjoy this
property, and Klebanov and Wacker (2022) show the same for some Gaussian
measures on sequence spaces.

Under an additional nesting assumption, intersection arguments can be applied
to AMFs to yield the existence of several kinds of modes.

######  Theorem 4.12 (AMFs and strong modes).

Let $X$ be a complete and separable metric space and let
$\mu\in\mathscr{P}(X)$. Let $(x_{r})_{r>0}$ be any AMF, i.e. any net
satisfying (4.10), and let
$I\coloneqq\bigcap_{r>0}\mathop{\uparrow}\nolimits_{r}x_{r}$. Then

  1. (a)

$I\subseteq\operatorname*{supp}(\mu)$;

  2. (b)

every $x^{\star}\in I$ is a strong (and hence weak and generalised strong)
mode for $\mu$;

  3. (c)

and if also

$0<r\leqslant
s\implies\mathop{\uparrow}\nolimits_{r}x_{r}\subseteq\mathop{\uparrow}\nolimits_{s}x_{s},$
(4.11)

then $I$ is non-empty and compact.

###### Proof.

Let $x^{\star}\in I$. For all sufficiently small $r>0$, it follows that
$\mu(B_{r}(x^{\star}))\geqslant\mu(B_{r}(x_{r}))\geqslant
M_{r}(1-\varepsilon(r))>0$, and so $x^{\star}\in\operatorname*{supp}(\mu)$,
which establishes (a). Furthermore, since
$x^{\star}\in\mathop{\uparrow}\nolimits_{r}x_{r}$ for each $r$,

$1\geqslant\frac{\mu(B_{r}(x^{\star}))}{M_{r}}\geqslant\frac{\mu(B_{r}(x_{r}))}{M_{r}}\geqslant
1-\varepsilon(r).$

Taking the limit as $r\to 0$ throughout shows that $x^{\star}$ is a strong
mode (and hence also a weak and generalised strong mode) for $\mu$,
establishing (b). (Alternatively, one may observe that $(x^{\star})_{r>0}$ is
a constant AMF and appeal to Theorem 4.11(a).)

For (c), let $(r_{n})_{n\in\mathbb{N}}$ be some null sequence of radii. The
nesting hypothesis (4.11) implies that
$I=\bigcap_{n\in\mathbb{N}}\mathop{\uparrow}\nolimits_{r_{n}}x_{r_{n}}$. For
each $n$, $\mathop{\uparrow}\nolimits_{r_{n}}x_{r_{n}}$ is non-empty and, by
Lemma 4.2, is closed and bounded with
$\gamma(\mathop{\uparrow}\nolimits_{r_{n}}x_{r_{n}})\leqslant 2r_{n}$. This,
together with the nesting hypothesis (4.11) and Kuratowski’s intersection
theorem (Theorem 2.1(c)), ensures that $I$ is non-empty and compact. ∎

######  Remark 4.13.

  1. (a)

The nesting hypothesis (4.11), in conjunction with Lemma 4.2, ensures that the
AMF $(x_{r})_{r>0}$ — and indeed any family of greatest elements
$(x_{r}^{\star})_{r>0}$ — must be bounded. This means that Theorem 4.12 does
not apply to measures such as Example 5.4(b), for which the radius-$r$ modes
“escape to infinity” as $r\to 0$. Hypothesis (4.11) also fails for measures
displaying oscillatory behaviour of the kind discussed in Example 5.7.

  2. (b)

Theorem 4.12 is not sharp, in the sense that there can exist modes
$x^{\star}\notin\bigcap_{r>0}\mathop{\uparrow}\nolimits_{r}x_{r}$. See Example
4.9 for an example of this situation with

$x_{r}\equiv{+1},\quad\mathop{\uparrow}\nolimits_{r}x_{r}=\\{+1\\},\quad
x^{\star}={-1}\notin\bigcap_{r>0}\mathop{\uparrow}\nolimits_{r}x_{r}.$

##  5 Preorders in the small-radius limit

One would like to think of $x^{\star}\in X$ as a mode of
$\mu\in\mathscr{P}(X)$ if $x^{\star}$ is a greatest or maximal element of $X$
with respect to the preorder $\preccurlyeq_{r}$ “in the limit as $r\to 0$” in
some sense. However, is such a limiting preorder well defined? Must this
preorder have greatest or maximal elements?

In fact, there are several candidates for a small-radius limiting preorder and
it appears that each of them has at least one undesirable feature. This work
will focus on the analytic small-radius limiting preorder $\preccurlyeq_{0}$,
to be defined shortly (Definition 5.1). This preorder has the advantage that
its greatest elements are weak modes; however, it has the disadvantage that it
is not total, i.e. the existence of greatest elements is not guaranteed, and
indeed the collection of incomparable elements may be rather large. We claim
that this is a small price to pay: we show in Appendix B that the alternative
definitions are even more ill behaved.

###  5.1 Definition and basic properties

######  Definition 5.1 (Small-radius limiting preorder).

Let $X$ be a metric space and let $\mu\in\mathscr{P}(X)$. Define a preorder
$\preccurlyeq_{0}$ on $X$ by

$\displaystyle x\preccurlyeq_{0}x^{\prime}$ $\displaystyle\iff\limsup_{r\to
0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(x^{\prime}))}\leqslant 1\iff\liminf_{r\to
0}\frac{\mu(B_{r}(x^{\prime}))}{\mu(B_{r}(x))}\geqslant 1,$ (5.1)

if both $x,x^{\prime}\in\operatorname*{supp}(\mu)$. Additionally, as
exceptional cases, $x\preccurlyeq_{0}x^{\prime}$ is defined to be false for
$x\in\operatorname*{supp}(\mu)$ and
$x^{\prime}\notin\operatorname*{supp}(\mu)$, and $x\preccurlyeq_{0}x^{\prime}$
is defined to be true for $x\notin\operatorname*{supp}(\mu)$ and
$x^{\prime}\in X$.

It is relatively straightforward to verify that $\preccurlyeq_{0}$, as defined
above, is a preorder on $X$; the only subtleties are correct handling of
points outside the support, and the use of the upper bound (but not equality)

$\limsup_{r\to
0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(y))}\frac{\mu(B_{r}(y))}{\mu(B_{r}(z))}\leqslant\limsup_{r\to
0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(y))}\limsup_{r\to
0}\frac{\mu(B_{r}(y))}{\mu(B_{r}(z))}$ (5.2)

when verifying transitivity. As usual, we will write $x\asymp_{0}x^{\prime}$
if both $x\preccurlyeq_{0}x^{\prime}$ and $x\succcurlyeq_{0}x^{\prime}$ hold,
and $x\mathrel{\|}_{0}x^{\prime}$ if neither $x\preccurlyeq_{0}x^{\prime}$ nor
$x\succcurlyeq_{0}x^{\prime}$ hold.

The appeal of the preorder $\preccurlyeq_{0}$ is that its greatest elements
are exactly the weak modes of $\mu$, as defined in (3.7), as the next two
results show.

######  Lemma 5.2 (Properties of $\preccurlyeq_{0}$-maximal elements).

Let $X$ be separable and let $\mu\in\mathscr{P}(X)$.

  1. (a)

If $x^{\star}$ is $\preccurlyeq_{0}$-maximal, then
$x^{\star}\in\operatorname*{supp}(\mu)$.

  2. (b)

The point $x^{\star}\in\operatorname*{supp}(\mu)$ is
$\preccurlyeq_{0}$-maximal if and only if any $x\in X$ satisfies either

$\liminf_{r\to 0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(x^{\star}))}<1\text{ or
}\lim_{r\to 0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(x^{\star}))}=1.$ (5.3)

###### Proof.

  1. (a)

Suppose that $x^{\star}$ is maximal but, for a contradiction, suppose also
that $x^{\star}\notin\operatorname*{supp}(\mu)$. As $X$ is separable, take
$x\in\operatorname*{supp}(\mu)\neq\varnothing$. By the exceptional cases of
Definition 5.1, $x^{\star}\prec_{0}x$, contradicting the assumption that
$x^{\star}$ is maximal.

  2. (b)

Let $x,x^{\prime}\in\operatorname*{supp}(\mu)$. It is straightforward to
verify from the definitions that

$\displaystyle x\prec_{0}x^{\prime}$ $\displaystyle\iff\liminf_{r\to
0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(x^{\prime}))}<1,$ (5.4) $\displaystyle
x\asymp_{0}x^{\prime}$ $\displaystyle\iff\lim_{r\to
0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(x^{\prime}))}=1.$ (5.5)

Suppose first that $x^{\star}$ is maximal, and let $x\in X$ be arbitrary. If
$x\notin\operatorname*{supp}(\mu)$, then $\mu(B_{r}(x))=0$ for all
sufficiently small $r$, so (5.3) holds. If $x\in\operatorname*{supp}(\mu)$,
then maximality of $x^{\star}$ implies that either $x\prec_{0}x^{\star}$ or
$x\asymp_{0}x^{\star}$, from which (5.3) follows.

Conversely, suppose that $x\in X$ satisfies $x^{\star}\preccurlyeq_{0}x$. The
exceptional cases in Definition 5.1 imply that
$x\in\operatorname*{supp}(\mu)$. Hence, by (5.5), $x^{\star}\asymp_{0}x$,
proving that $x^{\star}$ is $\preccurlyeq_{0}$-maximal. ∎

######  Lemma 5.3 (Characterisation of weak modes).

Let $X$ be separable and let $\mu\in\mathscr{P}(X)$. Then the following are
equivalent:

  1. (a)

$x^{\star}\in X$ is a weak mode for $\mu$;

  2. (b)

$x^{\star}\in X$ is a $\preccurlyeq_{0}$-greatest element;

  3. (c)

$x^{\star}\in X$ is a $\preccurlyeq_{0}$-maximal element that is comparable
with every other $x^{\prime}\in X$.

###### Proof.

((a)$\implies$(b)) Suppose that $x^{\star}$ is a weak mode for $\mu$. Then, by
definition (see (3.7)), it follows that $x\preccurlyeq_{0}x^{\star}$ for each
$x\in\operatorname*{supp}(\mu)$. As $x^{\star}\in\operatorname*{supp}(\mu)$,
any point $x\notin\operatorname*{supp}(\mu)$ satisfies
$x\preccurlyeq_{0}x^{\star}$ by the special cases in the definition of
$\preccurlyeq_{0}$.

((b)$\implies$(a)) Suppose that $x^{\star}$ is a $\preccurlyeq_{0}$-greatest
element. Then $x^{\star}\in\operatorname*{supp}(\mu)$ by Lemma 5.2. Hence, for
$x^{\prime}\in\operatorname*{supp}(\mu)$, (3.7) holds because
$x^{\prime}\preccurlyeq_{0}x^{\star}$. For
$x^{\prime}\notin\operatorname*{supp}(\mu)$, we obtain

$\frac{\mu(B_{r}(x^{\prime}))}{\mu(B_{r}(x^{\star}))}=0\text{ for sufficiently
small }r,$

proving that $x^{\star}$ is a weak mode.

((b)$\iff$(c)) This is obvious, since the defining property of being greatest
is exactly the property of being maximal and globally comparable. ∎

The preorder $\preccurlyeq_{0}$ does have some shortcomings. One is that, in
contrast to $\preccurlyeq_{r}$ with $r>0$ (Lemma 4.2), upward closures under
$\preccurlyeq_{0}$ need be neither closed nor bounded.

######  Example 5.4.

  1. (a)

For an example of a non-closed upward closure under $\preccurlyeq_{0}$,
similar in spirit to the example of Clason et al. (2019) mentioned in Section
3, let $\mu\in\mathscr{P}(\mathbb{R})$ have the Lebesgue density
$\rho\colon\mathbb{R}\to\mathbb{R}$, $\rho(x)\coloneqq 2x\mathds{1}[0\leqslant
x\leqslant 1]$, with $\operatorname*{supp}(\mu)=[0,1]$, and consider
$y\in\mathbb{R}$. If $y<0$ or $y>1$, then $y\notin\operatorname*{supp}(\mu)$
and $\mathop{\uparrow}\nolimits_{0}y=\mathbb{R}$. For $0\leqslant
y\leqslant\nicefrac{{1}}{{2}}$, $\mathop{\uparrow}\nolimits_{0}y=[y,1]$, which
is closed. However, for $\nicefrac{{1}}{{2}}<y<1$,

$\lim_{r\to 0}\frac{\mu(B_{r}(1))}{\mu(B_{r}(y))}=\lim_{r\to
0}\frac{2r-r^{2}}{4yr}=\frac{1}{2y}<1$

and so $\mathop{\uparrow}\nolimits_{0}y=[y,1)$, which is not closed. Finally,
$\mathop{\uparrow}\nolimits_{0}1=[\nicefrac{{1}}{{2}},1]$, which is closed.

  2. (b)

For an example of an unbounded upward closure under $\preccurlyeq_{0}$, let
$\mu\in\mathscr{P}(\mathbb{R})$ have the unbounded Lebesgue density
$\rho\colon\mathbb{R}\to\mathbb{R}$,

$\rho(x)\coloneqq\sum_{n\in\mathbb{N}}n\mathds{1}\bigl{[}n-\tfrac{2^{-n-1}}{n}\leqslant
x\leqslant n+\tfrac{2^{-n-1}}{n}\bigr{]}.$

That is, $\rho$ consists of a sum of disjoint indicator functions centred on
the natural numbers $n\in\mathbb{N}$, each having mass $2^{-n}$ and height
$n$. Then, for any $x,y\in\mathbb{N}$ with $x>y$,

$\lim_{r\to 0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(y))}=\lim_{r\to
0}\frac{2xr}{2yr}=\frac{x}{y}>1$

and so $\mathop{\uparrow}\nolimits_{0}y\supseteq\mathbb{N}\cap[y,\infty)$.

Example 5.4 furnishes two examples of measures with no
$\preccurlyeq_{0}$-maximal element, let alone a $\preccurlyeq_{0}$-greatest
element (weak mode), or strong mode. However, these examples are relatively
tame: there is no mode simply because any candidate mode $x^{\star}$ is
dominated by some other point $x^{\prime}$. The _real_ shortcoming and
subtlety of $\preccurlyeq_{0}$ is that it is not total — that is, the order
admits incomparable elements — and we make this the topic of the next
subsection.

###  5.2 Criteria for incomparability and comparability

For $r>0$, totality of $\preccurlyeq_{r}$ followed immediately from Definition
4.1. This is certainly not so obvious for $\preccurlyeq_{0}$. Indeed, what is
immediate from Definition 5.1 is that $\preccurlyeq_{0}$-incomparable elements
can be characterised as follows:

######  Lemma 5.5 (Incomparability in the limiting preorder).

For $x,x^{\prime}\in X$,

$x\mathrel{\|}_{0}x^{\prime}\iff
x,x^{\prime}\in\operatorname*{supp}(\mu)\text{ and }\liminf_{r\to
0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(x^{\prime}))}<1<\limsup_{r\to
0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(x^{\prime}))}.$ (5.6)

In the other direction, we can give a (very strong) sufficient condition for
two points to be comparable under $\preccurlyeq_{0}$:

######  Lemma 5.6.

Let $X$ be any metric space and let $\mu\in\mathscr{P}(X)$. Suppose that on
some interval $(0,r^{\star})$, the function
$r\mapsto\nicefrac{{\mu(B_{r}(x))}}{{\mu(B_{r}(x^{\prime}))}}$ is uniformly
continuous for $x$, $x^{\prime}\in\operatorname*{supp}(\mu)$. Then $x$ and
$x^{\prime}$ are $\preccurlyeq_{0}$-comparable.

###### Proof.

The ratio function $\nicefrac{{\mu(B_{r}(x))}}{{\mu(B_{r}(x^{\prime}))}}$ can
be uniquely extended to a uniformly continuous function on $[0,r^{\star}]$
(Aliprantis and Border, 2006, Lemma 3.11). By continuity, the limit of the
ratio function as $r\to 0$ must exist; the result follows by Lemma 5.5. ∎

The previous two lemmas hint at a way to construct concrete examples of
measures with incomparable points under $\preccurlyeq_{0}$: one must choose
the masses around two points such that the ratio of the masses of balls around
such points oscillates as $r\to 0$.

We now construct such a measure on $\mathbb{R}$ with a Lebesgue density and
two $\preccurlyeq_{0}$-incomparable maximal points, neither of which is
$\preccurlyeq_{0}$-greatest. (Section 5.3 will supply even more extreme and
general examples, but it is pedagogically useful to consider a simpler
construction first.) The idea is to construct a density so that the measure
$\mu\in\mathscr{P}(\mathbb{R})$ induced by it has a specific behaviour around
the points $x=\pm 1$. In this case, the density is chosen so that
$r\mapsto\mu(B_{r}(x))$ piecewise linearly interpolates the function
$r\mapsto\sqrt{r}$ through either the interpolation knots $r=a^{-n}$ with
$n\in\mathbb{N}$ even or the interpolation knots $r=a^{-n}$ with
$n\in\mathbb{N}$ odd, where $a>1$ is chosen arbitrarily. It turns out that
these mild perturbations of the integrable singularity $\rho(x)\propto\lvert
x\rvert^{-1/2}$ produce “incomparable modes”.

(a) The RCDFs $\mu^{\textup{e}}(B_{r}(0))$ and $\mu^{\textup{o}}(B_{r}(0))$
(shown on a linear scale) interpolate between the knots $a^{-n}$ to give mild
perturbations of the function $r\mapsto\sqrt{r}$.

(b) The RCDFs $\mu^{\textup{e}}(B_{r}(0))$ and $\mu^{\textup{o}}(B_{r}(0))$
(shown with a logarithmic scale for $r$) agree with the function
$r\mapsto\sqrt{r}$ at the knots $a^{-n}$ for even $n$ and odd $n$
respectively.

(c) The probability density functions $\rho^{\textup{e}}(\hbox
to5.14375pt{\hss$\cdot$\hss}+1)$ and $\rho^{\textup{o}}(\hbox
to5.14375pt{\hss$\cdot$\hss}-1)$ have singularities which behave like
$\lvert\hbox to5.14375pt{\hss$\cdot$\hss}\rvert^{-\nicefrac{{1}}{{2}}}$ at
$-1$ and $+1$ respectively.

(d) The ratio $\nicefrac{{\mu(B_{r}(-1))}}{{\mu(B_{r}(+1))}}$ (shown with a
logarithmic scale for $r$) oscillates between $\alpha$ and $\alpha^{-1}$ as
$r\to 0$, so the $\liminf$ of the ratio is below $1$ and the $\limsup$ is
above $1$.

Figure 5.1: Illustration of the measures defined in Example 5.7 for the
parameter choice $a=2$.

######  Example 5.7 (An absolutely continuous measure on $\mathbb{R}$ with
incomparable maximal points and neither weak nor generalised modes; after an
example of I. Klebanov).

Let $X$ be any Borel-measurable subset of $\mathbb{R}$ containing $[-2,2]$.
Fix $a>1$ and, as illustrated in Figure 5.1, define
$\mu^{\textup{e}},\mu^{\textup{o}}\in\mathscr{P}(X)$ via their Lebesgue
densities $\rho^{\textup{e}},\rho^{\textup{o}}\colon X\to[0,\infty]$,

$\rho^{\textup{e}}(x)\coloneqq\begin{cases}0,&\text{if $\lvert x\rvert>1$,}\\\
\dfrac{a^{\nicefrac{{n}}{{2}}}(1-a^{-1})}{2(1-a^{-2})},&\text{if
$a^{-2-n}\leqslant\lvert x\rvert\leqslant a^{-n}$ for even
$n\in\mathbb{N}_{0}\coloneqq\mathbb{N}\cup\\{0\\}$,}\\\ \infty,&\text{if
$x=0$,}\end{cases}$

and

$\rho^{\textup{o}}(x)\coloneqq\begin{cases}0,&\text{if $\lvert x\rvert>1$,}\\\
\dfrac{1-a^{-\nicefrac{{1}}{{2}}}}{2(1-a^{-1})},&\text{if
$a^{-1}\leqslant\lvert x\rvert\leqslant 1$,}\\\
\dfrac{a^{\nicefrac{{n}}{{2}}}(1-a^{-1})}{2(1-a^{-2})},&\text{if
$a^{-2-n}\leqslant\lvert x\rvert\leqslant a^{-n}$ for odd
$n\in\mathbb{N}$,}\\\ \infty,&\text{if $x=0$,}\end{cases}$

so that the RCDFs are

$\mu^{\textup{e}}(B_{r}(0))=\begin{cases}1,&\text{if $r\geqslant 1$,}\\\
a^{-1-\nicefrac{{n}}{{2}}}+(r-a^{-2-n})\dfrac{a^{\nicefrac{{n}}{{2}}}(1-a^{-1})}{1-a^{-2}},&\text{if
$a^{-2-n}\leqslant r\leqslant a^{-n}$ for even $n\in\mathbb{N}_{0}$,}\\\
0,&\text{if $r=0$,}\end{cases}$

and

$\mu^{\textup{o}}(B_{r}(0))=\begin{cases}1,&\text{if $r\geqslant 1$,}\\\
a^{-\nicefrac{{1}}{{2}}}+(r-a^{-1})\dfrac{1-a^{-\nicefrac{{1}}{{2}}}}{1-a^{-1}},&\text{if
$a^{-1}\leqslant r\leqslant 1$,}\\\
a^{-1-\nicefrac{{n}}{{2}}}+(r-a^{-2-n})\dfrac{a^{\nicefrac{{n}}{{2}}}(1-a^{-1})}{1-a^{-2}},&\text{if
$a^{-2-n}\leqslant r\leqslant a^{-n}$ for odd $n\in\mathbb{N}$,}\\\
0,&\text{if $r=0$.}\end{cases}$

We now consider the probability measure
$\mu\coloneqq\tfrac{1}{2}\mu^{\textup{e}}(\hbox
to5.71527pt{\hss$\cdot$\hss}+1)+\tfrac{1}{2}\mu^{\textup{o}}(\hbox
to5.71527pt{\hss$\cdot$\hss}-1)\in\mathscr{P}(X)$ with Lebesgue density
$\rho\coloneqq\tfrac{1}{2}\rho^{\textup{e}}(\hbox
to5.71527pt{\hss$\cdot$\hss}+1)+\tfrac{1}{2}\rho^{\textup{o}}(\hbox
to5.71527pt{\hss$\cdot$\hss}-1)$.

We first observe that $\pm 1\succcurlyeq_{0}x$ for any $x\neq\pm 1$. For
sufficiently small $r>0$, both $\rho^{\textup{e}}$ and $\rho^{\textup{o}}$ are
bounded above by a constant on $B_{r}(x)=[x-r,x+r]$, so that
$\mu(B_{r}(x))\leqslant cr$ for some $c\geqslant 0$. On the other hand, by
construction, both $\mu(B_{r}(-1))$ and $\mu(B_{r}(+1))$ are asymptotically
equivalent to $\nicefrac{{\sqrt{r}}}{{2}}$ as $r\to 0$, from which it follows
that ${\pm 1}\succcurlyeq_{0}x$.

However, ${-1}$ and ${+1}$ are incomparable. Observe that, for $r=a^{-n}$ with
$n\in\mathbb{N}$ even,

$\frac{\mu(B_{r}(-1))}{\mu(B_{r}(+1))}=\alpha\coloneqq\frac{a+1}{2a^{1/2}}>1,$

whereas for $r=a^{-n}$ with $n\in\mathbb{N}$ odd, this ratio of ball masses
takes the value $\alpha^{-1}$, and, for all $r>0$, it lies in the interval
$[\alpha^{-1},\alpha]$, all of which can be verified easily from the
interpolation formulae for $\mu^{\textup{e}}(B_{r}(0))$ and
$\mu^{\textup{o}}(B_{r}(0))$. Lemma 5.5 now implies that
${-1}\mathrel{\|}_{0}{+1}$, since

$\alpha^{-1}=\liminf_{r\to
0}\frac{\mu(B_{r}(-1))}{\mu(B_{r}(+1))}<1<\limsup_{r\to
0}\frac{\mu(B_{r}(-1))}{\mu(B_{r}(+1))}=\alpha.$

Thus, the preorder $\preccurlyeq_{0}$ induced by $\mu$ has two incomparable
maximal elements, namely $\pm 1$, has no greatest elements, and hence $\mu$
has no weak modes (Lemma 5.3).

We now check that $+1$ and $-1$ are not generalised modes. Let $r_{n}\coloneqq
a^{-2n}$, and suppose that $x_{n}\to 1$ as $n\to\infty$. Choose $N$ large
enough that, for all $n\geqslant N$, $\lvert
x_{n}-1\rvert<\nicefrac{{1}}{{2}}$ and $r_{n}<\nicefrac{{1}}{{2}}$. As the
density $\rho^{\textup{o}}(\hbox to5.71527pt{\hss$\cdot$\hss}-1)$ is a
symmetric singularity around $+1$, it follows that
$\mu(B_{r_{n}}(x_{n}))\leqslant\mu(B_{r_{n}}(+1))$. As
$M_{r_{n}}=\mu(B_{r_{n}}(-1))$, we obtain that

$\liminf_{n\to\infty}\frac{\mu(B_{r_{n}}(x_{n}))}{M_{r_{n}}}\leqslant\liminf_{n\to\infty}\frac{\mu(B_{r_{n}}(+1))}{M_{r_{n}}}=\alpha^{-1}<1.$

This proves that $+1$ is not a generalised mode; a similar argument with
$(r_{n})_{n\in\mathbb{N}}=a^{-2n+1}$ proves that $-1$ is not a generalised
mode.

Finally, suppose that $x\neq\pm 1$, and let $(r_{n})_{n\in\mathbb{N}}$ be any
null sequence. Let $\varepsilon\coloneqq\min\bigl{\\{}\lvert x-1\rvert,\lvert
x+1\rvert\bigr{\\}}$. Suppose that $x_{n}\to x$ as $n\to\infty$. There must
exist $N\in\mathbb{N}$ such that, for all $n\geqslant N$, $\lvert
x_{n}-1\rvert>\nicefrac{{\varepsilon}}{{2}}$ and $\lvert
x_{n}+1\rvert>\nicefrac{{\varepsilon}}{{2}}$. The Lebesgue density of $\mu$ is
bounded on
$\mathbb{R}\setminus\bigl{(}B_{\nicefrac{{\varepsilon}}{{2}}}(+1)\cup
B_{\nicefrac{{\varepsilon}}{{2}}}(-1)\bigr{)}$ by some constant $C>0$, so
$\mu(B_{r_{n}}(x_{n}))\leqslant Cr_{n}$ for $n\geqslant N$. As
$M_{r_{n}}\in\Theta\bigl{(}r_{n}^{\nicefrac{{1}}{{2}}}\bigr{)}$ as
$n\to\infty$, it follows that

$\liminf_{n\to\infty}\frac{\mu(B_{r_{n}}(x_{n}))}{M_{r_{n}}}=0,$

so $x$ is not a generalised mode.

Example 5.7 illustrates a difficulty with weak modes, and one whose cause can
be traced to incomparability: if the space $X$ is partitioned into disjoint
positive-mass sets $A$ and $B$, existence of modes for $\mu$ restricted to (or
conditioned upon) $A$ and $B$ individually cannot ensure existence of a mode
for $\mu$, since the modes of $\mu|_{A}$ and $\mu|_{B}$ may be
$\preccurlyeq_{0}$-incomparable.

Thus, while $\pm 1$ are intuitively modes and have Lebesgue density $+\infty$,
the measure $\mu$ has no modes in any of the senses defined in Section 3. We
emphasise that one cannot simply declare all points with Lebesgue density
$+\infty$ to be modes, since this would place all singularities of the density
on the same footing, which is clearly undesirable if one singularity is
genuinely “smaller” than the other in the sense that the RCDFs around these
points are, say, $\sqrt{r}$ and $2\sqrt{r}$, and so the smaller one ought not
to be considered a mode.

As suggested in the introduction, this example could be interpreted as
evidence that maximal — rather than greatest — elements of a preorder are good
candidates for modes. Indeed, from the order-theoretic perspective, maximal
elements appear to be just as reasonable as greatest elements, and we hope
that this encourages further study of whether maximal elements are sufficient
for applications.

The extension theorems of Szpilrajn, Arrow, and Hansson (Hansson, 1968;
Szpilrajn, 1930) assert that any non-total preorder $\preccurlyeq$ can be
extended to a total preorder $\preccurlyeq^{\prime}$. Thus, given the non-
totality of $\preccurlyeq_{0}$, one might hope to resolve all these issues by
defining a mode of $\mu$ to be a $\preccurlyeq^{\prime}_{0}$-greatest element.
Unfortunately, such a total extended preorder is not uniquely determined and
so such a definition of a mode would not be well defined: for the measure
$\mu$ of Example 5.7, there are total extensions $\preccurlyeq^{\prime}_{0}$
of $\preccurlyeq_{0}$ yielding each of the three situations

$-1\preccurlyeq^{\prime}_{0}+1\not\preccurlyeq^{\prime}_{0}-1,\quad+1\preccurlyeq^{\prime}_{0}-1\not\preccurlyeq^{\prime}_{0}+1,\quad\text{and
}-1\asymp^{\prime}_{0}+1.$

That is, which (if any) of $\pm 1$ counts as a mode would seem to be a matter
of personal choice.

Finally, we note that similar ideas could be used to construct incomparable
points that are not $\preccurlyeq_{0}$-maximal, but such examples have less
importance for the theory of modes.

###  5.3 Absolutely continuous measures with dense antichains

Example 5.7 can be easily extended to construct a measure
$\mu\in\mathscr{P}(\mathbb{R})$ with any finite number of mutually
incomparable $\preccurlyeq_{0}$-maximal elements, none of which are greatest
elements. Indeed, it is natural to wonder how bad the situation of
incomparability can be, and in particular how large an antichain can be. This
section’s main result, Theorem 5.11, shows that $\mu$ may have a topologically
dense antichain consisting of maximal elements (and mutually incomparable
would-be modes are “nearly everywhere”), even when $\mu$ has a Lebesgue
density; from the perspective of geometric measure theory, the notable point
here is that there is no need to resort to singular measures.

We begin with the following straightforward proposition:

######  Proposition 5.8.

Let $X$ be a finite or discrete metric space and let $\mu\in\mathscr{P}(X)$.
Then $\preccurlyeq_{0}$ has no incomparable elements.

###### Proof.

Let $x,x^{\prime}\in\operatorname*{supp}(\mu)$. As $X$ is discrete, the
measure $\mu$ must be atomic, so

$\lim_{r\to
0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(x^{\prime}))}=\frac{\mu(\\{x\\})}{\mu(\\{x^{\prime}\\})}.$

As the limit exists, the ratio does not oscillate on either side of unity as
$r\to 0$, so comparability of $x$ and $x^{\prime}$ follows from Lemma 5.5. ∎

Proposition 5.8 shows that any measure on a finite metric space induces a
total order $\preccurlyeq_{0}$. We now show that incomparability can arise
even in very simple settings, such as in a countable metric space or on the
real line with a continuous, bounded Lebesgue density.

######  Example 5.9.

  1. (a)

Let $X$ be the closure of the set $\\{-1+2^{-n}\mid
n\in\mathbb{N}\\}\cup\\{1-2^{-n}\mid n\in\mathbb{N}\\}$ with the Euclidean
metric inherited from $\mathbb{R}$. Define the measure $\mu\in\mathscr{P}(X)$
by

$\mu\coloneqq\frac{1}{Z}\sum_{k=1}^{\infty}2^{-4k+1}\delta_{1-2^{-4k+1}}+2^{-4k-1}\delta_{-1+2^{-4k}},$

where $Z>0$ is a normalisation constant. Then $+1$ and $-1$ are incomparable
because

$\displaystyle\frac{\mu(B_{2^{-4k+1}}(+1))}{\mu(B_{2^{-4k+1}}(-1))}$
$\displaystyle=\frac{(15Z)^{-1}\times 2^{-4k+1}}{4\times(15Z)^{-1}\times
2^{-4k+1}}=\frac{1}{4},$ (5.7)
$\displaystyle\frac{\mu(B_{2^{-4k-1}}(+1))}{\mu(B_{2^{-4k-1}}(-1))}$
$\displaystyle=\frac{\mu(B_{2^{-4(k+1)+3}}(+1))}{\mu(B_{2^{-4k-1}}(-1))}=\frac{4\times(15Z)^{-1}\times
2^{-4(k+1)+3}}{(15Z)^{-1}\times 2^{-4k-1}}=4.$ (5.8)

  2. (b)

Let $X=\mathbb{R}$ and define the densities

$\Delta_{w,h}(x)=\begin{cases}h\left(1-\frac{|x|}{w}\right)&|x|\leqslant w\\\
0&\text{otherwise,}\end{cases}$

which have total mass $wh$ and are supported on the interval $[-w,w]$. By
analogy with part (a), let $\mu\in\mathscr{P}(\mathbb{R})$ have the
continuous, bounded Lebesgue density

$\rho(x)=\frac{1}{Z}\sum_{k\in\mathbb{N}}\Delta_{2^{-4k+1},1}\bigl{(}x-(1-2^{-4k+2})\bigr{)}+\Delta_{2^{-4k-1},1}\bigl{(}x-(-1+2^{-4k})\bigr{)},$

where $Z>0$ is a normalisation constant. As (5.7) and (5.8) remain true for
the measure $\mu$ in this example, the points $\pm 1$ are incomparable.

While Example 5.9(b) shows that even a measure with a continuous, bounded
Lebesgue density may have an antichain, we show in Proposition 5.14 that this
antichain is never at the “top” of the order as in Example 5.7.

Examples such as Examples 5.7 and 5.9 can be extended to show that an
antichain may be countably infinite. To do so, we first introduce a family of
“coprime” oscillatory RCDFs to generalise the RCDFs $\mu^{\textup{e}}$ and
$\mu^{\textup{o}}$ of Example 5.7:

(a) As in Example 5.7, the density $\rho_{k,m}$ is based on perturbations of
the singularity $\lvert\hbox
to5.14375pt{\hss$\cdot$\hss}\rvert^{-\nicefrac{{1}}{{2}}}$ and has mass $m$.

(b) The RCDF $\mu_{k,m}(B_{\hbox to3.60062pt{\hss$\cdot$\hss}}(0))$ linearly
interpolates between the knots $r=a^{-n}$, $n\in\mathbb{N}$ (marked as
circles) to obtain the desired perturbations of the “growth rate” $\sqrt{r}$.

Figure 5.2: Example of a density $\rho_{k,m}$ and RCDF $\mu_{k,m}$ from
Proposition 5.10 with $a=2$.

######  Proposition 5.10 (A family of oscillatory RCDFs).

Fix $a>1$ and a natural number $k\geqslant 2$. Construct the Lebesgue
densities $\rho_{k}\colon\mathbb{R}\to[0,\infty]$ as in Figure 5.2(a), defined
by

$\rho_{k}(x)\coloneqq\begin{cases}0,&\text{ if }\lvert x\rvert>a^{-1},\\\
\frac{1}{2}a^{\frac{n+1}{2}},&\text{ if }a^{-n-1}<\lvert x\rvert\leqslant
a^{-n}\text{ for }n\in\mathbb{N}\text{ with }k\mid n,\\\ 0,&\text{ if
}a^{-n-1}<\lvert x\rvert\leqslant a^{-n}\text{ for }n\in\mathbb{N}\text{ with
}k\mid n+1,\\\
\frac{1}{2}a^{\frac{n}{2}}\left(\frac{1-a^{-\nicefrac{{1}}{{2}}}}{1-a^{-1}}\right),&\text{
if }a^{-n-1}<\lvert x\rvert\leqslant a^{-n}\text{ for }n\in\mathbb{N}\text{
with }k\nmid n\text{ and }k\nmid n+1,\\\ \infty,&\text{ if }x=0,\end{cases}$

and, given $m>0$, define the corresponding truncated densities
$\rho_{k,m}(x)\coloneqq\rho_{k}(x)\mathds{1}\left[|x|\leqslant r(m)\right]$,
with the truncation radius $r(m)$ chosen such that

$r(m)\coloneqq\inf\left\\{s\,\middle|\,\int_{B_{s}(0)}\rho_{k}(t)\,\mathrm{d}t=m\right\\}.$

Write $\mu_{k,m}$ for the measure on the real line with
$\mu_{k,m}(\mathbb{R})=m$ and Lebesgue density $\rho_{k,m}$.

  1. (a)

The RCDF $s\mapsto\mu_{k,m}(B_{s}(0))$ linearly interpolates between the knots

$\bigl{\\{}(a^{-n},a^{-\nicefrac{{n}}{{2}}})\,\big{|}\,n\in\mathbb{N},~{}k\nmid
n\bigr{\\}}\cup\bigl{\\{}(a^{-n},a^{\nicefrac{{1}}{{2}}-\nicefrac{{n}}{{2}}})\,\big{|}\,n\in\mathbb{N},~{}k\mid
n\bigr{\\}}\cup\bigl{\\{}(0,0)\bigr{\\}},$

until truncated at radius $r(m)$ (Figure 5.2(b)) and has formula

$\mu_{k,m}(B_{s}(0))=\begin{cases}\mu_{k,m}(B_{r(m)}(0)),&\text{if }s>r(m),\\\
a^{\frac{n+1}{2}}s,&\text{if $a^{-n-1}<s\leqslant a^{-n}$ for $n\in\mathbb{N}$
with $k\mid n$,}\\\ a^{-\frac{n}{2}},&\text{if $a^{-n-1}<s\leqslant a^{-n}$
for $n\in\mathbb{N}$ with $k\mid n+1$,}\\\
\frac{1-a^{-\nicefrac{{1}}{{2}}}}{1-a^{-1}}\left(a^{\frac{n}{2}}s+a^{-\frac{n}{2}}\right),&\text{if
$a^{-n-1}<s\leqslant a^{-n}$ for $n\in\mathbb{N}$ with $k\nmid n$ and $k\nmid
n+1$,}\\\ 0,&\text{if $s=0$.}\end{cases}$

  2. (b)

In particular, if $a^{-n}\leqslant r(m)$,

$\mu_{k,m}(B_{a^{-n}}(0))=\begin{cases}a^{-\nicefrac{{n}}{{2}}},&\text{if
$k\nmid n$,}\\\ a^{\nicefrac{{1}}{{2}}-\nicefrac{{n}}{{2}}},&\text{if $k\mid
n$.}\end{cases}$

  3. (c)

Given distinct coprime integers $k,k^{\prime}\geqslant 2$ and arbitrary
$m,m^{\prime}>0$,

$\liminf_{s\to
0}\frac{\mu_{k,m}(B_{s}(0))}{\mu_{k^{\prime},m^{\prime}}(B_{s}(0))}<1<\limsup_{s\to
0}\frac{\mu_{k,m}(B_{s}(0))}{\mu_{k^{\prime},m^{\prime}}(B_{s}(0))}.$

  4. (d)

Provided $s\leqslant r(m)$, we have
$\sqrt{\nicefrac{{s}}{{a}}}\leqslant\mu_{k,m}(B_{s}(0))\leqslant\sqrt{as}$.

  5. (e)

The truncation radius satisfies $r(m)\leqslant am^{2}$.

  6. (f)

The density $\rho_{k}$ satisfies $\rho_{k}(t)\leqslant
t^{-\nicefrac{{1}}{{2}}}$ for all $t\in\mathbb{R}$.

###### Proof.

  1. (a)

The formula for the RCDF follows by integrating the density $\rho_{k,m}$.

  2. (b)

The value at the knots $a^{-n}$ follows from (a).

  3. (c)

We exploit the fact that $k$ and $k^{\prime}$ are coprime, so the sequence
$(n_{i})_{i\in\mathbb{N}}=(ik^{\prime}-1)k\nearrow\infty$ is divisible by $k$
but not $k^{\prime}$, and the sequence
$(m_{i})_{i\in\mathbb{N}}=(ik-1)k^{\prime}\nearrow\infty$ is divisible by
$k^{\prime}$ but not $k$. For sufficiently large $i$,
$a^{-n_{i}}\leqslant\min\\{r(m),r(m^{\prime})\\}$, and hence by (b) we obtain

$\frac{\mu_{k,m}(B_{a^{-n_{i}}}(0))}{\mu_{k^{\prime},m^{\prime}}(B_{a^{-n_{i}}}(0))}=\frac{a^{\nicefrac{{1}}{{2}}-\nicefrac{{n_{i}}}{{2}}}}{a^{-\nicefrac{{n_{i}}}{{2}}}}=a^{\frac{1}{2}}.$

Similarly, for $i$ sufficiently large such that
$a^{-m_{i}}\leqslant\min\\{r(m),r(m^{\prime})\\}$,

$\frac{\mu_{k,m}(B_{a^{-m_{i}}}(0))}{\mu_{k^{\prime},m^{\prime}}(B_{a^{-m_{i}}}(0))}=\frac{a^{-\nicefrac{{m_{i}}}{{2}}}}{a^{\nicefrac{{1}}{{2}}-\nicefrac{{m_{i}}}{{2}}}}=a^{-\frac{1}{2}}.$

As these hold for all $i$ sufficiently large, and $a^{-n_{i}}$ and
$a^{-m_{i}}$ converge to zero, the desired inequality follows.

  4. (d)

The lower bound follows because, for $s\leqslant r(m)$,

$\mu_{k,m}(B_{s}(0))\geqslant\mu_{k,m}(B_{a^{-\lfloor-\log_{a}(s)\rfloor}}(0))\geqslant
a^{-\lfloor-\log_{a}(s)\rfloor/2}\geqslant\sqrt{\nicefrac{{s}}{{a}}},$

where the penultimate inequality uses (b); the upper bound is easily verified
from the construction of $\mu_{k,m}$ as a linear interpolation of the knots.

  5. (e)

As
$\int_{B_{s}(0)}\rho_{k}(t)\,\mathrm{d}t\geqslant\sqrt{\nicefrac{{s}}{{a}}}$,
it follows that $\int_{B_{am^{2}}(0)}\rho_{k}(t)\,\mathrm{d}t\geqslant m$, and
hence $r\leqslant am^{2}$.

  6. (f)

This is easily verified from the expression for $\rho_{k}$. ∎

We now use Proposition 5.10 to show that a maximal antichain of a measure can
be topologically dense even in the apparently well-behaved case of an
absolutely continuous probability measure on the real line. Our example shows
that the set of $\preccurlyeq_{0}$-maximal elements might be very different to
the set of $\preccurlyeq_{0}$-greatest elements: the measure we construct has
a dense set of maximal elements, yet it does not possess any greatest element
because none of those maximal elements is globally comparable.

In spirit, the idea is much the same as Example 5.7: centre mutually
incomparable compactly supported singularities at a dense collection of points
$\\{q_{k}\\}_{k\in\mathbb{N}}$. This is much more subtle, however, as one must
take care to ensure that the points $q_{k}$ are distant enough from one
another that the singularities neither interfere with each other nor
accumulate too much mass at a point outside of the dense set. Here, this is
achieved by taking the $q_{k}$ to be multiples of powers of two, a case that
is easily analysed but quite sparse. Indeed, we write $D$ for the set of
dyadic rationals, which we write as the disjoint union over the _levels_
$D_{\ell}\coloneqq\left\\{(2i-1)2^{-\ell}\,\middle|\,1\leqslant i\leqslant
2^{\ell-1}\right\\}$. By a slight abuse of terminology, we also describe the
sum of the densities centred at points in $D_{\ell}$ as the
$\ell\textsuperscript{th}$ level of the measure.

######  Theorem 5.11 (An absolutely continuous measure on $\mathbb{R}$ with a
countable dense antichain).

Let $\mu\in\mathscr{P}(\mathbb{R})$ have the Lebesgue density
$\rho\colon\mathbb{R}\to\mathbb{R}$ as shown in Figure 5.3, defined by

$\rho(x)\coloneqq\sum_{\ell=1}^{\infty}\sum_{i=1}^{2^{\ell-1}}\rho_{k(\ell,i),m(\ell)}(x-q_{\ell,i}),$

where $\rho_{k,m}$ is the density constructed in Proposition 5.10 with
parameter $a=2$; $k(\ell,i)$ is the $(2^{\ell-1}+i-1)\textsuperscript{th}$
prime; $m(\ell)\coloneqq 2^{-2\ell+1}$; and
$q_{\ell,i}\coloneqq(2i-1)2^{-\ell}\in D_{\ell}$. Then:

  1. (a)

the $\ell\textsuperscript{th}$ level of the measure $\mu$, consisting of all
densities centred at points in $D_{\ell}$, has mass $2^{-\ell}$, and hence
$\mu$ is a probability measure;

  2. (b)

the set of dyadic rationals
$D=\left\\{(2i-1)2^{-\ell}\,\middle|\,\ell\in\mathbb{N},~{}1\leqslant
i\leqslant 2^{\ell-1}\right\\}$ is a $\preccurlyeq_{0}$-antichain;

  3. (c)

every element of $D$ is $\preccurlyeq_{0}$-maximal.

###### Proof.

  1. (a)

By construction, each density in level $\ell$ has mass $m(\ell)=2^{-2\ell+1}$,
and there are $2^{\ell-1}$ densities, giving a total mass of $2^{-\ell}$. It
follows that $\mu$ is a probability measure as
$\int_{\mathbb{R}}\rho(x)\,\mathrm{d}x=\sum_{\ell\in\mathbb{N}}2^{-\ell}=1$.

  2. (b)

Take distinct elements $q_{\ell,i},q_{\ell^{\prime},i^{\prime}}\in D$. It is
sufficient to check that
$q_{\ell,i}\not\preccurlyeq_{0}q_{\ell^{\prime},i^{\prime}}$, as one can swap
$q_{\ell,i}$ and $q_{\ell^{\prime},i^{\prime}}$ to obtain that
$q_{\ell,i}\mathrel{\|}_{0}q_{\ell^{\prime},i^{\prime}}$. Asymptotically,
$\mu(B_{r}(q_{\ell,i}))\sim\mu_{k(\ell,i),m(\ell)}(B_{r}(0))$ as $r\to 0$, and
likewise
$\mu(B_{r}(q_{\ell^{\prime},i^{\prime}}))\sim\mu_{k(\ell^{\prime},i^{\prime}),m(\ell^{\prime})}(B_{r}(0))$
(Lemma A.12(a)). Using the identity $\limsup_{r\to 0}f(r)g(r)=\limsup_{r\to
0}f(r)\lim_{r\to 0}g(r)$, we obtain that

$\displaystyle\limsup_{r\to
0}\frac{\mu(B_{r}(q_{\ell,i}))}{\mu(B_{r}(q_{\ell^{\prime},i^{\prime}}))}$
$\displaystyle\quad=\limsup_{r\to
0}\frac{\mu_{k(\ell,i),m(\ell)}(B_{r}(0))}{\mu_{k(\ell^{\prime},i^{\prime}),m(\ell^{\prime})}(B_{r}(0))}\lim_{r\to
0}\frac{\mu(B_{r}(q_{\ell,i}))}{\mu_{k(\ell,i),m(\ell)}(B_{r}(0))}\frac{\mu_{k(\ell^{\prime},i^{\prime}),m(\ell^{\prime})}(B_{r}(0))}{\mu(B_{r}(q_{\ell^{\prime},i^{\prime}}))}$
$\displaystyle\quad=\limsup_{r\to
0}\frac{\mu_{k(\ell,i),m(\ell)}(B_{r}(0))}{\mu_{k(\ell^{\prime},i^{\prime}),m(\ell^{\prime})}(B_{r}(0))}>1$

where the final line follows by the construction of the oscillatory RCDFs in
Proposition 5.10(c) as $k(\ell,i)$ and $k(\ell^{\prime},i^{\prime})$ are
distinct primes. This proves that
$q_{\ell,i}\not\preccurlyeq_{0}q_{\ell^{\prime},i^{\prime}}$ as claimed, from
which incomparability follows.

  3. (c)

To show that $q\in D$ is maximal, it suffices to check that
$q\not\preccurlyeq_{0}x$ for any $x\in[0,1]\setminus D$; part (b) proves that
$q\not\preccurlyeq_{0}x$ when $x\in D$. To prove this, we must characterise
the behaviour of the RCDF $\mu(B_{r}(x))$; this depends on the properties of
the binary representation of $x$ and in particular on a quantity we call the
_dyadic irrationality exponent_ $\beta_{2}(x)\in[1,\infty)$ (Definition A.10).
If $\beta_{2}(x)<4$, then $\mu(B_{r}(x))\in o(r^{\nicefrac{{1}}{{2}}})$ (Lemma
A.12(b)); as $\mu(B_{r}(q))\in\Theta(r^{\nicefrac{{1}}{{2}}})$ by the
construction of the density centred at $q$, it follows that
$q\not\preccurlyeq_{0}x$ because

$\limsup_{r\to 0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(q))}=0.$

If $\beta_{2}(x)>4$, then $x$ is approximated particularly well by a sequence
of dyadic rationals, so there exists a sequence of scales as $r\to 0$ such
that the RCDF $\mu(B_{r}(x))$ behaves much like its approximating dyadic
rational. In fact, this approximation is so good that $q\mathrel{\|}_{0}x$
(Lemma A.12(d)) for the same reason that two dyadic rationals are
incomparable. In the critical case $\beta_{2}(x)=4$, there exist examples with
$q\mathrel{\|}_{0}x$ and examples where $x\prec_{0}q$, but in either case we
can still verify that $q\not\preccurlyeq_{0}x$ (Lemma A.12(c)) as required.
This proves that no $x\in[0,1]\setminus D$ can dominate any $q\in D$,
completing the proof. ∎

######  Remark 5.12.

  1. (a)

The proof shows that the dyadic rationals do not form a maximal antichain in
the sense of setwise inclusion: points with $\beta_{2}(x)>4$ are also
incomparable with the dyadic rationals; thus, the cardinality of a maximal
antichain is at least $\aleph_{0}$. On the other hand, the Lebesgue
differentiation theorem implies that any antichain has Lebesgue measure zero
(see also Proposition 5.14(a)), so one cannot expect to find a larger
antichain in a measure-theoretic sense.

  2. (b)

Our construction is not limited to this specific dense set and enumeration, or
even to absolutely continuous measures on the real line; for example, one can
reweight a Gaussian measure on a separable Hilbert space $H$ to have a similar
RCDF to our prototypical measures $\mu_{k,m}$ at the point $0$, then place
such measures at points in a dense subset of $H$. Another possibility is to
argue as in Theorem 5.11 using $\mathbb{Q}\cap[0,1]$ as the dense set; the
behaviour then depends on the usual number-theoretic irrationality
exponent777For further details about the irrationality exponent, traditionally
denoted $\mu(x)$, see e.g. Fel′dman and Nesterenko (1998). instead of the
dyadic irrationality exponent $\beta_{2}(x)$, but one still obtains a dense
antichain containing all rationals in $[0,1]$. Some of the technical steps are
described in more detail in Lambley (2022, Section 7.3 and Appendix A).

(a) The density $\rho$ is constructed as a sum of the prototype densities
$\rho_{k,m}$. The orange density is $\rho_{2,2^{-1}}(\hbox
to5.14375pt{\hss$\cdot$\hss}-\nicefrac{{1}}{{2}})$, and the grey densities are
$\rho_{3,2^{-3}}(\hbox to5.14375pt{\hss$\cdot$\hss}-\nicefrac{{1}}{{4}})$ and
$\rho_{5,2^{-3}}(\hbox to5.14375pt{\hss$\cdot$\hss}-\nicefrac{{3}}{{4}})$.

(b) Approximation of the density, truncated at the fifth level (i.e. with the
densities centred at all dyadic rationals of the form $c2^{-n}$ with
$n\leqslant 5$).

Figure 5.3: The density $\rho$ from Theorem 5.11 for which the dyadic
rationals are an antichain.

###  5.4 Essential totality

The need for a $\preccurlyeq_{0}$-greatest element to be globally comparable
is a non-trivial one, and it can fail rather dramatically, e.g. when the
maximal elements form a dense antichain as in Theorem 5.11. Such examples
could be criticised as somewhat artificial, but we feel that they highlight
the importance of checking for incomparability and developing technical
conditions on the measure which prevent it.

One could rule out incomparability if $\preccurlyeq_{0}$ were total, but this
is not true in general, and checking this condition is often difficult in
practice. We propose a somewhat weaker condition, where one can tolerate
incomparability away from the “top” of the preorder, as long as any candidate
for a maximal element is also globally comparable.

Our condition of _essential totality_ can be interpreted as an order-theoretic
generalisation of the $M$-property of Ayanbayev et al. (2022a); recall (3.8).
A motivating example is that of a Gaussian measure $\mu$ on an infinite-
dimensional space $X$: the Cameron–Martin space $H(\mu)$ is an essentially
total subspace where a maximal element must lie, and any element of the
Cameron–Martin space is globally comparable using the OM functional and
property $M(\mu,H(\mu))$.

######  Definition 5.13.

Let $X$ be a metric space and let $\mu\in\mathscr{P}(X)$. A non-empty subset
$E\subseteq X$ is _$\mu$ -essentially total_ if:

  1. (a)

any two elements of $E$ are comparable (i.e. $E$ is a
$\preccurlyeq_{0}$-chain);

  2. (b)

for any $x\in E$ and $x^{\prime}\in X\setminus E$,
$x^{\prime}\preccurlyeq_{0}x$; and

  3. (c)

for any $x^{\prime}\in X\setminus E$, there exists $x\in E$ such that
$x^{\prime}\prec_{0}x$.

Condition (b) says that if $x^{\star}\in E$ is an upper bound on $E$, then it
is $\preccurlyeq_{0}$-greatest; (c) says that no element in $X\setminus E$ can
be greatest. We emphasise, though, that there is no need for $E$ to be a large
set in any measure-theoretic or topological sense.

######  Proposition 5.14 (Examples of essentially total subsets).

  1. (a)

Suppose that $X\subseteq\mathbb{R}^{n}$ is open and that
$\mu\in\mathscr{P}(X)$ has continuous density $\rho\colon X\to[0,\infty)$ with
respect to $\lambda^{n}$. Then $E\coloneqq\\{x\in X\mid\rho(x)>0\\}$ is
$\mu$-essentially total, and $I_{\mu}(x)\coloneqq-\log\rho(x)$ is an OM
functional with domain $E$.

  2. (b)

Suppose that $\mu\in\mathscr{P}(X)$ has an OM functional $I_{\mu}\colon
E\to\mathbb{R}$ and property $M(\mu,E)$ holds. Then $E$ is $\mu$-essentially
total.

  3. (c)

Suppose more generally that $\mu_{0}\in\mathscr{P}(X)$ has an OM functional
$I_{\mu_{0}}\colon E\to\mathbb{R}$ and property $M(\mu_{0},E)$ holds, and that
$\mu\in\mathscr{P}(X)$ has Radon–Nikodym derivative

$\frac{\mathrm{d}\mu}{\mathrm{d}\mu_{0}}(x)\propto\exp\bigl{(}-\Phi(x)\bigr{)}$

for some locally uniformly continuous _potential_ $\Phi\colon X\to\mathbb{R}$.
Then $E$ is $\mu$-essentially total, and $I_{\mu}(x)\coloneqq
I_{\mu_{0}}(x)+\Phi(x)$ is an OM functional for $\mu$.

###### Proof.

  1. (a)

The Lebesgue differentiation theorem implies that for any $x\in X$,

$\lim_{r\to 0}\frac{\mu(B_{r}(x))}{\lambda^{n}(B_{r}(x))}=\rho(x).$

For any $x$ and $x^{\prime}\in E$, one can pick $r$ sufficiently small such
that $B_{r}(x)$ and $B_{r}(x^{\prime})$ lie in the open set $X$. This implies
that $\lambda^{n}(B_{r}(x))=\lambda^{n}(B_{r}(x^{\prime}))$, and so

$\lim_{r\to 0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(x^{\prime}))}=\lim_{r\to
0}\frac{\mu(B_{r}(x))}{\lambda^{n}(B_{r}(x))}\lim_{r\to
0}\frac{\lambda^{n}(B_{r}(x^{\prime}))}{\mu(B_{r}(x^{\prime}))}=\frac{\rho(x)}{\rho(x^{\prime})}.$
(5.9)

Hence, $E$ is a chain and $I_{\mu}$ is an OM functional on $E$. When
$x^{\prime}\in X\setminus E$, one can still apply the Lebesgue differentiation
theorem to obtain

$\lim_{r\to
0}\frac{\mu(B_{r}(x^{\prime}))}{\lambda^{n}(B_{r}(x^{\prime}))}=0,$

so an argument similar to that in (5.9) proves that $x^{\prime}\prec_{0}x$ for
any $x\in E$.

  2. (b)

The existence of an OM functional $I_{\mu}$ proves that $E$ is a chain. Using
the $M$-property and Ayanbayev et al. (2022a, Lemma B.1), for $x^{\prime}\in
X\setminus E$ and $x\in E$, we must have $x^{\prime}\prec_{0}x$, because

$\lim_{r\to 0}\frac{\mu(B_{r}(x^{\prime}))}{\mu(B_{r}(x))}=0.$

  3. (c)

By Ayanbayev et al. (2022a, Lemma B.8), $I_{\mu}$ is an OM functional for
$\mu$ and property $M(\mu,E)$ holds. The result follows by (b). ∎

######  Proposition 5.15.

Let $X$ be a metric space and let $\mu\in\mathscr{P}(X)$. Suppose that
$\varnothing\neq E\subseteq X$ is $\mu$-essentially total.

  1. (a)

Any $\preccurlyeq_{0}$-maximal element must lie in $E$ and is
$\preccurlyeq_{0}$-greatest.

  2. (b)

If $\mu$ admits an OM functional $I_{\mu}\colon E\to\mathbb{R}$, then

$x^{\star}\text{ is $\preccurlyeq_{0}$-greatest}\iff x^{\star}\in E\text{ and
}x^{\star}\text{ minimises }I_{\mu}.$

###### Proof.

  1. (a)

A maximal element $x^{\star}$ must lie in $E$, or else one could find $x\in E$
such that $x^{\star}\prec_{0}x$ by essential totality, contradicting the
maximality of $x^{\star}$. Conditions (a) and (b) of essential totality
together imply that $x^{\star}$ is globally comparable, so it must be greatest
(Lemma 5.3).

  2. (b)

Using the OM functional for $E$, one finds that

$\displaystyle x^{\star}\in E\text{ is an upper bound for }E$
$\displaystyle\iff\lim_{r\to
0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(x^{\star}))}\leqslant 1\text{ for all }x\in
E$ $\displaystyle\iff\frac{e^{-I_{\mu}(x)}}{e^{-I_{\mu}(x^{\star})}}\leqslant
1\text{ for all }x\in E$ $\displaystyle\iff x^{\star}\text{ minimises
}I_{\mu}.$

If $x^{\star}$ is $\preccurlyeq_{0}$-greatest, then $x^{\star}\in E$ by (a),
and the previous implications prove that $x^{\star}$ minimises $I_{\mu}$.
Conversely, the definition of essential totality ensures that an upper bound
for $E$ is $\preccurlyeq_{0}$-greatest, proving the reverse implication. ∎

The variational characterisation of weak modes as minimisers of the OM
functional generalises the result of Ayanbayev et al. (2022a, Proposition 4.1)
to essentially total subsets. Specialising to the case of a continuous
Lebesgue density on an open set (Proposition 5.14(a)) recovers the intuitive
result that $x^{\star}$ is a weak mode if and only if it is a global maximiser
of $\rho$. The situation is more subtle if $X$ is not open: the measure in
Example 5.4(a) restricted to $X=[0,1]$ has a continuous Lebesgue density
maximised at $x^{\star}=1$, but $x^{\star}$ is not a weak mode.

As a consequence of our result on reweightings of well-behaved measures
(Proposition 5.14(c)), we obtain the significant corollary that maximal
elements are always greatest when the measure is a Bayesian posterior as in
(3.1) arising from a Gaussian prior. This is highly reassuring from the
perspective of applications: pathological examples in the style of Theorem
5.11 with non-greatest maximal elements do not occur in Bayesian posteriors
for well-behaved inverse problems.

##  6 Closing remarks

This article has proposed that modes of probability measures should be
understood as greatest or maximal elements of preorders that are defined using
the masses of metric balls.

At fixed radius $r>0$, there is an obvious choice of total preorder, and the
order-theoretic point of view opens up attractive proof techniques for the
existence of maximal/greatest elements (radius-$r$ modes) (Theorem 4.6).
However, we have also seen that such radius-$r$ modes can fail to exist
(Examples 4.7 and 4.8), which provides further justification for the use of
asymptotic maximising families as proposed by Klebanov and Wacker (2022), and
we are able to contribute to the convergence analysis of such families as
$r\to 0$ (Theorems 4.11 and 4.12).

In the limit as $r\to 0$, there are several limiting preorders that one could
consider. The one on which we have focussed, whose greatest elements are weak
modes, is a non-total preorder. Indeed, we have shown that even absolutely
continuous measures can admit topologically dense antichains (Theorem 5.11),
indicating that a measure must satisfy stringent regularity conditions to be
certain of having greatest elements, i.e. weak modes.

As remarked in the introduction, we hope that this article will stimulate
further discussion in the community about the “correct” definition of a mode.
We argue that there is a tension between the order-theoretic desire for modes
to be merely _maximal_ elements of some preorder and an application-driven
desire for modes to be _greatest_ elements. To some extent, this tension can
be avoided if one works only with particularly nice measures that display no
oscillatory properties or that satisfy criteria such as essential totality,
thus keeping all pathologies away from the “top” of the preorder.

Further useful new definitions of modes may be introduced and one would hope
that they correspond to preorders. However, as explored in Appendix B, it may
well be that such definitions only induce non-transitive _relations_. In such
cases, the loss of transitivity is not necessarily fatal, so long as it is
kept away from the “top” of the relation, so that maximal/greatest elements
may be defined.

On a high level, it would be interesting to know whether or not there can
exist a function assigning to every (sufficiently well-behaved) measure
$\mu\in\mathscr{P}(X)$ a total preorder $\preccurlyeq^{\mu}$ whose maximal or
greatest elements are useful modes for $\mu$. This would appear to be a major
open question that will involve much further investigation.

##  Appendix A Technical supporting results

###  A.1 Radial cumulative distribution functions

######  Lemma A.1 (Properties of RCDFs).

Let $X$ be a metric space and let $\mu\in\mathscr{P}(X)$.

  1. (a)

For each $r>0$, $x\mapsto\mu(B_{r}(x))$ is upper semicontinuous.

  2. (b)

For each $x\in X$, $r\mapsto\mu(B_{r}(x))$ is monotonically increasing, is
continuous from the right, has limits from the left, and is upper
semicontinuous. Furthermore, $r\mapsto\mu(B_{r}(x))$ is differentiable
$\lambda^{1}$-a.e.

###### Proof.

For (a), fix $r>0$ and let $(x_{n})_{n\in\mathbb{N}}$ converge in $X$ to some
$x\in X$. Then

$\displaystyle\mu(B_{r}(x))$
$\displaystyle=\lim_{n\to\infty}\mu(B_{r+d(x,x_{n})}(x))$
$\displaystyle\text{since
}B_{r}(x)=\bigcap_{n\in\mathbb{N}}B_{r+d(x,x_{n})}(x)$
$\displaystyle=\limsup_{n\to\infty}\mu(B_{r+d(x,x_{n})}(x))$
$\displaystyle\geqslant\limsup_{n\to\infty}\mu(B_{r}(x_{n}))$
$\displaystyle\text{since $B_{r+d(x,x_{n})}(x)\supseteq B_{r}(x_{n})$}.$

For (b), monotonicity follows from the monotonicity of probability. To examine
continuity, fix $x\in X$ and let $(r_{n})_{n\in\mathbb{N}}$ be a convergent
sequence in $[0,\infty)$ with limit $r\geqslant 0$. If
$(r_{n})_{n\in\mathbb{N}}$ is decreasing, then
$\bigcap_{n\in\mathbb{N}}B_{r_{n}}(x)=B_{r}(x)$ and so the continuity of
probability along monotone sequences implies that
$\mu(B_{r_{n}}(x))\searrow\mu(B_{r}(x))$, which establishes continuity from
the right. If $(r_{n})_{n\in\mathbb{N}}$ is increasing, then
$\bigcup_{n\in\mathbb{N}}B_{r_{n}}(x)=\mathring{B}_{r}(x)$, and continuity of
probability implies that
$\mu(B_{r_{n}}(x))\nearrow\mu(\mathring{B}_{r}(x))\leqslant\mu(B_{r}(x))$, and
this establishes existence of a limit from the left. Now let $r_{n}\to r$, and
make no assumption that this convergence is monotone. By the above,

$\limsup_{n\to\infty}\mu(B_{r_{n}}(x))\in\\{\mu(\mathring{B}_{r}(x)),\mu(B_{r}(x))\\},$

i.e. the $\limsup$ is at most $\mu(B_{r}(x))$, which establishes upper
semicontinuity. Finally, a.e.-differentiability of $r\mapsto\mu(B_{r}(x))$
follows from monotonicity and Lebesgue’s theorem on differentiability of
monotone functions. ∎

######  Corollary A.2.

Let $X$ be a separable metric space, let $\mu\in\mathscr{P}(X)$, and fix
$r>0$. Then

$M_{r}\coloneqq\sup_{x\in X}\mu(B_{r}(x))>0$

and every sequence $(x_{n})_{n\in\mathbb{N}}$ such that $\mu(B_{r}(x_{n}))\to
M_{r}$ as $n\to\infty$ is bounded.

###### Proof.

The separability of $X$ implies that
$\operatorname*{supp}(\mu)\neq\varnothing$ (Aliprantis and Border, 2006,
Theorem 12.14), and so there must exist at least one $x\in X$ with
$\mu(B_{r}(x))>0$. Hence, $M_{r}>0$.

Now let $(x_{n})_{n\in\mathbb{N}}$ be any sequence such that
$\mu(B_{r}(x_{n}))\to M_{r}$ as $n\to\infty$. Then there must exist
$N\in\mathbb{N}$ such that

$n\geqslant N\implies\mu(B_{r}(x_{n}))\geqslant\nicefrac{{M_{r}}}{{2}}>0,$

i.e. $x_{n}$ eventually lies in $\\{x\in
X\mid\mu(B_{r}(x_{n}))\geqslant\nicefrac{{M_{r}}}{{2}}\\}$, which is a bounded
set by Lemma 4.2(c), and so $(x_{n})_{n\in\mathbb{N}}$ is a bounded sequence.
∎

######  Definition A.3.

Let $X$ be a metric space. A probability measure $\mu\in\mathscr{P}(X)$ will
be called _spherically non-atomic_ if every metric sphere has zero $\mu$-mass,
i.e., for all $r\geqslant 0$ and all $x\in X$,
$\mu(\mathring{B}_{r}(x))=\mu(B_{r}(x))$.

######  Corollary A.4 (RCDFs of spherically non-atomic measures).

Let $X$ be a metric space and assume that $\mu\in\mathscr{P}(X)$ is
spherically non-atomic.

  1. (a)

For each $r>0$, $x\mapsto\mu(B_{r}(x))$ is continuous.

  2. (b)

For each $x\in X$, $r\mapsto\mu(B_{r}(x))$ is monotonically increasing and
continuous.

  3. (c)

$(r,x)\mapsto\mu(B_{r}(x))$ is continuous.

  4. (d)

For each $x\in X$, $\mu(\\{x\\})=\mu(B_{0}(x))=\lim_{r\to 0}\mu(B_{r}(x))=0$.

###### Proof.

Easy modification of the proof of Lemma A.1 shows that

  * •

for each $r>0$, $x\mapsto\mu(\mathring{B}_{r}(x))$ is lower semicontinuous;

  * •

for each $x\in X$, $r\mapsto\mu(\mathring{B}_{r}(x))$ is monotonically
increasing, is continuous from the left, has limits from the right, and is
lower semicontinuous.

For a spherically non-atomic measure $\mu$, each occurrence of
$\mu(\mathring{B}_{r}(x))$ can be replaced with $\mu(B_{r}(x))$, and this
together with the original statement of Lemma A.1 proves parts (a) and (b).

An easy modification of the classical theorem of Young (1910) on the joint
continuity of separately continuous functions (see e.g. Grushka (2019, Theorem
3.1)) establishes (c).

Finally, (d) follows from
$\mu(\\{x\\})=\mu(B_{0}(x))=\mu(\mathring{B}_{0}(x))=\mu(\varnothing)=0$; the
claim regarding the limit follows from the continuity of
$r\mapsto\mu(B_{r}(x))$, as proven in (b). ∎

###  A.2 Radius-$r$ modes in sequence spaces

Given $p\in[1,\infty)$ and $\alpha\in\mathbb{R}^{\mathbb{N}}_{>0}$, we define
the corresponding weighted $\ell^{p}$ space and its norm by

$\ell^{p}_{\alpha}\coloneqq\left\\{x=(x_{n})_{n\in\mathbb{N}}\in\mathbb{R}^{\mathbb{N}}\,\middle|\,\lVert
x\rVert_{\ell^{p}_{\alpha}}\coloneqq\left(\sum_{n\in\mathbb{N}}\left|\frac{x_{n}}{\alpha_{n}}\right|^{p}\right)^{\nicefrac{{1}}{{p}}}<\infty\right\\}.$

We also equip $\mathbb{R}^{\mathbb{N}}$ and its subspaces with the finite-
dimensional projections

$P_{n}\colon\mathbb{R}^{\mathbb{N}}\to\mathbb{R}^{n},\quad
x=(x_{k})_{k\in\mathbb{N}}\mapsto(x_{1},\dots,x_{n}),$

and denote the ball of radius $r>0$ centred at $x\in\mathbb{R}^{n}$ by

$B_{r}^{n}(x)\coloneqq\left\\{y\in\mathbb{R}^{n}\,\middle|\,\left(\sum_{k=1}^{n}\frac{\lvert
y_{k}-x_{k}\rvert^{p}}{\alpha_{k}^{p}}\right)^{\nicefrac{{1}}{{p}}}\leqslant
r\right\\}.$

######  Lemma A.5.

Let $X=\ell^{p}_{\alpha}$ for some $p\in[1,\infty)$,
$\alpha\in\mathbb{R}^{\mathbb{N}}_{>0}$ and let $\mu\in\mathscr{P}(X)$. Define
the set function $\mu_{n}(A)\coloneqq(\mu\circ P_{n}^{-1})(P_{n}A)$ (This
function is not necessarily a measure.)

  1. (a)

For any $n\in\mathbb{N}$, $x\in X$, and $r>0$, the projection maps satisfy
$P_{n}B_{r}(x)=B_{r}^{n}(P_{n}x)$.

  2. (b)

For any $x\in X$ and $r>0$,

$\bigcap_{n\in\mathbb{N}}P_{n}^{-1}(P_{n}B_{r}(x))=B_{r}(x).$

  3. (c)

For any $n\in\mathbb{N}$ and $A\in\mathscr{B}(X)$, the projection maps satisfy
$P_{n+1}^{-1}\bigl{(}P_{n+1}A\bigr{)}\subseteq
P_{n}^{-1}\bigl{(}P_{n}A\bigr{)}$.

  4. (d)

For any $n\in\mathbb{N}$ and $A\in\mathscr{B}(X)$, the set functions $\mu_{n}$
satisfy $\mu_{n}(A)\geqslant\mu(A)$.

  5. (e)

For any $x\in X$ and $r>0$, one has
$\lim_{n\to\infty}\mu_{n}(B_{r}(x))=\mu(B_{r}(x))$.

###### Proof.

  1. (a)

Use that

$\displaystyle P_{n}B_{r}(x)$
$\displaystyle=\left\\{y\in\mathbb{R}^{n}\,\middle|\,\text{there exists
}\tilde{y}\in X\text{ such that
}\lVert\tilde{y}-x\rVert_{\ell^{p}_{\alpha}}\leqslant r\text{ and
}y=P_{n}\tilde{y}\right\\}$
$\displaystyle=\left\\{y\in\mathbb{R}^{n}\,\middle|\,\text{there exists
}\tilde{y}\in X\text{ such that
}\sum_{k=1}^{\infty}\frac{\lvert\tilde{y}_{k}-x_{k}\rvert^{p}}{\alpha_{k}^{p}}\leqslant
r^{p}\text{ and }y=P_{n}\tilde{y}\right\\}$
$\displaystyle=\left\\{y\in\mathbb{R}^{n}\,\middle|\,\sum_{k=1}^{n}\frac{\lvert
y_{k}-x_{k}\rvert^{p}}{\alpha_{k}^{p}}\leqslant
r^{p}\right\\}=B_{r}^{n}(P_{n}x).$

  2. (b)

Observe that, by (a),

$\displaystyle\bigcap_{n\in\mathbb{N}}P_{n}^{-1}(P_{n}B_{r}(x))$
$\displaystyle=\bigcap_{n\in\mathbb{N}}P_{n}^{-1}(B_{r}^{n}(P_{n}x))$
$\displaystyle=\left\\{y\in X\,\middle|\,P_{n}y\in B_{r}^{n}(P_{n}x)\text{ for
all }n\in\mathbb{N}\right\\}$ $\displaystyle=\left\\{y\in
X\,\middle|\,\sum_{k=1}^{n}\frac{\lvert
y_{k}-x_{k}\rvert^{p}}{\alpha_{k}^{p}}\leqslant r^{p}\text{ for all
}n\in\mathbb{N}\right\\}=B_{r}(x).$

  3. (c)

This is a straightforward consequence of the definitions.

  4. (d)

This follows from the inclusion $A\subseteq P_{n}^{-1}(P_{n}A)$ and
monotonicity of $\mu$.

  5. (e)

As $\bigl{(}P_{n}^{-1}(P_{n}B_{r}(x))\bigr{)}_{n\in\mathbb{N}}$ is a
decreasing sequence of sets, it follows that

$\lim_{n\to\infty}\mu_{n}(A)=\lim_{n\to\infty}\mu\bigl{(}P_{n}^{-1}(P_{n}B_{r}(x))\bigr{)}=\mu\left(\bigcap_{n\in\mathbb{N}}P_{n}^{-1}(P_{n}B_{r}(x))\right)=\mu(B_{r}(x))$

by continuity of measure. ∎

######  Lemma A.6 (Spherical non-atomicity and weak upper semicontinuity in
sequence spaces).

Let $X=\ell^{p}_{\alpha}$, $1\leqslant p<\infty$,
$\alpha\in\mathbb{R}^{\mathbb{N}}_{>0}$, and let $\mu\in\mathscr{P}(X)$.
Suppose that $\mu\circ P_{n}^{-1}\in\mathscr{P}(\mathbb{R}^{n})$ is
spherically non-atomic for each $n\in\mathbb{N}$. Then, for each fixed $r>0$,
the map $x\mapsto\mu(B_{r}(x))$ is weakly upper semicontinuous.

###### Proof.

Suppose that $x_{k}\rightharpoonup x^{\star}$ as $k\to\infty$, and let
$\mu_{n}(A)\coloneqq(\mu\circ P_{n}^{-1})(P_{n}A)$. As
$\mu_{n}(B_{r}(x^{\star}))\searrow\mu(B_{r}(x^{\star}))$ (Lemma A.5), it
follows that, for any $\varepsilon>0$, there exists $N\in\mathbb{N}$ such
that, for all $n\geqslant N$,
$\mu_{n}(B_{r}(x^{\star}))-\mu(B_{r}(x^{\star}))<\varepsilon$. Using this and
the inequality $\mu_{n}(B_{r}(x_{k}))\geqslant\mu(B_{r}(x_{k}))$ for any
$k\in\mathbb{N}$, we obtain

$\mu(B_{r}(x_{k}))-\mu(B_{r}(x^{\star}))\leqslant\mu_{n}(B_{r}(x_{k}))-\mu_{n}(B_{r}(x^{\star}))+\varepsilon.$
(A.1)

By hypothesis, $x_{k}\rightharpoonup x^{\star}$, so $P_{n}x_{k}\to
P_{n}x^{\star}$ as $k\to\infty$. As $\mu\circ P_{n}^{-1}$ is assumed to be
spherically non-atomic, $x\mapsto(\mu\circ P_{n}^{-1})(B_{r}^{n}(x))$ is
continuous (Corollary A.4). Hence,

$\displaystyle\lim_{k\to\infty}(\mu\circ P_{n}^{-1})(P_{n}B_{r}(x_{k}))$
$\displaystyle=\lim_{k\to\infty}(\mu\circ P_{n}^{-1})(B_{r}^{n}(P_{n}x_{k}))$
(Lemma A.5(a)) $\displaystyle=(\mu\circ
P_{n}^{-1})(B_{r}^{n}(P_{n}x^{\star}))$ (by continuity)
$\displaystyle=(\mu\circ P_{n}^{-1})(P_{n}B_{r}(x^{\star}))$ (Lemma A.5(a)).

Hence, $\lim_{k\to\infty}\mu_{n}(B_{r}(x_{k}))=\mu_{n}(B_{r}(x^{\star}))$.
Taking limits as $k\to\infty$ in (A.1) yields that

$\limsup_{k\to\infty}\mu(B_{r}(x_{k}))-\mu(B_{r}(x^{\star}))\leqslant\lim_{k\to\infty}\mu_{n}(B_{r}(x_{k}))-\mu_{n}(B_{r}(x^{\star}))+\varepsilon=\varepsilon.$

As $\varepsilon>0$ was arbitrary, this shows that $x\mapsto\mu(B_{r}(x))$ is
weakly upper semicontinuous. ∎

We now state an explicit version of Anderson’s inequality following the
inequalities of Dashti et al. (2013, Lemma 3.6) for Gaussian measures and
Agapiou et al. (2018, Lemma 6.2) for Besov measures with $p=1$.

Fix parameters888In the original setting of real analysis, $s$ and $d$ were
interpreted as smoothness and spatial dimension respectively, but for us only
the ratio $\nicefrac{{s}}{{d}}$ is important. $1\leqslant p<\infty$,
$s\in\mathbb{R}$, and $d\in\mathbb{N}$; the (_sequence space_) _Besov space_
$X_{p}^{s}$ is defined to be $\ell^{p}_{\gamma}$ for the weighting sequence
$\gamma_{k}\coloneqq k^{-(s/d+1/2)+1/p}$, and the (_sequence space_) _Besov
measure_ $B_{p}^{s}$ is defined to be the countable product measure
$\bigotimes_{k\in\mathbb{N}}\mu_{k}$, where
$\mu_{k}\in\mathscr{P}(\mathbb{R})$ has Lebesgue density proportional to
$\exp(-\lvert x_{k}/\gamma_{k}\rvert^{p})$. It is known that $B_{p}^{s}$
charges $X_{p}^{t}$ with full mass when $t=s-(1+\eta)\nicefrac{{d}}{{p}}$ and
$\eta>0$ (e.g. Lassas et al., 2009, Lemma 2).

######  Lemma A.7 (Explicit Anderson inequality for Besov-$p$ priors,
$1\leqslant p\leqslant 2$).

Let $s\in\mathbb{R}$, $d\in\mathbb{N}$, $\eta>0$ and let $t\coloneqq
s-(1+\eta)\nicefrac{{d}}{{p}}$. Suppose that $X=X_{p}^{t}$ and let
$\mu=B_{p}^{s}\in\mathscr{P}(X)$ be a sequence-space Besov measure. Then, for
any $0<r<\|x\|_{X_{p}^{t}}$ and $x\in X$,

$\frac{\mu(B_{r}(x))}{\mu(B_{r}(0))}\leqslant\exp\left(-\frac{1}{2}\left(\lVert
x\rVert_{X_{p}^{t}}-r\right)^{p}\right).$ (A.2)

###### Proof.

The space $X_{p}^{t}$ can be written as the sequence space $\ell^{p}_{\delta}$
with the weighting sequence
$\delta_{k}=k^{-(s/d+1/2)+(2+\eta)/p}>\gamma_{k}=k^{-(s/d+1/2)+1/p}$. The
formula for the unnormalised marginal density of the Besov measure then yields

$\displaystyle\frac{\mu_{n}(B_{r}(x))}{\mu_{n}(B_{r}(0))}$
$\displaystyle\quad=\frac{\int_{P_{n}B_{r}(x)}\exp\left(-\sum_{i=1}^{n}\lvert\nicefrac{{u_{i}}}{{\gamma_{i}}}\rvert^{p}\right)\,\mathrm{d}u}{\int_{P_{n}B_{r}(0)}\exp\left(-\sum_{i=1}^{n}\lvert\nicefrac{{u_{i}}}{{\gamma_{i}}}\rvert^{p}\right)\,\mathrm{d}u}$
$\displaystyle\quad\leqslant\frac{\sup_{y\in
P_{n}B_{r}(x)}\exp(-\frac{1}{2}\sum_{i=1}^{n}\lvert\nicefrac{{y_{i}}}{{\delta_{i}}}\rvert^{p})\int_{P_{n}B_{r}(x)}\exp\left(-\sum_{i=1}^{n}\lvert\nicefrac{{u_{i}}}{{\gamma_{i}}}\rvert^{p}+\frac{1}{2}\sum_{i=1}^{n}\lvert\nicefrac{{u_{i}}}{{\delta_{i}}}\rvert^{p}\right)\,\mathrm{d}u}{\int_{P_{n}B_{r}(0)}\exp\left(-\sum_{i=1}^{n}\lvert\nicefrac{{u_{i}}}{{\gamma_{i}}}\rvert^{p}+\frac{1}{2}\sum_{i=1}^{n}\lvert\nicefrac{{u_{i}}}{{\delta_{i}}}\rvert^{p}\right)\,\mathrm{d}u}$
$\displaystyle\quad\leqslant\sup_{y\in
P_{n}B_{r}(x)}\exp\left(-\frac{1}{2}\sum_{i=1}^{n}\left|\nicefrac{{y_{i}}}{{\delta_{i}}}\right|^{p}\right),$

where the ratio of integrals is bounded above by $1$ using Anderson’s
inequality (Anderson, 1955). Hence, as
$\lim_{n\to\infty}\mu_{n}(B_{r}(x))=\mu(B_{r}(x))$ (Lemma A.5),

$\frac{\mu(B_{r}(x))}{\mu(B_{r}(0))}=\lim_{n\to\infty}\frac{\mu_{n}(B_{r}(x))}{\mu_{n}(B_{r}(0))}\leqslant\lim_{n\to\infty}\sup_{y\in
P_{n}B_{r}(x)}\exp\left(-\frac{1}{2}\sum_{i=1}^{n}\left|\nicefrac{{y_{i}}}{{\delta_{i}}}\right|^{p}\right)=\exp\left(-\frac{1}{2}\left(\lVert
x\rVert_{X_{p}^{t}}-r\right)^{p}\right),$

which establishes (A.2). ∎

######  Theorem A.8 (Radius-$r$ modes for product measures on weighted
$\ell^{p}$ spaces).

Let $X=\ell_{\alpha}^{p}$, $1<p<\infty$,
$\alpha\in\mathbb{R}^{\mathbb{N}}_{>0}$. Let
$\mu_{0}=\bigotimes_{n\in\mathbb{N}}\mu_{n}\in\mathscr{P}(X)$ with each
$\mu_{n}\ll\lambda^{1}$ on $\mathbb{R}$. If $\mu\ll\mu_{0}$, then $\mu$ has a
radius-$r$ mode for any $r>0$.

###### Proof.

As $\mu_{0}$ is a product of the measures $\mu_{n}$, which are all absolutely
continuous with respect to $\lambda^{1}$, the pushforward measures
$\mu_{0}\circ P_{n}^{-1}$ are absolutely continuous with respect to
$\lambda^{n}$. As $\mu\ll\mu_{0}$, it follows that $\mu\circ
P_{n}^{-1}\ll\mu_{0}\circ P_{n}^{-1}$, so the pushforwards of $\mu$ are also
absolutely continuous with respect to $\lambda^{n}$. Hence, the measure $\mu$
has spherically non-atomic pushforwards $\mu\circ P_{n}^{-1}$, and so the map
$x\mapsto\mu(B_{r}(x))$ is weakly upper semicontinuous for any $r>0$ (Lemma
A.6). As any sequence $(x_{n})_{n\in\mathbb{N}}$ with
$\mu(B_{r}(x_{n}))\nearrow M_{r}$ is bounded (Corollary A.2), there must exist
a weakly convergent subsequence $(x_{n_{k}})_{k\in\mathbb{N}}\rightharpoonup
x^{\star}$ by the reflexivity of $\ell^{p}_{\alpha}$, $p>1$. The weak upper
semicontinuity of $x\mapsto\mu(B_{r}(x))$ implies that $x^{\star}$ is a
radius-$r$ mode, because
$M_{r}=\lim_{k\to\infty}\mu(B_{r}(x_{n_{k}}))\leqslant\mu(B_{r}(x^{\star}))$.
∎

######  Corollary A.9.

Suppose that $X=\ell^{p}_{\alpha}$, $1<p<\infty$,
$\alpha\in\mathbb{R}^{\mathbb{N}}_{>0}$. If $\mu\ll\mu_{0}$ and
$\mu_{0}=\bigotimes_{n\in\mathbb{N}}\mu_{n}$ is

  1. (a)

a Gaussian measure;

  2. (b)

a Besov measure; or

  3. (c)

a Cauchy measure,

then $\mu$ has a radius-$r$ mode for any $r>0$.

###  A.3 Small-ball probabilities for the countable dense antichain

The measure in Theorem 5.11 places variants of the prototype densities
$\rho_{k,m}$ at each dyadic rational. While a variety of constructions are
possible (see Remark 5.12), we choose to use the dyadic rationals in $[0,1]$
as the dense set for simplicity. The advantage of using the dyadic rationals
is that one can exploit the natural “level” structure, writing
$D_{\ell}\coloneqq\left\\{(2i-1)2^{-\ell}\,\middle|\,1\leqslant i\leqslant
2^{\ell-1}\right\\}$ for those dyadic rationals which, in their simplest form,
can be written as $c2^{-\ell}$. From this level structure, one can explicitly
compute the distance between terms and bound the support of the densities
$\rho_{k(\ell,i),m(\ell)}$ centred at points in $D_{\ell}$.

As the dyadic rationals are precisely the points in $[0,1]$ with a finite
binary expansion, the behaviour of the RCDF $\mu(B_{r}(x))$ at an arbitrary
point $x\in[0,1]$ depends on a quantity which we call the _dyadic
irrationality exponent_ , and denote $\beta_{2}(x)$, which can be thought of
as a quantitative estimate on the length of runs of $0$s or $1$s in the binary
expansion of $x$. This quantity is very much analogous to the number-theoretic
_irrationality measure_ $\varphi(x,n)\coloneqq\min_{1<p<q,q\leqslant
n}|x-\nicefrac{{p}}{{q}}|$ and corresponding _irrationality exponent_
$\beta(x)$ (Fel′dman and Nesterenko, 1998). We choose the notation $\beta(x)$
for the irrationality exponent and not the more usual $\mu(x)$ to avoid
confusion with the measure $\mu$.

######  Definition A.10.

  1. (a)

The _dyadic irrationality measure_ of $x\in[0,1]$ is given by
$\varphi_{2}(x,\ell)\coloneqq\min_{q\in\biguplus_{i=1}^{\ell}D_{i}}\lvert
x-q\rvert$.

  2. (b)

The _dyadic irrationality exponent_ of $x\in[0,1]\setminus D$ is given by

$\beta_{2}(x)\coloneqq\inf\left\\{\beta\geqslant
1\,\middle|\,\liminf_{\ell\to\infty}\frac{\varphi_{2}(x,\ell)}{2^{-\beta\ell}}>0\right\\}=\sup\left\\{\beta\geqslant
1\,\middle|\,\liminf_{\ell\to\infty}\frac{\varphi_{2}(x,\ell)}{2^{-\beta\ell}}<\infty\right\\}.$

The dyadic irrationality exponent $\beta_{2}(x)$ is well defined, and indeed

$\liminf_{\ell\to\infty}\frac{\varphi_{2}(x,\ell)}{2^{-\beta\ell}}=\begin{cases}0,&\beta<\beta_{2}(x),\\\
+\infty,&\beta>\beta_{2}(x).\end{cases}$ (A.3)

In general, it is not possible to say anything about the limit in (A.3) in the
critical case $\beta=\beta_{2}(x)$; the value could be anything in the range
$[0,+\infty]$. Furthermore, as
$\varphi(x,2^{\ell})\leqslant\varphi_{2}(x,\ell)$, it immediately follows that
$\beta_{2}(x)\leqslant\beta(x)$, but the quantities are not equal in general —
for example, any irrational number must satisfy $\beta(x)\geqslant 2$ by
Dirichlet’s approximation theorem, but one can construct irrational numbers
with $\beta_{2}(x)=1$.

######  Lemma A.11 (Properties of the measure in Theorem 5.11).

Let $\mu\in\mathscr{P}(\mathbb{R})$ be the measure in Theorem 5.11 and fix
$\ell\in\mathbb{N}$.

  1. (a)

Given $q_{\ell,i}\in D_{\ell}$, the support of the density
$\rho_{k(\ell,i),m(\ell)}(\hbox to5.71527pt{\hss$\cdot$\hss}-q_{\ell,i})$ is
contained in $B_{2^{-4\ell+3}}(q_{\ell,i})$.

  2. (b)

For distinct $q_{\ell,i},q_{\ell,i^{\prime}}\in D_{\ell}$, the densities
$\rho_{k(\ell,i),m(\ell)}(\hbox to5.71527pt{\hss$\cdot$\hss}-q_{\ell,i})$ and
$\rho_{k(\ell,i^{\prime}),m(\ell)}(\hbox
to5.71527pt{\hss$\cdot$\hss}-q_{\ell,i^{\prime}})$ have disjoint support, and
the supports are a distance at least $2^{-\ell}-2^{-4\ell+4}$ apart.

  3. (c)

Fix $\delta,r>0$ and $x\in[0,1]$, and suppose that $\inf_{q\in D_{\ell}}\lvert
x-q\rvert>\delta+r$. Then

$\sum_{i=1}^{2^{\ell-1}}\int_{x-r}^{x+r}\rho_{k(\ell,i),m(\ell)}(t-q_{\ell,i})\,\mathrm{d}t\leqslant
2r\delta^{-\nicefrac{{1}}{{2}}}.$

###### Proof.

  1. (a)

By construction, $\rho_{k(\ell,i),m(\ell)}$ has mass $m(\ell)=2^{-2\ell+1}$.
Hence, the truncation radius of this singularity is at most $2m(\ell)^{2}$
(Proposition 5.10(e)) and therefore the support is contained in a ball of
radius $2\times 2^{-4\ell+2}\leqslant 2^{-4\ell+3}$.

  2. (b)

Distinct points in $D_{\ell}$ must be a distance at least $2^{-\ell}$ apart,
and by (a) the supports of the densities $\rho_{k(\ell,i),m(\ell)}$ and
$\rho_{k(\ell,i^{\prime}),m(\ell)}$ are contained in a ball of radius
$2^{-4\ell+3}$. Hence, their supports must be at least a distance
$2^{-\ell}-2\times 2^{-4\ell+3}$ apart.

  3. (c)

By Proposition 5.10(f), outside of $B_{\delta}(q_{\ell,i})$, the density
$\rho_{k(\ell,i),m(\ell)}$ is bounded above by
$\delta^{-\nicefrac{{1}}{{2}}}$, and the supports of the densities are
disjoint, so the upper bound follows immediately. ∎

######  Lemma A.12 (Behaviour of RCDFs in Theorem 5.11).

Let $\mu\in\mathscr{P}(\mathbb{R})$ be the measure in Theorem 5.11.

  1. (a)

Suppose that $q_{\ell,i}\in D_{\ell}$. Then
$\mu(B_{r}(q_{\ell,i}))\sim\mu_{k(\ell,i),m(\ell)}(B_{r}(0))$ as $r\to 0$.

  2. (b)

Suppose that $x\in[0,1]\setminus D$ and that $\beta_{2}(x)<4$. Then, for any
$\beta\in(\beta_{2}(x),4)$, it follows that $\mu(B_{r}(x))\in
O(r^{\min\\{1,\nicefrac{{2}}{{\beta}}\\}})$ as $r\to 0$, and in particular
$\mu(B_{r}(x))\in o(r^{\nicefrac{{1}}{{2}}})$.

  3. (c)

Suppose that $x\in[0,1]\setminus D$. Then, for any $q\in D$,

$\liminf_{r\to 0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(q))}<1.$

  4. (d)

Suppose that $x\in[0,1]\setminus D$ and that $\beta_{2}(x)>4$. Then, for any
$q\in D$,

$\limsup_{r\to 0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(q))}>1,$

and therefore $x\mathrel{\|}_{0}q$.

###### Proof.

  1. (a)

For any $r<2^{-\ell}$, the ball $B_{r}(q_{\ell,i})$ does not contain any
element of $\biguplus_{i=1}^{\ell}D_{i}$ except $q_{\ell,i}$. Furthermore, for
$m\in\mathbb{N}$, if $r<2^{-m}-2^{-4m+3}$, then $B_{r}(q_{\ell,i})$ does not
intersect the support of any singularity centred at $q^{\prime}\in D_{m}$.
(Lemma A.11(a)).

As $2^{-m}-2^{-4m+3}\in\Omega(2^{-m})$ as $m\to\infty$, there exists
$M\in\mathbb{N}$ and $c>0$ such that

$2^{-m}-2^{-4m+3}\geqslant c2^{-m}\text{ for all $m\geqslant M$.}$

Picking $\ell_{1}(r)\coloneqq\lfloor-\log_{2}(\nicefrac{{r}}{{c}})\rfloor$, we
observe that $B_{r}(q_{\ell,i})$ is disjoint from the supports of any
singularities in $\biguplus_{i=M}^{\ell_{1}(r)}D_{i}$. Hence, we bound the
mass from the first $M$ levels using Lemma A.11(c), then note that there is no
contribution from levels $M,\dots,\ell_{1}(r)$, and finally bound the total
mass from level $\ell_{1}(r)+1$ onwards crudely. Fix
$\delta\coloneqq\inf_{q^{\prime}\in\biguplus_{i=1}^{M}D_{i}}|q_{\ell,i}-q^{\prime}|$
and suppose that $r\leqslant\nicefrac{{\delta}}{{2}}$; then

$\displaystyle\mu(B_{r}(q_{\ell,i}))$
$\displaystyle\leqslant\mu_{k(\ell,i),m(\ell)}(B_{r}(0))$
$\displaystyle\quad\quad+\sum_{i=1}^{M}\sum_{j=1}^{2^{i-1}}\int_{x-r}^{x+r}\rho_{k(i,j),m(i)}(t-q_{i,j})\,\mathrm{d}t+\sum_{i=\ell_{1}(r)+1}^{\infty}2^{-i}$
$\displaystyle\leqslant\mu_{k(\ell,i),m(\ell)}(B_{r}(0))+2M(\nicefrac{{\delta}}{{2}})^{-\nicefrac{{1}}{{2}}}r+\sum_{i=\ell_{1}(r)+1}^{\infty}2^{-i}$
(Lemma A.11(c)) $\displaystyle=\mu_{k(\ell,i),m(\ell)}(B_{r}(0))+O(r)\text{ as
$r\to 0$.}$

As $\mu_{k(\ell,i),m(\ell)}(B_{r}(0))\in\Theta(r^{\nicefrac{{1}}{{2}}})$ as
$r\to 0$ (Proposition 5.10(c)), the $O(r)$ term is negligible and hence
$\mu(B_{r}(q_{\ell,i}))\sim\mu_{k(\ell,i),m(\ell)}(B_{r}(0))$.

  2. (b)

Take $\beta\in(\beta_{2}(x),4)$; (A.3) implies that

$\liminf_{\ell\to\infty}\frac{\inf_{q\in\biguplus_{i=1}^{\ell}D_{i}}\lvert
x-q\rvert}{2^{-\beta\ell}}=\infty.$

Hence, for $r_{\ell}\coloneqq 2^{-4\ell+3}$, it follows that
$\inf_{q\in\biguplus_{i=1}^{\ell}D_{i}}\lvert x-q\rvert-
r_{\ell}\in\Omega(2^{-\beta\ell})$ as $\ell\to\infty$. Furthermore, as the
supports of the densities centred at distinct elements of $D_{\ell}$ are
disjoint and at least a distance $2^{-\ell}-2^{-4\ell+4}\in\Omega(2^{-\ell})$
apart (Lemma A.11(b)), there must exist $L\in\mathbb{N}$ and $c>0$ such that,
for all $\ell\geqslant L$,

$\displaystyle\inf_{q\in\biguplus_{i=1}^{\ell}D_{i}}\lvert x-q\rvert-r_{\ell}$
$\displaystyle>c2^{-\beta\ell}$ $\displaystyle 2^{-\ell}-2^{-4\ell+4}$
$\displaystyle>c2^{-\ell}.$

Defining
$\ell_{1}(r)\coloneqq\lfloor-\frac{1}{\beta}\log_{2}(\nicefrac{{r}}{{c}})\rfloor$
and $\ell_{2}(r)\coloneqq\lfloor-\log_{2}(\nicefrac{{r}}{{c}})\rfloor$, we see
that if $L\leqslant\ell\leqslant\ell_{1}(r)$, then $B_{r}(x)$ is disjoint from
the support of every density centred at a point of $D_{\ell}$, and if
$\ell_{1}(r)<\ell\leqslant\ell_{2}(r)$, then $B_{r}(x)$ intersects the support
of at most one density centred at a point in $D_{\ell}$. For
$\ell>\ell_{2}(r)$, it is sufficient to bound $\mu(B_{r}(x))$ by counting the
total mass added in the $\ell\textsuperscript{th}$ level.

Hence, let $\delta\coloneqq\inf_{q\in\biguplus_{i=1}^{L}D_{i}}\lvert
x-q\rvert>0$ and pick $r<\nicefrac{{\delta}}{{2}}$ so that we may bound the
mass from the first $L$ levels using Lemma A.11(c). Using this and the claims
above,

$\displaystyle\mu(B_{r}(x))$
$\displaystyle\leqslant\sum_{\ell=1}^{L}\sum_{i=1}^{2^{\ell-1}}\int_{x-r}^{x+r}\rho_{k(\ell,i),m(\ell)}(t-q_{\ell,i})\,\mathrm{d}t+\sum_{\ell=\ell_{1}(r)+1}^{\ell_{2}(r)}m(\ell)$
$\displaystyle\quad\quad+\sum_{\ell=\ell_{2}(r)+1}^{\infty}2^{-\ell}$
$\displaystyle\leqslant
2L(\nicefrac{{\delta}}{{2}})^{-\nicefrac{{1}}{{2}}}r+\sum_{\ell=\ell_{1}(r)+1}^{\ell_{2}(r)}2^{-2\ell+1}+2^{-\ell_{2}(r)}$
(Lemma A.11(c)) $\displaystyle\leqslant
2L(\nicefrac{{\delta}}{{2}})^{-\nicefrac{{1}}{{2}}}r+\frac{8}{3}2^{-2(\ell_{1}(r)+1)}+\frac{2r}{c}$
$\displaystyle\leqslant
2L(\nicefrac{{\delta}}{{2}})^{-\nicefrac{{1}}{{2}}}r+\frac{8}{3}\left(\frac{r}{c}\right)^{\nicefrac{{2}}{{\beta}}}+\frac{2r}{c}\in
O(r^{\min\\{1,\nicefrac{{2}}{{\beta}}\\}})\text{ as $r\to 0$.}$

  3. (c)

The case $\beta_{2}(x)<4$ follows from (b). Hence, without loss of generality,
suppose that $\beta_{2}(x)>1$ and pick $\beta\in(1,\beta_{2}(x))$; (A.3)
implies that

$\liminf_{\ell\to\infty}\frac{\inf_{q\in\biguplus_{i=1}^{\ell}D_{i}}\lvert
x-q\rvert}{2^{-\beta\ell}}=0.$

Hence, there must exist a sequence $(\ell_{k})_{k\in\mathbb{N}}\nearrow\infty$
and a sequence $(q_{\ell_{k}})_{k\in\mathbb{N}}$ with $q_{\ell_{k}}\in
D_{\ell_{k}}$ such that $|x-q_{\ell_{k}}|<2^{-\beta\ell_{k}-1}$. This implies
that any $q_{\ell_{k}}\neq q\in\biguplus_{i=1}^{\ell_{k}}D_{i}$ must satisfy
$\lvert x-q\rvert>2^{-\ell_{k}}-2^{-\beta\ell_{k}-1}$. As it suffices to bound
$\mu(B_{r}(x))$ at the radii $s_{k}\coloneqq 2^{-\beta\ell_{k}-1}\searrow 0$,
we proceed by bounding the mass contributed by the first $\ell_{k}$ levels by
the total mass from the density centred at $q_{\ell_{k}}$ plus a $\Theta(r)$
term given by Lemma A.11(c).

For $\ell>\ell_{k}$, by a similar argument to that used above, any $q\in
D_{\ell}$ satisfies $\lvert x-q\rvert>2^{-\ell}-2^{-\beta\ell_{k}-1}$. As the
density centred at $q$ is truncated at a radius at most $r_{\ell}\coloneqq
2^{-4\ell+3}$, and $2^{-\ell}-2^{-4\ell+3}\in\Omega(2^{-\ell})$ as
$\ell\to\infty$, there must exist $L\in\mathbb{N}$ and $c\in(0,1)$ such that
for $L\leqslant\ell\leqslant\beta\ell_{k}$,

$\lvert x-q\rvert-
r_{\ell}>2^{-\ell}-2^{-\beta\ell_{k}-1}-2^{-4\ell+3}>c2^{-\ell}-2^{-\beta\ell_{k}-1}.$

So, $B_{s_{k}}(x)$ does not intersect the support of any density centred at a
point of $D_{\ell}$ if $s_{k}<c2^{-\ell}-2^{-\beta\ell_{k}-1}$; hence, if
$L\leqslant\ell<\beta\ell_{k}+\log_{2}(c)$, then $B_{s_{k}}(x)$ does not
intersect the support of any density in the $\ell\textsuperscript{th}$ level.

Combining these two claims and taking $k$ large enough that $\ell_{k}\geqslant
L+1$ yields the bound

$\displaystyle\mu(B_{s_{k}}(x))$
$\displaystyle\leqslant\sum_{\ell=1}^{\ell_{k}-1}\sum_{i=1}^{2^{\ell-1}}\int_{x-r}^{x+r}\rho_{k(\ell,i),m(\ell)}(t-q_{\ell,i})\,\mathrm{d}t$
$\displaystyle\quad\quad+\mu(B_{s_{k}}(q_{\ell_{k}}))+\sum_{\ell=\lfloor\beta\ell_{k}+\log_{2}(c)\rfloor}^{\infty}2^{-\ell}$
$\displaystyle\leqslant
2\ell_{k}\left(2^{-\ell_{k}}-2^{-\beta\ell_{k}}\right)^{-\nicefrac{{1}}{{2}}}s_{k}+\mu(B_{s_{k}}(q_{\ell_{k}}))+\frac{8s_{k}}{c}$
(Lemma A.11(c)) $\displaystyle\leqslant
2\ell_{k}\left((2s_{k})^{\nicefrac{{1}}{{\beta}}}-2s_{k}\right)^{-\nicefrac{{1}}{{2}}}s_{k}+\mu(B_{s_{k}}(q_{\ell_{k}}))+\frac{8s_{k}}{c}.$

As
$(2s_{k})^{\nicefrac{{1}}{{\beta}}}-2s_{k}\in\Omega(s_{k}^{\nicefrac{{1}}{{\beta}}})$
as $k\to\infty$, we may pick $k$ sufficiently large that
$(2s_{k})^{\nicefrac{{1}}{{\beta}}}-2s_{k}\geqslant
Cs_{k}^{\nicefrac{{1}}{{\beta}}}$ for some $C>0$. Hence, using that
$\ell_{k}=-\frac{1}{\beta}\left(\log_{2}(s_{k})-1\right)$,

$\displaystyle\mu(B_{s_{k}}(x))$
$\displaystyle\leqslant\mu(B_{s_{k}}(q_{\ell_{k}}))+2\ell_{k}(Cs_{k}^{\nicefrac{{1}}{{\beta}}})^{-\frac{1}{2}}s_{k}+\frac{8s_{k}}{c}$
$\displaystyle\leqslant\mu(B_{s_{k}}(q_{\ell_{k}}))-\frac{2C^{-\nicefrac{{1}}{{2}}}}{\beta}\bigl{(}\log_{2}(s_{k})-1\bigr{)}s_{k}^{1-\nicefrac{{1}}{{2\beta}}}+\frac{8s_{k}}{c}$
$\displaystyle=\mu(B_{s_{k}}(q_{\ell_{k}}))+o(s_{k}^{\nicefrac{{1}}{{2}}})\text{
as $k\to\infty$.}$

The claim follows because

$\liminf_{r\to
0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(q))}\leqslant\liminf_{k\to\infty}\frac{\mu(B_{s_{k}}(q_{\ell_{k}}))}{\mu(B_{s_{k}}(q))}<1,$

and the RCDFs at distinct dyadic rationals are chosen so that their ratio
oscillates on either side of unity.

  4. (d)

Take $\beta\in(4,\beta_{2}(x))$. Then, by (A.3), there exists a sequence of
levels $(\ell_{k})_{k\in\mathbb{N}}\nearrow\infty$ and a sequence
$(q_{\ell_{k}})_{k\in\mathbb{N}}$ with $q_{\ell_{k}}=q_{\ell_{k},i_{k}}\in
D_{\ell_{k}}$ with $|x-q_{\ell_{k}}|<c2^{-\beta\ell_{k}}$. Ignoring the
contribution from densities centred at points other than $q_{\ell_{k}}$, we
observe that

$\mu(B_{r}(x))\geqslant\int_{x-r}^{x+r}\rho_{k(\ell_{k},i_{k}),m(\ell_{k})}(t-q_{\ell_{k}})\,\mathrm{d}t.$

Either $x>q_{\ell_{k}}$ or $x<q_{\ell_{k}}$; we deal with the first case as
the second is almost identical. Fix $s_{k}\coloneqq 2^{-4\ell_{k}+3}$. By
translating the density, we see that

$\displaystyle\mu(B_{s_{k}}(x))$
$\displaystyle\geqslant\int_{-s_{k}+(x-q_{\ell_{k}})}^{s_{k}}\rho_{k(\ell_{k},i_{k}),m(\ell_{k})}(t)\,\mathrm{d}t$
$\displaystyle=\int_{-s_{k}}^{s_{k}}\rho_{k(\ell_{k},i_{k}),m(\ell_{k})}(t)\,\mathrm{d}t-\int_{-s_{k}}^{-s_{k}+(x-q_{\ell_{k}})}\rho_{k(\ell_{k},i_{k}),m(\ell_{k})}(t)\,\mathrm{d}t.$

Indeed, as the density $\rho_{k(\ell_{k},i_{k}),m(\ell_{k})}$ is truncated at
a radius at most $2^{-4\ell+3}$ (Proposition 5.10(e)), and as
$|x-q_{\ell_{k}}|\in o(s_{k})$, one sees that the ball mass around $x$
asymptotically approaches the ball mass around the approximant $q_{\ell_{k}}$.
By a similar argument to (c),

$\limsup_{r\to
0}\frac{\mu(B_{r}(x))}{\mu(B_{r}(q))}\geqslant\limsup_{k\to\infty}\frac{\mu(B_{s_{k}}(q_{\ell_{k}}))}{\mu(B_{s_{k}}(q))}>1,$

i.e. $x\not\preccurlyeq_{0}q$, because the ratio of RCDFs at two distinct
dyadic rationals oscillates on either side of one. The claim on
incomparability then follows because $q\not\preccurlyeq_{0}x$ (Lemma A.12(c)).
∎

##  Appendix B Alternative small-radius preorders

This section briefly outlines some alternatives to Definition 5.1 of
$\preccurlyeq_{0}$ and their shortcomings.

The main difficulty that one encounters with alternative definitions is that
the corresponding relation may not be transitive. We claim that transitivity
is an essential property for any small-radius relation: without transitivity,
it is not meaningful to talk about maximal and greatest elements, and so the
characterisation of modes as greatest elements of an order fails.

Of course, for a small-radius preorder to be relevant to us, its greatest
elements must have some natural interpretation as “points of maximum
probability”. In some sense, determining what characterises a point of maximum
probability is the main challenge, but, motivated by the examples considered
throughout the paper, we believe that none of the alternative small-radius