disposition pdfshort=PDF, long=probability density function gs-
modeshort=gs:-mode, long=generalised strong mode pgs-modeshort=pgs:-mode,
long=partial generalised strong mode gps-modeshort=gps:-mode, long=generalised
partial strong mode ps-modeshort=ps:-mode, long=partial strong mode
s-modeshort=s:-mode, long=strong mode gw-modeshort=gw-mode, long=generalised
weak mode pgw-modeshort=pgw-mode, long=partial generalised weak mode pw-
modeshort=pw-mode, long=partial weak mode w-modeshort=w-mode, long=weak mode
wp-modeshort=wp-mode, long=weak partial mode wg-modeshort=wg-mode, long=weak
generalised mode gwp-modeshort=gwp-mode, long=generalised weak partial mode
gpw-modeshort=gpw-mode, long=generalised partial weak mode pwg-modeshort=pwg-
mode, long=partial weak generalised mode wpg-modeshort=wpg-mode, long=weak
partial generalised mode wgp-modeshort=wgp-mode, long=weak generalised partial
mode wap-modeshort=wap-mode, long=weak approximating partial mode gwa-
modeshort=gwa-mode, long=generalised weak approximating mode wga-
modeshort=wga-mode, long=weak generalised approximating mode wag-
modeshort=wag-mode, long=weak approximating generalised mode pgwa-
modeshort=pgwa-mode, long=partial generalised weak approximating mode gwpa-
modeshort=gwpa-mode, long=generalised weak partial approximating mode pwga-
modeshort=pwga-mode, long=partial weak generalised approximating mode gwap-
modeshort=gwap-mode, long=generalised weak approximating partial mode wpga-
modeshort=wpga-mode, long=weak partial generalised approximating mode wgpa-
modeshort=wgpa-mode, long=weak generalised partial approximating mode pwag-
modeshort=pwag-mode, long=partial weak approximating generalised mode wgap-
modeshort=wgap-mode, long=weak generalised approximating partial mode wpag-
modeshort=wpag-mode, long=weak partial approximating generalised mode wapg-
modeshort=wapg-mode, long=weak approximating partial generalised mode

###### 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

###### A ‘periodic table’ of modes and
maximum a posteriori estimators

###### Abstract

Abstract. The last decade has seen many attempts to generalise the definition
of modes, or MAP estimators, of a probability distribution $\mu$ on a space
$X$ to the case that $\mu$ has no continuous Lebesgue density, and in
particular to infinite-dimensional Banach and Hilbert spaces $X$. This paper
examines the properties of and connections among these definitions. We
construct a systematic taxonomy — or ‘periodic table’ — of modes that includes
the established notions as well as large hitherto-unexplored classes. We
establish implications between these definitions and provide counterexamples
to distinguish them. We also distinguish those definitions that are merely
‘grammatically correct’ from those that are ‘meaningful’ in the sense of
satisfying certain ‘common-sense’ axioms for a mode, among them the correct
handling of discrete measures and those with continuous Lebesgue densities.
However, despite there being 17 such ‘meaningful’ definitions of mode, we show
that none of them satisfy the ‘merging property’, under which the modes of
$\mu|_{A}$, $\mu|_{B}$ and $\mu|_{A\cup B}$ enjoy a straightforward
relationship for well-separated positive-mass events $A,B\subseteq X$.

Keywords. Bayesian inverse problems $\bullet$ maximum a posteriori estimation
$\bullet$ generalised modes $\bullet$ weak modes $\bullet$ local behaviour of
measures

2020 Mathematics Subject Classification. 28C15 $\bullet$ 60B05 $\bullet$ 62F10
$\bullet$ 62F15 $\bullet$ 62R20

FUBFreie Universität Berlin, Arnimallee 6, 14195 Berlin, Germany ()
WarwickMathematics Institute and School of Engineering, The University of
Warwick, Coventry, CV4 7AL, United Kingdom () TuringAlan Turing Institute, 96
Euston Road, London, NW1 2DB, United Kingdom

##  1 Introduction

Modes, or maximum a posteriori (MAP) estimators, of a probability measure
$\mu$ constitute important and widely-used point estimators in Bayesian
statistics, since they are ‘points of maximal probability’ and are the objects
calculated by many regularised optimisation-based approaches to inverse
problems — at least heuristically. Such points also arise as ‘most likely’
minimum-action transition paths for random dynamical systems. If $\mu$ has a
continuous Lebesgue pdf, then modes can be unambiguously defined as maximisers
of this pdf (when they exist). The generalisation to measures with no such
pdf, and in particular to measures on Hilbert or Banach spaces, is non-trivial
and ambiguous. Most of the definitions suggested over the last decade
incorporate the idea of a mode as a maximiser $u\in X$ of the ‘small ball
probability’ $\mu(B_{r}(u))$ in the limit as $r\to 0$, where $X$ is (at least)
a separable metric space equipped with its Borel $\sigma$-algebra
$\mathcal{B}(X)$ and $B_{r}(u)$ denotes the open ball of radius $r>0$ centred
at $u\in X$.

To briefly summarise, and reserving precise definitions until Section 3,
_strong modes_ were introduced by Dashti et al. (2013) in the context of
Bayesian inverse problems, and they compare $\mu(B_{r}(u))$ to the overall
supremum $M_{r}^{\mu}\coloneqq\sup_{x\in X}\mu(B_{r}(x))$ in the limit as
$r\to 0$. In contrast, the _weak mode_ 111While we work with global weak modes
introduced by Ayanbayev et al. (2022, Definition 3.7) rather than $E$-weak
modes Helin and Burger (2015, Definition 4), most of the results transfer
similarly to $E$-weak modes. In particular, this is true for all
counterexamples when setting $E\coloneqq X$. of Helin and Burger (2015)
involves pairwise comparisons between $\mu(B_{r}(u))$ and $\mu(B_{r}(v))$ for
$u,v\in X$ in the limit as $r\to 0$. The _generalised strong mode_ of Clason
et al. (2019) deals with discontinuities of the pdf and with compactly-
supported measures by comparing $\mu(B_{r_{n}}(u_{n}))$ to $M_{r_{n}}^{\mu}$
along any null sequence $(r_{n})_{n\in\mathbb{N}}$ and some approximating
sequence $(u_{n})_{n\in\mathbb{N}}$ with limit $u$. One starting point for
this article is to investigate what happens when ‘along any null sequence’ is
replaced by ‘along some null sequence’; this change leads to a new class of
modes that we term _partial modes_. It turns out that this new notion is
inferior to none of the existing mode definitions (at least, under the set of
axioms that we study in this paper), and, in fact, it is surprisingly hard to
construct examples (cf. Example 7.5) that distinguish between partial and non-
partial modes.

strong mode ‘s:-mode’ of Dashti et al. (2013) generalised strong mode
‘gs:-mode’ of Clason et al. (2019) global weak mode ‘w-mode’ of Ayanbayev et
al. (2022) $E$-weak mode
---
of Helin and Burger (2015)
$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$(Clason
et al., 2019, Ex.
2.2)$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$(Lie
and Sullivan, 2018, Ex. 4.4)

Figure 1.1: Established relations among various notions of modes. The
abbreviated terminology in quotation marks is new to this paper. Unlabelled
implications follow immediately from the relevant definitions. The colour
scheme used here and throughout the paper will be explained in Section 6.2 and
can be safely ignored upon first reading.

The connections, i.e. implications and counterexamples, among the above-
mentioned modes have partially been established and are summarised in Figure
1.1. However, all these types of modes are unsatisfactory in the following
sense: It can happen that $\mu|_{A}$ and $\mu|_{B}$ each have a mode, while
$\mu|_{A\cup B}$ has none, where $A,B\in\mathcal{B}(X)$ are positive-mass sets
with positive distance between them and $\mu|_{C}$ denotes the
restriction/conditional measure of $\mu$ conditional upon $C$. Intuitively,
one would hope that either the modes within $A$ dominate those within $B$, or
vice versa, or that all those modes are equally dominant. In any case, we
expect at least one of the modes of $\mu|_{A}$ and $\mu|_{B}$ to be a mode of
$\mu|_{A\cup B}$. We call this desirable property of modes the _merging
property_ (MP) and define it rigorously in Section 5.4. The reason why weak
modes do not satisfy the MP is that the binary relation $\succcurlyeq_{0}$ on
$X$ defined222We will allow the denominator in (1.1) and similar expressions
to be zero and we set $\nicefrac{{c}}{{0}}\coloneqq\infty$ for $c>0$ and
$\nicefrac{{0}}{{0}}\coloneqq 1$; we comment on the consequences of the choice
$\nicefrac{{0}}{{0}}\coloneqq 1$, rather than $\nicefrac{{0}}{{0}}\coloneqq
0$, in Appendix C.  by

$u\succcurlyeq_{0}v\iff\liminf_{r\to
0}\frac{\mu(B_{r}(u))}{\mu(B_{r}(v))}\geqslant 1,\qquad u,v\in X,$ (1.1)

is a _non-total_ preorder. While the greatest elements under
$\succcurlyeq_{0}$ are precisely the weak modes (and note that points outside
the support of $\mu$ can never be modes), it is also possible for points $u$
and $u^{\prime}$ to be incomparable maximal elements under $\succcurlyeq_{0}$,
satisfying, for example,

$\begin{array}[]{c}\text{for all $z\in X\setminus\\{u,u^{\prime}\\}$,
}u\succcurlyeq_{0}z\text{ and }u^{\prime}\succcurlyeq_{0}z,\\\ \text{but
}u\not\succcurlyeq_{0}u^{\prime}\text{ and
}u^{\prime}\not\succcurlyeq_{0}u.\end{array}$ (1.2)

Example 7.5 gives an explicit example of this situation; the same example
refutes the MP for strong and generalised modes. This order-theoretic
viewpoint on modes has been studied in detail in the recent work of Lambley
and Sullivan (2023), but it does not provide a remedy regarding the MP. The
novel notion of partial modes mentioned above gives hope for solving the MP
issue, since Example 7.5 does not refute the MP for these types of modes.
Unfortunately, the more sophisticated Example 7.7 shows that partial modes
also fail to satisfy the MP. Thus, one significant challenge left open by the
present work is the formulation of a definition of modes satisfying the MP.

While this paper does not solve the MP problem, it does discuss in detail the
relations among all those notions of modes (and also their combinations) and
provides a large number of insightful counterexamples that fill several gaps
in our understanding of the differences among various definitions of modes and
their properties, and hopefully will shape the future development of the
theory of modes.

#### Contributions and outline.

Section 2 establishes the general notation used in the paper.

Section 3 recalls more precisely the notions of strong, weak, and generalised
modes from the existing literature, and introduces the concepts of _partial
modes_ , which are dominant in terms of small ball probabilities only along
_some_ null sequence $(r_{n})_{n\in\mathbb{N}}$, and weak _approximating
modes_ , for which the comparison point $v\in X$ is also endowed with an
approximating sequence $v_{n}\to v$. This section also studies some of the
ways in which the elementary concepts of strong/weak, generalised, partial,
and approximating modes can be combined and the relations among them.

Building upon Section 3, Section 4 sets out a general structured definition
that results in an exhaustive list of 282 possible mode definitions based on
dominance of small-ball probabilities, of which many are logically equivalent,
leaving 144 possibilities. Since even this is clearly excessive, in Sections 5
and 6 we set out to keep only those definitions that are ‘meaningful’ rather
than merely ‘grammatically correct’; we introduce several ‘common-sense’
axioms that any reasonable definition of modes should satisfy. After
dismissing those definitions that violate at least one of these axioms (except
for the MP; see below) we end up with a list of 30 meaningful mode definitions
(Table 6.1). We study in detail the properties of and causal relations among
these 30 mode definitions, and the resulting classification into 17
equivalence classes333Under Conjecture 5.8, three additional equivalences hold
and the 17 equivalence classes reduce to 14. Our counterexamples show that no
further equivalences are possible. To avoid repeated awkward turns of phrase,
we shall usually simply write that there are 17 distinct meaningful
definitions of a mode. is illustrated in Figure 6.1. We stress that there seem
to be no objective axiomatic reasons to prefer one of these definitions over
another, even though some of them might seem somewhat exotic.

One of the axioms introduced in Section 5 is the _merging property_ (MP),
which states that, if $\mu|_{A}$ and $\mu|_{B}$ each have a mode for positive-
mass sets $A,B\in\mathcal{B}(X)$ with positive distance between them, then, at
the very least, $\mu|_{A\cup B}$ must also have a mode; see the discussion in
Section 5.4. However, _none_ of the remaining 17 mode definitions satisfies
MP. In our opinion, this situation is rather unsatisfactory and calls for
further research into the definition of modes.

We stress the large number of illuminating counterexamples provided in Section
7, which will hopefully shape the future development of the theory.
Surprisingly, all of these examples could be constructed in $X=\mathbb{R}$
with $\mu$ possessing a (possibly discontinuous) pdf. Some readers may wish to
read Section 7 immediately after Section 3, before proceeding with the rest of
the paper.

Section 8 gives some closing remarks.

Appendix A contains some supporting technical results; Appendix B discusses
possible additional axioms for modes that could be topics for future research;
Appendix C elaborates on the consequences of our convention
$\nicefrac{{0}}{{0}}=1$ instead of $\nicefrac{{0}}{{0}}=0$; and Appendix D
gives a complete enumeration of all mode types considered in the paper.

##  2 Setup and notation

Throughout this paper, we consider a separable metric space $(X,d)$ equipped
with its Borel $\sigma$-algebra $\mathcal{B}(X)$. In the case
$X=\mathbb{R}^{m}$ for $m\in\mathbb{N}$, we work with the Euclidean metric
unless stated otherwise. We write $B_{r}(u)\coloneqq\\{x\in X\mid d(x,u)<r\\}$
for the open ball with centre $u\in X$ and radius $r>0$, and
$\textup{dist}(A,B)\coloneqq\inf\\{d(a,b)\mid a\in A,\,b\in B\\}$ denotes the
distance between $A,B\subseteq X$.

We write $\mathcal{P}(X)$ for the set of all probability measures $\mu$
defined on $\mathcal{B}(X)$. The topological support of $\mu\in\mathcal{P}(X)$
is

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

For $r>0$, we write $M_{r}^{\mu}\coloneqq\sup_{x\in X}\mu(B_{r}(x))\in(0,1]$
and, for $u,v\in X$, we define the ratios

$\displaystyle\,\mathfrak{R}_{r}^{\mu}(u,v)$
$\displaystyle\coloneqq\frac{\mu(B_{r}(u))}{\mu(B_{r}(v))},$ (2.1)
$\displaystyle\,\mathfrak{R}_{r}^{\mu}(u,\sup)$
$\displaystyle\coloneqq\frac{\mu(B_{r}(u))}{M_{r}^{\mu}}.$ (2.2)

Note that, since $X$ is separable, $\operatorname{supp}(\mu)\neq\varnothing$
and $M_{r}^{\mu}>0$ for all $r>0$ (Aliprantis and Border, 2006, Theorem
12.14). As noted in Footnote 2, we will allow the denominator in (2.1) to
attain the value zero and follow the conventions
$\nicefrac{{c}}{{0}}\coloneqq\infty$ for $c>0$ and
$\nicefrac{{0}}{{0}}\coloneqq 1$; see Appendix C for further discussion of
this choice. The restriction of $\mu$ to $C\in\mathcal{B}(X)$ with $\mu(C)>0$
is $\mu|_{C}\in\mathcal{P}(X)$ defined by

$\mu|_{C}(E)\coloneqq\frac{\mu(C\cap E)}{\mu(C)},\qquad E\in\mathcal{B}(X).$

We write $\chi_{A}$ for the indicator function of a subset $A\subseteq X$,
i.e.

$\chi_{A}(x)\coloneqq\begin{cases}1&\text{if $x\in A$,}\\\ 0&\text{if $x\notin
A$.}\end{cases}$

In this manuscript, all radii $r,r_{n}$ etc. are assumed to be strictly
positive, and when we write $r\to 0$ or similar only positive null sequences
are meant.

##  3 Strong, weak, generalised, and partial modes and their relations

As mentioned in Section 1, in the ‘small balls’ approach to mode theory, all
notions of mode characterise a mode $u\in X$ as being a ‘dominant’ point in
terms of its small ball probability $\mu(B_{r}(u))$ in some limit as $r\to 0$.
This section makes this idea precise using the terms _strong_ , _weak_ , and
_generalised_ that are already well-established in the literature. This
section also introduces a new term, _partial_ , and begins the study of modes
that are naturally described by combinations of these four basic adjectives,
e.g. partial strong modes — and we stress at the outset that _these adjectives
do not generally commute_. The investigations of this section will open the
door to a combinatorially large system of mode definitions, one which will be
introduced and analysed in a systematic way in Sections 4, 5, and 6.

###  3.1 Elementary strong and weak modes

Let us first summarise four basic ideas:

  1. (i)

A _strong_ mode $u\in X$ dominates the supremum over all small ball
probabilities, i.e. we compare $\mu(B_{r}(u))$ with $M_{r}^{\mu}$ as $r\to 0$.
This concept can be traced to Dashti et al. (2013).

  2. (ii)

In contrast, a _weak_ mode $u\in X$ dominates every other comparison point
$v\in X$ separately, i.e. we compare $\mu(B_{r}(u))$ with $\mu(B_{r}(v))$ as
$r\to 0$. This concept can be traced to Helin and Burger (2015).

  3. (iii)

A _partial_ mode $u\in X$ allows for $u$ to be dominant only along a certain
null sequence $(r_{n})_{n\in\mathbb{N}}$. This concept is new to this paper.

  4. (iv)

A _generalised_ mode $u\in X$ allows for dominance along certain approximating
sequences $(u_{n})_{n\in\mathbb{N}}$ of $u$ with $u_{n}\to u$, i.e. we
consider balls centred at $u_{n}$ rather than $u$. Such an approximating
sequence has to exist either for every null sequence
$(r_{n})_{n\in\mathbb{N}}$, or, in the case of partial generalised modes, just
for one specific null sequence, cf. (iii). Regarding approximation of the mode
$u$, this concept can be traced to Clason et al. (2019); however, in the
context of weak modes, we will expand the concept to approximation of
comparison points $v$ as well.

In what follows, whenever we refer to an approximating sequence
($\mathsf{as}$) of $u\in X$, we simply mean a convergent sequence
$(u_{n})_{n\in\mathbb{N}}$ in $X$ with $\lim_{n\to\infty}u_{n}=u$, and a
comparison point ($\mathsf{cp}$) simply means a point $v\in X$ (and we include
the possibility $v=u$).

The above ideas result in the following eight definitions, subdivided into
four strong and four weak versions of modes:

######  Definition 3.1 (Strong modes).

Let $\mu\in\mathcal{P}(X)$. A point $u\in X$ is called a

  1. (a)

s-mode if, for every null sequence $(r_{n})_{n\in\mathbb{N}}$,
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u,\sup)\geqslant 1$ (in
other words, if $\liminf_{r\to 0}\,\mathfrak{R}_{r}^{\mu}(u,\sup)\geqslant
1$);

  2. (b)

ps-mode if there exists a null sequence $(r_{n})_{n\in\mathbb{N}}$ such that
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u,\sup)\geqslant 1$ (in
other words, if $\limsup_{r\to 0}\,\mathfrak{R}_{r}^{\mu}(u,\sup)\geqslant
1$);

  3. (c)

gs-mode if, for every null sequence $(r_{n})_{n\in\mathbb{N}}$, there exists
an approximating sequence $(u_{n})_{n\in\mathbb{N}}$ of $u$ such that
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},\sup)\geqslant 1$;

  4. (d)

pgs-mode if there exist a null sequence $(r_{n})_{n\in\mathbb{N}}$ and an
$\mathsf{as}$ $(u_{n})_{n\in\mathbb{N}}$ of $u$ such that
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},\sup)\geqslant 1$.

######  Remark 3.2.

Since $\mu(B_{r}(u))\leqslant M_{r}^{\mu}$ for any $u\in X$ and $r>0$ by
definition, we can equivalently replace all inequalities of the form
$\geqslant 1$ in Definition 3.1 by equalities of the form $=1$ and each
‘$\liminf$’ by ‘$\lim$’.

######  Remark 3.3.

The definitions of s-mode and ps-mode differ only in the replacement of a
$\forall$-quantifier by an $\exists$-quantifier. While this seems like a major
change, the difference between the two notions of mode turns out to be rather
subtle, and it is surprisingly hard to construct ps-mode that are not s-mode;
see Example 7.5.

######  Definition 3.4 (Weak modes).

Let $\mu\in\mathcal{P}(X)$. A point $u\in X$ is called a

  1. (a)

w-mode if, for any null sequence $(r_{n})_{n\in\mathbb{N}}$ and any comparison
point $v\in X$,
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u,v)\geqslant 1$ (i.e., if
$\liminf_{r\to 0}\,\mathfrak{R}_{r}^{\mu}(u,v)\geqslant 1$ for every $v\in
X$);

  2. (b)

pw-mode if there exists a null sequence $(r_{n})_{n\in\mathbb{N}}$ such that,
for any $\mathsf{cp}$ $v\in X$,
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u,v)\geqslant 1$;

  3. (c)

gw-mode if, for any null sequence $(r_{n})_{n\in\mathbb{N}}$, there exists an
$\mathsf{as}$ $(u_{n})_{n\in\mathbb{N}}$ of $u$, such that, for any
$\mathsf{cp}$ $v\in X$,
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v)\geqslant 1$;

  4. (d)

pgw-mode if there exist a null sequence $(r_{n})_{n\in\mathbb{N}}$ and an
$\mathsf{as}$ $(u_{n})_{n\in\mathbb{N}}$ of $u$, such that, for any
$\mathsf{cp}$ $v\in X$,
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v)\geqslant 1$.

s: ps: gs: pgs: w pw gw
---
pgw
---
non-generalised $\scriptstyle\leftrightarrow$ generalisedpartial
$\scriptstyle\leftrightarrow$ non-
partial$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.1$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.5$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.2$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.2$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.5weak $\scriptstyle\leftrightarrow$
strong$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.1$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.5$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.2$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.1$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.2$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.1$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.5

Figure 3.1: Relations among the various definitions of modes introduced in
Definitions 3.1 and 3.4. Those implications that hold, do so by definition,
while just three counterexamples (Examples 7.1, 7.2, and 7.5) suffice to
disprove the twelve reverse implications; see Theorem 3.6. Note also that gw-
and pgw-modes will turn out to be ‘unsatisfactory’ in view of Proposition
5.4(a).

######  Remark 3.5.

We have replaced ‘$\lim_{r\to 0}$’ in the original definition of weak modes
(Helin and Burger, 2015, Definition 4) by ‘$\liminf_{r\to 0}$’. As already
pointed out by Ayanbayev et al. (2022, Remark 3.8) and Lambley and Sullivan
(2023, Footnote 3), this change means the limit is not implicitly and
incorrectly assumed to exist, even when
$\,\mathfrak{R}_{r}^{\mu}(u,v)\geqslant 1$ for all $r$ and $v$; see Example
7.6 for a counterexample. Under the ‘$\liminf$’ definition, every
$\textswab{s:}$-mode is a (global) $\textswab{w}$-mode, which is clearly
desirable.

Again, the difference between (all versions of) w-modes and (the corresponding
versions of) s-modes is rather subtle. Helin and Burger (2015, Section 7)
posed the question of whether there even exist w-mode that are not s-mode or
whether these two concepts are equivalent. This question was answered by Lie
and Sullivan (2018), who gave an example (Lie and Sullivan, 2018, Example 4.4)
of a w-mode that was not an s-mode as well as sufficient criteria under which
the two concepts coincide. We give a simpler example of a measure with a
w-mode but no pgs-mode (and hence no s-mode, ps-mode, or gs-mode) in Example
7.1, which is a one-dimensional probability measure with a Lebesgue pdf.444The
measure $\mu_{0}$ from Example 7.10 also satisfies this property. This example
of weak but not strong modes shows that, when constructing counterexamples, we
do not have to cover all possible counterexamples separately, but a well-
constructed example can distinguish between all notions of strong and weak
modes simultaneously (Example 7.1), another between all notions of partial and
non-partial modes (Example 7.5), and yet another between all notions of
generalised and non-generalised modes (Example 7.2). Thus, only three
counterexamples, summarised in the theorem below, suffice to disprove the
twelve implications in Figure 3.1, which illustrates the relations among the
first eight notions of modes introduced above.

######  Theorem 3.6.

  1. (a)

There exists an absolutely continuous measure on $X=\mathbb{R}$ with a w-mode
that is not a pgs-mode.

  2. (b)

There exists an absolutely continuous measure on $X=\mathbb{R}$ with a gs-mode
that is not a pw-mode.

  3. (c)

There exists an absolutely continuous measure on $X=\mathbb{R}$ with a ps-mode
that is not a gw-mode.

###### Proof.

Example 7.1 shows (a), Example 7.2 shows (b), while Example 7.5 shows (c). ∎

###  3.2 Further weak modes: Approximating modes

Definitions 3.1 and 3.4 introduced a variety of modes based on the terms ‘s: =
strong’, ‘g = generalised = there exists an approximating sequence $(u_{n})$
of $u$’, ‘w = weak = for each comparison point $v$’, and ‘p = partial = there
exists a null sequence $(r_{n})$’ (with the ‘default’ initial quantifier ‘for
each null sequence $(r_{n})$’ for non-partial modes), which gave them their
names. In the process, we made at least two important choices that are quite
ambiguous:

  1. (i)

When defining gw-mode and pgw-mode (i.e. modes that are generalised and weak),
we could alternatively allow the comparison point $v$ to have its own
approximating sequence $(v_{n})_{n\in\mathbb{N}}$, and then compare the two
approximating sequences $(u_{n})_{n\in\mathbb{N}}$ and
$(v_{n})_{n\in\mathbb{N}}$ via $\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v_{n})$, in
place of $\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v)$. As we will see in Section 5
and Proposition 5.4 in particular, such definitions turn out to be more
meaningful than the ones we stated in Definition 3.4, however, they are _not_
more general than w-mode and pw-mode, respectively, which is why the term
‘generalised’ would be very misleading (that was the main reason for us to
state the definitions in their current form). We will introduce the
alternative version below under the less misleading name of _approximating
modes_ with the shorthand ‘a = approximating = for each approximating sequence
$(v_{n})$ of $v$’.

  2. (ii)

While for modes that are strong the order of the quantifiers does not matter,
it is crucial for modes that are weak — which is why we paid attention to the
order of the words ‘generalised’, ‘partial’ and ‘weak’ in Definition 3.4 — and
we could go on and define wp-modes, wg-modes, gwp-modes, pwg-modes and wpg-
modes and study their relations. However, as touched upon in (i), none of
these notions (except for wp-modes) are meaningful since they do not satisfy
even the most basic desiderata; see Section 5. For that reason, we will focus
on the alternative notion discussed in (i) for the rest of this section.

The additional quantifier a (which, naturally, can only appear for weak modes
and after the quantifier w) opens up a whole zoo of new mode definitions. We
will state three of them below for the reader to get the gist of how such
definitions work, as they are rather self-explanatory. For the full list of
definitions we refer the reader to Sections 4 and 6.

######  Definition 3.7 (Some further weak modes).

Let $\mu\in\mathcal{P}(X)$. A point $u\in X$ is called a

  1. (a)

wp-mode if, for any $\mathsf{cp}$ $v\in X$, there exists a null sequence
$(r_{n})_{n\in\mathbb{N}}$ such that
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u,v)\geqslant 1$ (i.e., for
any $v\in X$, $\limsup_{r\to 0}\,\mathfrak{R}_{r}^{\mu}(u,v)\geqslant 1$);

  2. (b)

gwa-mode if, for any null sequence $(r_{n})_{n\in\mathbb{N}}$, there exists an
$\mathsf{as}$ $(u_{n})_{n\in\mathbb{N}}$ of $u$ such that, for any
$\mathsf{cp}$ $v\in X$ and any approximating sequence
$(v_{n})_{n\in\mathbb{N}}$ of $v$,
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v_{n})\geqslant 1$;

  3. (c)

wpag-mode if, for any $\mathsf{cp}$ $v\in X$, there exists a null sequence
$(r_{n})_{n\in\mathbb{N}}$ such that, for any approximating sequence
$(v_{n})_{n\in\mathbb{N}}$ of $v$, there exists an $\mathsf{as}$
$(u_{n})_{n\in\mathbb{N}}$ of $u$ such that
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v_{n})\geqslant 1$.

Note that the quantifiers ‘g = generalised = _there exists_ an approximating
sequence $(u_{n})$ of $u$’ and ‘w = weak = _for each_ comparison point $v$’
normally would not commute. However, we have the following result:

######  Proposition 3.8.

Let $\mu\in\mathcal{P}(X)$. Then $u\in X$ is a

  1. (a)

gwa-mode if and only if $u$ is a wga-mode;

  2. (b)

pgwa-mode if and only if $u$ is a pwga-mode.

###### Proof.

It is clear that every gwa-mode is a wga-mode. For the other direction, let
$u\in X$ be a wga-mode and $(r_{n})_{n\in\mathbb{N}}$ an arbitrary null
sequence. Since $u$ is a wga-mode, there exists, for $v=u$, an $\mathsf{as}$
$(u_{n})_{n\in\mathbb{N}}$ of $u$, such that, for any $\mathsf{as}$
$(v_{n})_{n\in\mathbb{N}}$ of $v$, $n\in\mathbb{N}$,
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v_{n})\geqslant 1$. We
will now show that this $\mathsf{as}$ $(u_{n})_{n\in\mathbb{N}}$ of $u$
dominates any other $\mathsf{as}$ $(w_{n})_{n\in\mathbb{N}}$ of any other
$w\in X$, i.e.
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},w_{n})\geqslant 1$,
proving that $u$ is a gwa-mode.

For this purpose, let $w\in X$ be arbitrary and $(w_{n})_{n\in\mathbb{N}}$ be
any $\mathsf{as}$ of $w$. Since $u$ is a wga-mode, there exists an
$\mathsf{as}$ $(\tilde{u}_{n})_{n\in\mathbb{N}}$ of $u$ (which may depend on
$w$), such that
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(\tilde{u}_{n},w_{n})\geqslant
1$. But then, by the choice of $(u_{n})_{n\in\mathbb{N}}$ above and setting
$v=u$, $v_{n}\coloneqq\tilde{u}_{n}$, we obtain

$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},w_{n})\geqslant\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v_{n})\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(\tilde{u}_{n},w_{n})\geqslant
1,$

proving (a). The proof of (b) goes similarly, except that ‘for every null
sequence’ is replaced by ‘there exists a null sequence’. ∎

w pw gwa wp pgwa = gpwa wga gwpa pwga wag gwap wpga = wgpa pwag wgap wpag wapg
= wagp
---
$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.10(a)$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.10(a)$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.5Proposition
3.8(a)$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.2$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.8$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.10(a)Proposition
3.8(b)$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.2$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.5$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.8$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.2$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.5$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.10(a)$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.9$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.8$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.5$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.9$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.10(a)$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.8$\scriptstyle{\definecolor[named]{.}{rgb}{1,0,0}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}/}$Example
7.10(a)

Figure 3.2: Relations among all definitions of modes that are weak and result
from different orders of the quantifiers w, g, p, and a, where for clarity of
illustration we have omitted definitions that are generalised (g), but not
approximating (a) (since they are not meaningful; see above as well as Section
5). As mentioned above, modes with the quantifiers g and a are _not_
generalisations of their counterparts without these quantifiers by Example
7.10(a) (gwa-modes do not generalise w-modes, pgwa-modes do not generalise pw-
modes, and gwpa-modes do not generalise wp-modes). The references to Example
7.5 use the existence of a ps:-mode that is not a wag-mode and the fact that
ps:-modes are pgwa-modes. All unlabelled implications that hold, do so by
definition. Note also that wapg- and wagp-modes will turn out to be
‘meaningless’ in view of Proposition 5.4(d).

Figures 3.1 and 3.2 make clear that any comprehensive classification of mode
types will not be linearly ordered in strength, and so it would be fruitless
to rely on terms such as ‘strong’, ‘weak’, ‘ultraweak’ etc. that suggest such
a simple linear structure. The quantifiers s:, w, g, p, and a introduced in
this section go some way towards providing a rich enough language to describe
all ‘meaningful’ notions of mode. In the next section, albeit with some
notational overhead, we take a fully systematic approach; inter alia, this
system will produce a new notion of mode strictly intermediate between strong
and weak modes and that cannot be expressed using the quantifiers s:, w, g, p,
and a.

##  4 Structured definitions of modes

We can now introduce a general framework that encompasses all the modes
discussed hitherto: the existing notions of strong, weak, and generalised
modes, as well as the new notion of partial modes, and their combinations. Our
taxonomic approach carries some notational overhead but offers a systematic
viewpoint on different notions of modes and which variations upon the
definitions actually produce ‘meaningful’ objects of study.

We call the resulting system (Table 6.1 and Figure 6.1) of meaningful mode
types a ‘periodic table’ — by analogy with Mendeleev’s periodic table of the
elements — for two reasons: first, it provides a structured descriptive
classification of modes; secondly, while we do not explore this large table in
its entirety, our ‘periodic table’ offers a roadmap for further research into
modes, much as the periodic table of the elements was used to predict the
existence and properties of elements before their isolation/synthesis.

Note that the definitions of modes introduced in Section 3, which stemmed from
four different conceptual ideas, involved several choices and not all possible
quantifiers and orders thereof have been considered. This section aims to give
a comprehensive list of all possible definitions stemming from these concepts.
For this purpose, we introduce the following notation:

######  Notation 4.1.

Let $X$ be a separable metric space and let $u\in X$. We will use the
following abbreviations:

  * •

$\mathsf{ns}$: positive null sequence $(r_{n})_{n\in\mathbb{N}}$, $r_{n}\to 0$
as $n\to\infty$;

  * •

$\mathsf{as}$: approximating sequence $(u_{n})_{n\in\mathbb{N}}$ of $u$,
$u_{n}\to u$ as $n\to\infty$;

  * •

$\mathsf{cp}$: comparison point $v\in X$;

  * •

$\mathsf{cs}$: comparison sequence, i.e. an approximating sequence
$(v_{n})_{n\in\mathbb{N}}$ of the $\mathsf{cp}$ $v$, $v_{n}\to v$ as
$n\to\infty$.

This notation allows us to introduce a vast number of self-explanatory mode
definitions in a structured way:

######  Definition 4.2.

Let $X$ be a separable metric space. Using Notation 4.1, we call $u\in X$ a
$[Q_{1}\ldots Q_{K}]$-mode of $\mu\in\mathcal{P}(X)$ with quantifiers

$Q_{k}\in\\{\forall\mathsf{ns},\exists\mathsf{ns},\forall\mathsf{cp},\exists\mathsf{cp},\forall\mathsf{as},\exists\mathsf{as},\forall\mathsf{cs},\exists\mathsf{cs}\\},\qquad
k=1,\dots,K,\qquad K\in\\{1,\dots,4\\},$

if

$Q_{1}\ldots
Q_{K}\colon\qquad\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(\star_{1},\star_{2})\geqslant
1,$ (4.1)

where

  1. (a)

$\star_{1}=\begin{cases}u_{n}&\text{if
}Q_{k}\in\\{\forall\mathsf{as},\exists\mathsf{as}\\}\text{ for some
$k=1,\dots,K$},\\\ u&\text{otherwise;}\end{cases}$

  2. (b)

$\star_{2}=\begin{cases}v_{n}&\text{if
}Q_{k}\in\\{\forall\mathsf{cs},\exists\mathsf{cs}\\}\text{ for some
$k=1,\dots,K$},\\\ \sup&\text{if
}Q_{k}\notin\\{\forall\mathsf{cp},\exists\mathsf{cp}\\}\text{ for each
$k=1,\dots,K$},\\\ v&\text{otherwise;}\end{cases}$

  3. (c)

$Q_{k}\in\\{\forall\mathsf{ns},\exists\mathsf{ns}\\}$ for some $k=1,\dots,K$
(i.e. $\mathsf{ns}$ has to appear);

  4. (d)

$Q_{k}\in\\{\forall\mathsf{cs},\exists\mathsf{cs}\\}$ for some $k=1,\dots,K$
implies $Q_{j}\in\\{\forall\mathsf{cp},\exists\mathsf{cp}\\}$ for some
$j\leqslant k$ (i.e. $\mathsf{cs}$ can only appear if $\mathsf{cp}$ appears
beforehand); and

  5. (e)

each of the objects $\mathsf{ns},\mathsf{cp},\mathsf{as},\mathsf{cs}$ appears
at most once, i.e.

     * •

$Q_{k}\in\\{\forall\mathsf{ns},\exists\mathsf{ns}\\}$ for some $k=1,\dots,K$
implies $Q_{j}\notin\\{\forall\mathsf{ns},\exists\mathsf{ns}\\}$ for each
$j\neq k$;

     * •

$Q_{k}\in\\{\forall\mathsf{cp},\exists\mathsf{cp}\\}$ for some $k=1,\dots,K$
implies $Q_{j}\notin\\{\forall\mathsf{cp},\exists\mathsf{cp}\\}$ for each
$j\neq k$;

     * •

$Q_{k}\in\\{\forall\mathsf{as},\exists\mathsf{as}\\}$ for some $k=1,\dots,K$
implies $Q_{j}\notin\\{\forall\mathsf{as},\exists\mathsf{as}\\}$ for each
$j\neq k$;

     * •

$Q_{k}\in\\{\forall\mathsf{cs},\exists\mathsf{cs}\\}$ for some $k=1,\dots,K$
implies $Q_{j}\notin\\{\forall\mathsf{cs},\exists\mathsf{cs}\\}$ for each
$j\neq k$.

We denote the corresponding mode map by $\mathfrak{M}[Q_{1}\ldots
Q_{K}]\colon\mathcal{P}(X)\to 2^{X}$:

$\mathfrak{M}[Q_{1}\ldots Q_{K}](\mu)\coloneqq\\{x\in X\mid x\text{ is a
$[Q_{1}\ldots Q_{K}]$-mode of }\mu\\}.$

######  Example 4.3.

A gs-mode is just a $[\forall\mathsf{ns}\exists\mathsf{as}]$-mode, while pgw-
modes are $[\exists\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cp}]$-modes.
All the other mode definitions in Section 3 can be expressed similarly, cf.
Table 6.1.

Definition 4.2 has been constructed so that it encompasses all, and precisely
all, those combinations of quantifiers $Q_{k}$ that contain no ‘grammatical
errors’ (such as $\mathsf{cs}$ before $\mathsf{cp}$, cf. (d)) and define a
mode to be some quantified limiting maximiser of small ball probabilities in
the sense of (4.1) (cf. (c)). In fact, there are 282 such ‘grammatically
correct’ definitions, and for the sake of completeness they are listed in
Appendix D.

However, many of these 282 definitions are logically equivalent due to
commutation of quantifiers,: for example, w-mode are both
$[\forall\mathsf{ns}\forall\mathsf{cp}]$-modes and
$[\forall\mathsf{cp}\forall\mathsf{ns}]$-modes. Such equivalences yield 144
logically distinct and grammatically correct mode definitions. Furthermore, it
seems quite clear that some of these 144 notions of mode are not ‘meaningful’
or ‘useful’. For example, it seems pointless to allow a point $u$ that
dominates merely one specific comparison point to qualify as a mode; hence, it
would seem reasonable to discard to discard definitions containing the
quantifier $\exists\mathsf{cp}$.

To take this programme further, and to rest the decision about which
definitions to keep and which to discard on a firm theoretical foundation
rather than mere heuristics, in the next sections we take an axiomatic
approach, defining natural and fundamental properties that ‘meaningful’ modes
ought to satisfy. These properties will then exclude a large number of mode
definitions from Definition 4.2, leaving us with the 30 definitions listed in
Table 6.1. Then, the non-trivial equivalences established analytically in
Propositions 3.8 and 6.1 will yield the 17 meaningful notions of a mode
illustrated in Figure 6.1.

######  Remark 4.4.

Our process of _excluding_ mode definitions rests on counterexamples for
$X=\mathbb{R}$, and we reason that any mode definition that fails in this
setting deserves to be discarded in general, even though it might satisfy the
axioms for much smaller spaces, e.g. $X=\mathbb{N}$ or even the extreme case
$X=\\{0\\}$. The _verification_ of axioms, on the other hand, must and will be
performed in full generality.

##  5 Meaningful axioms for modes and their consequences

As mentioned at the end of Section 4, not all ‘grammatically correct’ notions
of modes in Definition 4.2 are actually ‘meaningful’ in the sense of capturing
our intuitions of how modes ‘ought’ to behave. To clarify this restriction in
a mathematically rigorous way, we now define some formal axioms for modes.
Some possible additional axioms are discussed in Appendix B.

In the following, we first write down the axioms and then discuss their
motivation and their consequences. The Lebesgue property (LP), the atomic
property (AP), the self-dominating property (SP), and the cloning property
(CP) are ‘almost obvious’ criteria that a mode must satisfy. On the other
hand, the merging properties MP and rMP are more subtle and require some
discussion in Section 5.4. Indeed, none of the mode definitions considered in
the literature actually satisfies the merging properties even for
$X=\mathbb{R}$; cf. Proposition 5.11 below. Therefore, in our opinion, the
search for a reasonable definition of modes is far from over (which is one of
the main motivations for this paper). On these grounds, we only use the
properties LP, AP, SP, and CP to disqualify definitions of modes, yielding the
30 definitions listed in Table 6.1.

######  Definition 5.1.

Let $X$ be a separable metric space. We call any map
$\mathfrak{M}\colon\mathcal{P}(X)\to 2^{X}$ a mode map. We say that it
satisfies

  * (LP)

the Lebesgue property if $\mathfrak{M}(\mu)=\operatorname*{arg\,max}_{x\in
X}\rho(x)$ for any probability measure $\mu$ on $X=\mathbb{R}^{m}$ with
continuous Lebesgue pdf555Note that continuous pdfs $\rho$ on $\mathbb{R}^{m}$
can still be unbounded even in dimension $m=1$, cf. $\rho_{\beta}$ in Example
7.1, in which case $\mathfrak{M}(\mu)=\operatorname*{arg\,max}_{x\in
X}\rho(x)=\varnothing$. $\rho$;

  * (AP)

the atomic property if, for any $K\in\mathbb{N}$, any non-atomic measure
$\mu_{0}\in\mathcal{P}(X)$, any pairwise distinct $x_{1},\dots,x_{K}\in X$,
and any $\alpha_{0},\dots,\alpha_{K}\in[0,1]$ with
$\sum_{k=0}^{K}\alpha_{k}=1$,

$\textstyle\mathfrak{M}\big{(}\alpha_{0}\mu_{0}+\sum_{k=1}^{K}\alpha_{k}\delta_{x_{k}}\big{)}=\bigl{\\{}x_{k}\mid
k=1,\dots,K\text{ with
}\alpha_{k}=\max_{\tilde{k}=1,\dots,K}\alpha_{\tilde{k}}\bigr{\\}};$

  * (SP)

the self-dominating property if $\mathfrak{M}$ does not change if we replace
the definition of $\mathsf{cp}$ by ‘comparison point $v\neq u$’;

  * (CP)

the cloning property if $X$ is a normed space and, for any
$\mu\in\mathcal{P}(X)$, any $\alpha\in[0,1]$ and any $b\in X$ such that
$\textup{dist}(\operatorname{supp}(\mu),b+\operatorname{supp}(\mu))>0$,

$\mathfrak{M}(\alpha\mu+(1-\alpha)\mu(\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\displaystyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\textstyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\scriptstyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\scriptscriptstyle\bullet\;$}}}}}-b))=\begin{cases}\mathfrak{M}(\mu)&\text{if
}\alpha>\tfrac{1}{2},\\\ \mathfrak{M}(\mu)+b&\text{if }\alpha<\tfrac{1}{2},\\\
\mathfrak{M}(\mu)\cup(\mathfrak{M}(\mu)+b)&\text{if
}\alpha=\tfrac{1}{2}.\end{cases}$

As mentioned above, we consider the properties LP, AP, SP, and CP to be
natural criteria for definitions of modes, with the following reasoning:

  * (LP)

This property stems from the initial motivation for the entire theory, i.e.
generalising the term _mode_ from maximisers of continuous Lebesgue densities
of measures on $\mathbb{R}^{d}$ to more general spaces and measures.

  * (AP)

For probability measures $\mu\in\mathcal{P}(X)$ containing atoms, precisely
those atoms with the largest weight should be called modes.

  * (SP)

Every ‘potential’ mode should, at the very least, dominate itself. For certain
types of mode definitions (cf. Proposition 5.3) it can happen that a point
dominates every other point but fails to dominate itself. We argue that such
definitions should be dismissed altogether.

  * (CP)

If a probability measure on a normed space $X$ consists of two well-separated
and weighted copies of some other distribution $\mu$, then the copy with the
larger weight should inherit the modes of $\mu$ (and, in the case of equal
weights, the modes of both copies should be inherited).

###  5.1 Mode definitions that violate AP, SP, and CP

We will now analyse which notions of modes in Definition 4.2 violate AP, SP,
or CP for $X=\mathbb{R}$ (cf. Remark 4.4). We will consider such modes to be
‘not meaningful’ and exclude them from Table 6.1.

######  Proposition 5.2.

All mode maps $\mathfrak{M}[Q_{1}\ldots Q_{K}]$ of the following forms violate
AP for $X=\mathbb{R}$:

  1. (a)

$[\ldots\exists\mathsf{cp}\ldots]$; |  |
---|---|---

  2. (b)

$[\ldots\exists\mathsf{ns}\ldots\forall\mathsf{as}\ldots]$, |  $[\ldots\forall\mathsf{ns}\ldots\forall\mathsf{as}\ldots]$, |  $[\ldots\forall\mathsf{as}\ldots\forall\mathsf{ns}\ldots]$.
---|---|---

In other words, $[Q_{1}\ldots Q_{K}]$-modes violate AP for $X=\mathbb{R}$
whenever ($Q_{k}=\exists\mathsf{cp}$ for some $k$) or
($Q_{i}=\exists\mathsf{ns}$, $Q_{j}=\forall\mathsf{as}$ for some $i<j$) or
($Q_{i}=\forall\mathsf{ns}$, $Q_{j}=\forall\mathsf{as}$ for some $i,j$).

###### Proof.

We use the notation from Definition 4.2. Consider the discrete measure
$\mu=\tfrac{2}{3}\delta_{0}+\tfrac{1}{3}\delta_{1}$ on $X=\mathbb{R}$. If AP
holds, then $u=0$ is the only mode of $\mu$. For
$[\ldots\exists\mathsf{cp}\ldots]$-modes, $u=1$ would be a mode, violating AP
and proving (a): Since $v\coloneqq 2\notin\operatorname{supp}(\mu)$,
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(\star_{1},\star_{2})\geqslant
1$, no matter the order or type of quantifiers and whether we consider
approximating sequences of $u$ and/or $v$ or not.

In the following, we only consider mode types in (b) that are _not_ of the
form in (a), i.e. $Q_{k}\neq\exists\mathsf{cp}$ for each $k=1,\dots,K$.

Case 1: $Q_{k}\neq\exists\mathsf{cs}$ for each $k=1,\dots,K$.

Then, for each $\mathsf{ns}$ $(r_{n})_{n\in\mathbb{N}}$, the $\mathsf{as}$
$(u_{n})_{n\in\mathbb{N}}$ of the mode $u=0$ can always be chosen such that
$\mu(B_{r_{n}}(u_{n}))=0$ for each $n\in\mathbb{N}$ (e.g. $u_{n}=-2r_{n}$). On
the other hand, the denominator in the ratio
$\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},\star_{2})$ can be chosen to be non-zero,
which is clear in the case of $\star_{2}=\sup$ as well as for $\star_{2}=v=1$
and the (constant) $\mathsf{cs}$ $\star_{2}=v_{n}=1$. Therefore,
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},\star_{2})=0$ and
$u=0$ fails to be a
$[\ldots\exists\mathsf{ns}\ldots\forall\mathsf{as}\ldots]$-mode or a
$[\ldots\forall\mathsf{ns}\ldots\forall\mathsf{as}\ldots]$-mode, proving that
modes of these types violate AP. A similar argument holds for
$[\ldots\forall\mathsf{as}\ldots\forall\mathsf{ns}\ldots]$-modes since, for
the $\mathsf{as}$ $u_{n}=-2n^{-1}$ of $u=0$, the $\mathsf{ns}$ $r_{n}=n^{-1}$
guarantees $\mu(B_{r_{n}}(u_{n}))=0$ for each $n\in\mathbb{N}$.

Case 2: $Q_{k}=\exists\mathsf{cs}$ for some $k=1,\dots,K$ (and, therefore,
$Q_{j}=\forall\mathsf{cp}$ for some $j<k$).

For $[\ldots\exists\mathsf{ns}\ldots\forall\mathsf{as}\ldots]$-modes, $u=1$
would be a mode, violating AP: For any $\mathsf{cp}\ v$, choose the
$\mathsf{ns}$ $r_{n}=n^{-1}$ and the $\mathsf{cs}$ $v_{n}=v+2n^{-1}$,
guaranteeing $\mu(B_{r_{n}}(v_{n}))=0$ for sufficiently large
$n\in\mathbb{N}$.

For $[\ldots\forall\mathsf{ns}\ldots\forall\mathsf{as}\ldots]$\- and
$[\ldots\forall\mathsf{as}\ldots\forall\mathsf{ns}\ldots]$-modes, the argument
depends on the order of the quantifiers $Q_{i}=\forall\mathsf{ns}$,
$Q_{j}=\forall\mathsf{as}$ and $Q_{k}=\exists\mathsf{cs}$.

Case 2.1: $i<k$. Then $u=1$ would be a mode, violating AP: For any
$\mathsf{cp}\ v$ and $\mathsf{ns}$ $(r_{n})_{n\in\mathbb{N}}$, choose the
$\mathsf{cs}$ $v_{n}=v+2r_{n}$, guaranteeing $\mu(B_{r_{n}}(v_{n}))=0$ for
sufficiently large $n\in\mathbb{N}$.

Case 2.2: $k<i$ and $k<j$. Then $u=0$ would fail to be a mode, violating AP,
since, for the $\mathsf{cp}\ v=1$ and any $\mathsf{cs}$
$(v_{n})_{n\in\mathbb{N}}$, the $\mathsf{ns}$ $r_{n}=2|v_{n}-v|$ and the
$\mathsf{as}\ u_{n}=2r_{n}$ guarantee
$\mu(B_{r_{n}}(v_{n}))=\nicefrac{{1}}{{3}}$ and $\mu(B_{r_{n}}(u_{n}))=0$ for
sufficiently large $n\in\mathbb{N}$.

Case 2.3: $j<k<i$.
$[\forall\mathsf{as}\forall\mathsf{cp}\exists\mathsf{cs}\forall\mathsf{ns}]$-modes
(and, equivalently,
$[\forall\mathsf{cp}\forall\mathsf{as}\exists\mathsf{cs}\forall\mathsf{ns}]$-modes)
are the only ones satisfying this condition. Consider the measure
$\mu=\tfrac{1}{2}\delta_{0}+\tfrac{1}{2}\operatorname{Unif}([1,3])$ on
$X=\mathbb{R}$. Then, for $u=0$, the $\mathsf{cp}\ v=2$, the $\mathsf{as}$
$u_{n}=-2n^{-1}$ of $u$, an arbitrary $\mathsf{cs}$ $(v_{n})_{n\in\mathbb{N}}$
and the $\mathsf{ns}$ $r_{n}=n^{-1}$, the ratio
$\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v_{n})=0$, proving that $u=0$ is not a
$[\forall\mathsf{as}\forall\mathsf{cp}\exists\mathsf{cs}\forall\mathsf{ns}]$-mode
of $\mu$, violating AP. ∎

######  Proposition 5.3.

All mode maps $\mathfrak{M}[Q_{1}\ldots Q_{K}]$ of the following types violate
SP for $X=\mathbb{R}$:

  1. (a)

$[\forall\mathsf{ns}\forall\mathsf{cp}\forall\mathsf{cs}]$ (wa), |  $[\exists\mathsf{ns}\forall\mathsf{cp}\forall\mathsf{cs}]$ (pwa), |  $[\forall\mathsf{cp}\exists\mathsf{ns}\forall\mathsf{cs}]$ (wpa);
---|---|---

  2. (b)

$[\forall\mathsf{as}\exists\mathsf{ns}\forall\mathsf{cp}\forall\mathsf{cs}]$, |  $[\forall\mathsf{as}\forall\mathsf{cp}\exists\mathsf{ns}\forall\mathsf{cs}]$; |
---|---|---

  3. (c)

$[\exists\mathsf{as}\forall\mathsf{ns}\forall\mathsf{cp}\forall\mathsf{cs}]$, |  $[\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{ns}\forall\mathsf{cs}]$. |
---|---|---

###### Proof.

Consider the probability measure $\mu_{0}\in\mathcal{P}(\mathbb{R})$ from
Example 7.10 and recall that, for $r<\nicefrac{{1}}{{2}}$ and $-r\leqslant
t\leqslant r$, $\mu_{0}(B_{r}(t))=\sqrt{r+t}$, in particular (7.1) holds. We
will show for all mode maps above that $u=0$ is not a mode of $\mu_{0}$, since
it fails to dominate itself, while, being the only singularity of $\mu_{0}$,
it clearly666This is not entirely obvious for the mode maps in (b), but the
choice of the $\mathsf{ns}$ $r_{n}\coloneqq 3\lvert u_{n}\rvert$ for
$u_{n}\neq 0$, and $r_{n}\coloneqq n^{-1}$ otherwise, yields the desired
result. dominates each other $\mathsf{cp}\ v\neq u$, thus violating SP. Hence,
throughout the proof, we set $u=v=0$. Further, for a given $\mathsf{ns}$
$(r_{n})_{n\in\mathbb{N}}$, we choose the $\mathsf{cs}$ $v_{n}\coloneqq
r_{n}$. By (7.1), this already proves (a) and (b) (choosing the constant
$\mathsf{as}$ $u_{n}=u$ of $u$).

For (c), let $u_{n}\to u$ as $n\to\infty$. If $u_{n}=u$ infinitely often,
choose $r_{n}=\nicefrac{{1}}{{n}}$. Otherwise, choose $r_{n}=1$ for
$n\leqslant n_{0}\coloneqq\max\\{n\in\mathbb{N}\mid u_{n}=u\\}$ and
$r_{n}=2\lvert u_{n}-u\rvert$ for $n>n_{0}$. In both cases,

$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu_{2}}(u_{n},v_{n})\leqslant\liminf_{n\to\infty}\frac{\sqrt{3r_{n}/2}}{\sqrt{2r_{n}}}<1,$

showing that $u=0$ is neither a
$[\exists\mathsf{as}\forall\mathsf{ns}\forall\mathsf{cp}\forall\mathsf{cs}]$\-
nor a
$[\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{ns}\forall\mathsf{cs}]$-mode
of $\mu_{0}$. ∎

######  Proposition 5.4.

All mode maps $\mathfrak{M}[Q_{1}\ldots Q_{K}]$ of the following types violate
CP for $X=\mathbb{R}$:

  1. (a)

hide
$[\forall\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cp}]$ (gw),
$[\forall\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cp}\exists\mathsf{cs}]$,
$[\forall\mathsf{ns}\forall\mathsf{cp}\exists\mathsf{as}]$ (wg),
$[\exists\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cp}\exists\mathsf{cs}]$,
$[\exists\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cp}]$ (pgw),
$[\forall\mathsf{ns}\forall\mathsf{cp}\exists\mathsf{as}\exists\mathsf{cs}]$,
$[\exists\mathsf{ns}\forall\mathsf{cp}\exists\mathsf{as}]$ (pwg),
$[\exists\mathsf{ns}\forall\mathsf{cp}\exists\mathsf{as}\exists\mathsf{cs}]$,
$[\exists\mathsf{as}\forall\mathsf{cp}\exists\mathsf{ns}]$ (gwp),
$[\exists\mathsf{as}\forall\mathsf{cp}\exists\mathsf{ns}\exists\mathsf{cs}]$,
$[\forall\mathsf{cp}\exists\mathsf{ns}\exists\mathsf{as}]$ (wpg),
$[\forall\mathsf{cp}\exists\mathsf{ns}\exists\mathsf{as}\exists\mathsf{cs}]$,
$[\forall\mathsf{cp}\exists\mathsf{cs}\forall\mathsf{ns}\exists\mathsf{as}]$;

  2. (b)

hide
$[\forall\mathsf{ns}\forall\mathsf{cp}\exists\mathsf{cs}]$,
$[\exists\mathsf{as}\forall\mathsf{ns}\forall\mathsf{cp}\exists\mathsf{cs}]$,
$[\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{ns}\exists\mathsf{cs}]$,
$[\exists\mathsf{ns}\forall\mathsf{cp}\exists\mathsf{cs}]$,
$[\forall\mathsf{cp}\exists\mathsf{ns}\exists\mathsf{cs}]$;

  3. (c)

$[\forall\mathsf{as}\exists\mathsf{ns}\forall\mathsf{cp}\exists\mathsf{cs}]$, |  $[\forall\mathsf{as}\forall\mathsf{cp}\exists\mathsf{ns}\exists\mathsf{cs}]$, |  $[\forall\mathsf{cp}\exists\mathsf{cs}\forall\mathsf{as}\exists\mathsf{ns}]$;
---|---|---

  4. (d)

$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{ns}\exists\mathsf{as}]$ (wapg). |  |
---|---|---

###### Proof.

Consider the probability measures $\mu_{0},\mu_{2}\in\mathcal{P}(\mathbb{R})$
from Example 7.10. We will show for all mode maps above that $u=0$ is a mode
of both $\mu_{0}$ and $\mu_{2}$, contradicting CP.

Since $u=0$ is the only singularity of the pdf of $\mu_{0}$, it is a mode of
$\mu_{0}$. This is clear in all cases except the ones in (c), for which the
$\mathsf{ns}$ $r_{n}\coloneqq 2\lvert u_{n}\rvert$ if $u_{n}\neq u$, and
$r_{n}\coloneqq n^{-1}$ otherwise, guarantees that $(u_{n})_{n\in\mathbb{N}}$
dominates each $\mathsf{cs}$ $(v_{n})_{n\in\mathbb{N}}$ of each $\mathsf{cp}\
v\neq u$, while, for $v=u$, we can simply choose the same approximating
sequence for $v$ as for $u$ or vice versa. To see that $u=0$ is a mode of
$\mu_{2}$, we only have to consider the critical $\mathsf{cp}$ $v=2$. Choose
$u_{n}=r_{n}$ (and, potentially, $v_{n}=v$) for (a) and recall (7.1),
$v_{n}=v-r_{n}$ (and, potentially, $u_{n}=u$) for (b) and (c), where the
$\mathsf{ns}$ for (c) is chosen as above. For (d), choose the $\mathsf{ns}$
$r_{n}\coloneqq 2\lvert v_{n}\rvert$ if $v_{n}\neq v$, and $r_{n}\coloneqq
n^{-1}$ otherwise, and again $u_{n}=r_{n}$, recalling (7.1). ∎

###  5.2 Mode definitions that satisfy LP, AP, SP, and CP

After excluding all mode definitions from Propositions 5.2, 5.3, and 5.4,
which violate AP, SP, or CP for $X=\mathbb{R}$, we will now analyse the
remaining definitions from Definition 4.2 with respect to these axioms. These
definitions are listed in Table 6.1, following a classification scheme that we
have not yet introduced. While SP and CP are still unresolved for some of the
remaining mode definitions, no further definitions can be excluded by LP and
AP:

######  Proposition 5.5 (AP and LP for remaining mode definitions).

The mode maps $\mathfrak{M}[Q_{1}\ldots Q_{K}]$ of all types listed in Table
6.1 satisfy LP (for $X=\mathbb{R}^{m}$) and AP (for arbitrary metric spaces
$X$).

###### Proof.

The proof is straightforward for each definition separately, where for LP we
use standard continuity arguments. More precisely, for a probability measure
$\mu\in\mathcal{P}(\mathbb{R}^{m})$ with continuous pdf $\rho$, each
$\mathsf{ns}$ $(r_{n})_{n\in\mathbb{N}}$ and each $\mathsf{as}$
$(u_{n})_{n\in\mathbb{N}}$ of each point $u\in X$ satisfy

$\lim_{n\to\infty}\frac{\mu(B_{r_{n}}(u_{n}))}{\lambda_{m}(B_{r_{n}}(0))}=\rho(u),$

where $\lambda_{m}$ denotes the $m$-dimensional Lebesgue measure on
$\mathbb{R}^{m}$. ∎

######  Proposition 5.6 (SP for certain mode definitions).

All mode maps $\mathfrak{M}[Q_{1}\ldots Q_{K}]$ of the following types satisfy
SP for any separable metric space $X$ (note that further mode types, which are
equivalent to the ones stated here by Proposition 6.1, also satisfy SP):

  1. (a)

hide
$[\forall\mathsf{ns}]$ (s:), $[\exists\mathsf{ns}]$ (ps:),
$[\forall\mathsf{ns}\exists\mathsf{as}]$ (gs:),
$[\exists\mathsf{ns}\exists\mathsf{as}]$ (pgs:);

  2. (b)

$[\forall\mathsf{ns}\forall\mathsf{cp}]$ (w), |  $[\exists\mathsf{ns}\forall\mathsf{cp}]$ (pw), |  $[\forall\mathsf{cp}\exists\mathsf{ns}]$ (wp);
---|---|---

  3. (c)

hide
$[\forall\mathsf{ns}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}]$
(wag),
$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}\forall\mathsf{ns}]$,
$[\exists\mathsf{ns}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}]$
(pwag),
$[\forall\mathsf{cp}\exists\mathsf{ns}\forall\mathsf{cs}\exists\mathsf{as}]$
(wpag);

  4. (d)

$[\exists\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{ns}]$ (gwap), |  $[\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{cs}\exists\mathsf{ns}]$ (wgap). |
---|---|---

###### Proof.

Let $u$ be a $[Q_{1}\ldots Q_{K}]$-mode with respect to the alternative
definition of $\mathsf{cp}$ given in SP, namely $v\neq u$. We now have to show
that $u$ is also a $[Q_{1}\ldots Q_{K}]$-mode with respect to the original
definition of $\mathsf{cp}$, namely $v\in X$. Hence, throughout the proof, we
only have to consider the $\mathsf{cp}\ v=u$. For all mode types in (a), (b)
and (c) there is nothing to show (for those in (c) simply choose the
$\mathsf{as}$ to equal the $\mathsf{cs}$). For gwap- and wgap-modes in (d),
consider the constant $\mathsf{as}$ $u_{n}=u$ and an arbitrary $\mathsf{cs}$
$(v_{n})_{n\in\mathbb{N}}$ of $v=u$. Denoting $\Delta_{n}=d(v_{n},u)$, let
$(\tilde{r}_{n})_{n\in\mathbb{N}}$ and $(\varepsilon_{n})_{n\in\mathbb{N}}$ be
the corresponding null sequences from Lemma A.1. Since
$B_{\tilde{r}_{n}+\Delta_{n}}(u)\supseteq B_{\tilde{r}_{n}}(u_{n})$, we obtain

$\liminf_{n\to\infty}\,\mathfrak{R}_{\tilde{r}_{n}}^{\mu}(u_{n},v_{n})\geqslant\liminf_{n\to\infty}\frac{\mu(B_{\tilde{r}_{n}}(u))}{\mu(B_{\tilde{r}_{n}+\Delta_{n}}(u))}\geqslant\liminf_{n\to\infty}(1+\varepsilon_{n})^{-1}=1,$

proving SP for gwap- and wgap-modes. ∎

######  Proposition 5.7 (CP for certain mode definitions).

All mode maps $\mathfrak{M}[Q_{1}\ldots Q_{K}]$ of the following types satisfy
CP for any normed space $X$ (note that further mode types, which by
Propositions 3.8 and 6.1 are equivalent to the ones stated here, also satisfy
CP):

  1. (a)

hide
$[\forall\mathsf{ns}]$ (s:), $[\exists\mathsf{ns}]$ (ps:),
$[\forall\mathsf{ns}\exists\mathsf{as}]$ (gs:),
$[\exists\mathsf{ns}\exists\mathsf{as}]$ (pgs:);

  2. (b)

$[\forall\mathsf{ns}\forall\mathsf{cp}]$ (w), |  $[\exists\mathsf{ns}\forall\mathsf{cp}]$ (pw), |  $[\forall\mathsf{cp}\exists\mathsf{ns}]$ (wp);
---|---|---

  3. (c)

hide
$[\forall\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}]$
(gwa),
$[\forall\mathsf{cp}\exists\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cs}]$
(wpga),
$[\exists\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{ns}]$
(gwap),
$[\exists\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}]$
(pgwa),
$[\exists\mathsf{as}\forall\mathsf{cp}\exists\mathsf{ns}\forall\mathsf{cs}]$
(gwpa),
$[\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{cs}\exists\mathsf{ns}]$
(wgap);

  4. (d)

$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}\forall\mathsf{ns}]$. |  |
---|---|---

###### Proof.

Using the notation from Definition 5.1(CP), we write
$\bar{\mu}\coloneqq\alpha\mu+(1-\alpha)\mu(\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\displaystyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\textstyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\scriptstyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\scriptscriptstyle\bullet\;$}}}}}-b)$
and
$\Delta\coloneqq\textup{dist}(\operatorname{supp}(\mu),b+\operatorname{supp}(\mu))/2>0$.
Then
$\operatorname{supp}(\bar{\mu})=\operatorname{supp}(\mu)\cup(b+\operatorname{supp}(\mu))$
and, for any $0<r<\Delta$ and $x\in X$,

$\bar{\mu}(B_{r}(x))=\begin{cases}\alpha\,\mu(B_{r}(x))&\text{if
}x\in\operatorname{supp}(\mu)+B_{\Delta}(0),\\\
(1-\alpha)\,\mu(B_{r}(x-b))&\text{if
}x\in\operatorname{supp}(\mu)+B_{\Delta}(0)+b,\\\
0&\text{otherwise.}\end{cases}$

as well as $M_{r}^{\bar{\mu}}=\max(\alpha,(1-\alpha))\,M_{r}^{\mu}$. Further,
for any $u\in X$, any $\mathsf{as}$ $(u_{n})_{n\in\mathbb{N}}$ of $u$, any
$\mathsf{ns}$ $(r_{n})_{n\in\mathbb{N}}$, and for sufficiently large
$n\in\mathbb{N}$,

$B_{r_{n}}(u_{n})\subseteq\begin{cases}\operatorname{supp}(\mu)+B_{\Delta}(0)&\text{if
}u\in\operatorname{supp}(\mu),\\\
\operatorname{supp}(\mu)+B_{\Delta}(0)+b&\text{if
}u\in\operatorname{supp}(\mu)+b,\\\
X\setminus(\operatorname{supp}(\bar{\mu})+B_{\Delta}(0))\text{ with
}\bar{\mu}(B_{r_{n}}(u_{n}))=0&\text{otherwise.}\end{cases}$

These observations already imply CP for all mode definitions in (a) and (b).
For all remaining mode types, these observations imply
$\mathfrak{M}(\bar{\mu})\subseteq\mathfrak{M}(\mu)\cup(\mathfrak{M}(\mu)+b)$.
Further, for $\alpha\geqslant\nicefrac{{1}}{{2}}$, it follows that
$\mathfrak{M}(\bar{\mu})\subseteq\mathfrak{M}(\mu)$ and that these are the
only modes within $\operatorname{supp}(\mu)$. It remains to show that, for
$\alpha<\nicefrac{{1}}{{2}}$, all modes lie within
$\operatorname{supp}(\mu)+b$ and therefore
$\mathfrak{M}(\bar{\mu})=\mathfrak{M}(\mu)+b$. By symmetry, this completes the
proof. Hence, we will now assume $\alpha<\nicefrac{{1}}{{2}}$ and that there
is a mode $u\in\operatorname{supp}(\mu)$ of $\bar{\mu}$ and derive
contradictions for all remaining mode types.

For (c) consider the $\mathsf{cp}\ v=u+b$ and choose, for any $\mathsf{as}$
$(u_{n})_{n\in\mathbb{N}}$ of $u$, the $\mathsf{cs}$ $v_{n}=u_{n}+b$ of $v$.
Then, for any $\mathsf{ns}$ $(r_{n})_{n\in\mathbb{N}}$,
$\,\mathfrak{R}_{r_{n}}^{\bar{\mu}}(u_{n},v_{n})\leqslant\frac{\alpha}{1-\alpha}<1$
for sufficiently large $n\in\mathbb{N}$, contradicting the assumption.

For (d) consider the $\mathsf{cp}$ $v=u+b$ and the constant $\mathsf{cs}$
$v_{n}=v$. By assumption there exists an $\mathsf{as}$
$(u_{n})_{n\in\mathbb{N}}$ of $u$ such that, for any $\mathsf{ns}$
$(r_{n})_{n\in\mathbb{N}}$,
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\bar{\mu}}(u_{n},v)\geqslant 1$.
On the other hand, denoting $\Delta_{n}=d(u_{n},u)$ and using Lemma A.1, there
exist null sequences $(\tilde{r}_{n})_{n\in\mathbb{N}}$ and
$(\varepsilon_{n})_{n\in\mathbb{N}}$ such that
$\mu(B_{\tilde{r}_{n}+\Delta_{n}}(u))\leqslant(1+\varepsilon_{n})\,\mu(B_{\tilde{r}_{n}}(u))$.
Therefore, since $B_{\tilde{r}_{n}+\Delta_{n}}(u)\supseteq
B_{\tilde{r}_{n}}(u_{n})$,

$\liminf_{n\to\infty}\,\mathfrak{R}_{\tilde{r}_{n}}^{\mu}(u_{n},u)\leqslant\liminf_{n\to\infty}\frac{\mu(B_{\tilde{r}_{n}+\Delta_{n}}(u))}{\mu(B_{\tilde{r}_{n}}(u))}\leqslant\liminf_{n\to\infty}1+\varepsilon_{n}=1.$

Since
$\,\mathfrak{R}_{\tilde{r}_{n}}^{\bar{\mu}}(u_{n},v)=\frac{\alpha}{1-\alpha}\,\,\mathfrak{R}_{\tilde{r}_{n}}^{\mu}(u_{n},u)$
and $\alpha<\nicefrac{{1}}{{2}}$, this yields a contradiction. ∎

###  5.3 Unresolved instances of SP and CP

The main reason why SP and CP remain unresolved for certain mode types is the
open question of whether, for a potential mode $u\in X$, there exists an
optimal approximating sequence of $u$, meaning that it dominates any other
approximating sequence of $u$ in the following sense:

######  Conjecture 5.8 (optimal approximating sequences exist).

Let $X$ be a separable metric space, let $\mu\in\mathcal{P}(X)$, and let $u\in
X$ be a wpag-mode of $\mu$. Then, for each $\mathsf{ns}$
$(r_{n})_{n\in\mathbb{N}}$, there exists an ‘optimal’ $\mathsf{as}$
$(u_{n}^{\ast})_{n\in\mathbb{N}}$ of $u$, meaning that, for any other
approximating sequence $(u_{n})_{n\in\mathbb{N}}$ of $u$,
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n}^{\ast},u_{n})\geqslant
1$.

######  Proposition 5.9.

Let $X$ be a separable metric space and let $\mu\in\mathcal{P}(X)$. If
Conjecture 5.8 holds true, then

  1. (a)

SP holds for gwa-, wga-, pgwa-, pwga-, wpga-, and gwpa-modes;

  2. (b)

CP holds for wag-, pwag-, and wpag-modes;

  3. (c)

    1. (i)

each wag-mode is a wga-mode;

    2. (ii)

each pwag-mode is a pwga-mode;

    3. (iii)

each wpag-mode is a wpga-mode.

###### Proof.

Let $u$ be any of the mode types mentioned in the proposition. Then, by
definition, it is also a wpag-mode of $\mu$ and Conjecture 5.8 guarantees the
existence of an ‘optimal’ $\mathsf{as}$ of $u$. For (a) simply choose the
$\mathsf{as}$ of $u$ as the optimal one from Conjecture 5.8. For (b), under
the assumptions of Definition 5.1(CP), the only critical step is to show that,
if $\alpha<\nicefrac{{1}}{{2}}$, then there is no mode
$u\in\operatorname{supp}(\mu)$ of
$\bar{\mu}\coloneqq\alpha\mu+(1-\alpha)\mu(\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\displaystyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\textstyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\scriptstyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\scriptscriptstyle\bullet\;$}}}}}-b)$
(see the proof of Proposition 5.7 for this reasoning). Hence, assume that
$\alpha<\nicefrac{{1}}{{2}}$ and that there is a mode
$u\in\operatorname{supp}(\mu)$ of $\bar{\mu}$. Then $u$ dominates the
$\mathsf{cp}\ v=u+b$ with respect to some $\mathsf{ns}$
$(r_{n})_{n\in\mathbb{N}}$ (and to _each_ $\mathsf{ns}$ in the case of wag-
modes). Let $(u_{n}^{\ast})_{n\in\mathbb{N}}$ be the corresponding optimal
$\mathsf{as}$ and consider the $\mathsf{cs}$ $v_{n}^{\ast}=u_{n}^{\ast}+b$ of
$v$. Then, for any $\mathsf{as}\ (u_{n})_{n\in\mathbb{N}}$ of $u$,

$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\bar{\mu}}(u_{n},v_{n})=\frac{\alpha}{1-\alpha}\,\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},u_{n}^{\ast})\leqslant\frac{\alpha}{1-\alpha}<1,$

by optimality of $(u_{n}^{\ast})_{n\in\mathbb{N}}$, yielding a contradiction.
For (c), let $u\in X$ be a $\textswab{wag}$-mode of $\mu$,
$(r_{n})_{n\in\mathbb{N}}$ be any $\mathsf{ns}$ and
$(u_{n}^{\ast})_{n\in\mathbb{N}}$ the corresponding optimal $\mathsf{as}$.
Further, let $v\in X$ be any $\mathsf{cp}$ and $(v_{n})_{n\in\mathbb{N}}$ an
arbitrary $\mathsf{cs}$ of $v$. By assumption, there exists an $\mathsf{as}\
(u_{n})_{n\in\mathbb{N}}$ of $u$ such that
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v_{n})\geqslant 1$. It
follows from the optimality of $(u_{n}^{\ast})_{n\in\mathbb{N}}$ that

$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n}^{\ast},v_{n})\geqslant\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n}^{\ast},u_{n})\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v_{n})\geqslant
1.$

Hence, the $\mathsf{as}\ (u_{n}^{\ast})_{n\in\mathbb{N}}$ dominates any
$\mathsf{cs}$ of any $\mathsf{cp}$, proving that $u$ is a wga-mode. The proofs
of the other two implications go analogously. ∎

###  5.4 Merging properties

We originally constructed Example 7.5 in order to better understand the
difference between partial and non-partial modes. However, it has another
remarkable property: The points $-1$ and $+1$ are strong modes on the
intervals $A=(-\infty,-\nicefrac{{1}}{{2}})$ and
$B=(\nicefrac{{1}}{{2}},\infty)$, respectively, but are preventing one another
from being strong (or even generalised weak) modes on $A\cup B$. Thus, these
two modes of the unimodal restricted measures vanish when we take the union of
the two intervals, yielding a measure that is not bimodal, but _amodal_!

This behavior seems rather unsatisfactory given our intuition on modes —
either the modes of $\mu|_{A}$ ‘dominate’ the ones of $\mu|_{B}$, in which
case they should agree with the ones of $\mu|_{A\cup B}$, or the other way
around, or neither the modes of $\mu|_{A}$ nor of $\mu|_{B}$ are ‘superior’ in
which case $\mu|_{A\cup B}$ should have all of them as modes. At the very
least, if both $\mu|_{A}$ and $\mu|_{B}$ have at least one mode, then so
should $\mu|_{A\cup B}$.

With this motivation, we now formally introduce the merging properties MP and
rMP, which are two expressions of the idea that the modes of $\mu$ ought to
have some relationship with the modes of $\mu|_{A}$ and $\mu|_{B}$ for
positive-mass sets $A,B\in\mathcal{B}(X)$ with positive distance:

######  Definition 5.10.

Let $X$ be a separable metric space. We say that a mode map
$\mathfrak{M}\colon\mathcal{P}(X)\to 2^{X}$ satisfies

  * (MP)

the merging property if, for any $\mu\in\mathcal{P}(X)$ and any subsets
$A,B\in\mathcal{B}(X)$ with $\mu(A)>0$, $\mu(B)>0$, and
$\textup{dist}(A,B)>0$,

$\mathfrak{M}(\mu|_{A\cup
B})=\mathfrak{M}(\mu|_{A})\cup\mathfrak{M}(\mu|_{B})\quad\text{or}\quad\mathfrak{M}(\mu|_{A\cup
B})=\mathfrak{M}(\mu|_{A})\quad\text{or}\quad\mathfrak{M}(\mu|_{A\cup
B})=\mathfrak{M}(\mu|_{B});$

  * (rMP)

the relaxed merging property if, for any $\mu\in\mathcal{P}(X)$ and any
subsets $A,B\in\mathcal{B}(X)$ with $\mu(A)>0$, $\mu(B)>0$, and
$\textup{dist}(A,B)>0$, both

    1. (a)

$\mathfrak{M}(\mu|_{A\cup
B})\subseteq\mathfrak{M}(\mu|_{A})\cup\mathfrak{M}(\mu|_{B})$, and

    2. (b)

$\mathfrak{M}(\mu|_{A})\neq\varnothing,\mathfrak{M}(\mu|_{B})\neq\varnothing\implies\mathfrak{M}(\mu|_{A\cup
B})\neq\varnothing$.

A few remarks on these merging properties are in order:

Part (a) of the rMP is clearly777If $u\in X$ dominates every $\mathsf{cp}$,
$\mathsf{cs}$, or $\sup$ within $A\cup B$, then it also does so within $A$ and
$B$ separately. satisfied by all of the remaining definitions satisfying LP,
AP, SP, and CP and listed in Table 6.1. Thus, it is part (b) of the rMP, which
is satisfied by none of the remaining definitions, that is problematic.

The requirement that $\textup{dist}(A,B)>0$ is included in MP and rMP so that
these properties are not immediately ‘too strong to be useful’, as the
following example illustrates. Let
$\rho_{\triangle}(x)\coloneqq\max\\{0,1-\lvert x\rvert\\}$ and consider
$\mu\in\mathcal{P}(\mathbb{R})$ with pdf

$\rho(x)\coloneqq\tfrac{3}{10}\rho_{\triangle}(x+2)+\tfrac{4}{10}\rho_{\triangle}(x)+\tfrac{3}{10}\rho_{\triangle}(x-2).$

This is a unimodal measure whose unique mode (in any meaningful sense, since
$\rho$ is continuous) is at $u=0$. However, consider the case
$A\coloneqq(-\infty,0)$ and $B\coloneqq(0,\infty)$: $\mu|_{A}$ and $\mu|_{B}$
have unique modes (at $u_{A}={-2}$ and $u_{B}={+2}$ respectively) for a large
class of mode definitions, including s:- and w-modes. Thus, by the all-too-
easy expedient of ‘splitting’ the mode at $u=0$ between the sets $A$ and $B$,
this example would violate both MP and rMP if no separation requirement were
imposed on $A$ and $B$, and prohibiting such scenarios a priori is hardly an
objective axiom. In summary, allowing $\textup{dist}(A,B)=0$ in Definition
5.10 would make MP and rMP excessively strict.

However, the separation of the sets $A$ and $B$ is not the key issue for
merging properties. Rather, the issue is oscillation of the ratio of ball
masses, as exemplified by the above-mentioned Example 7.5, which shows that
neither s:- nor gs:- nor w- nor gw-modes satisfy rMP, let alone MP. Partial
modes would seem to remedy this deficiency, but the even more sophisticated
Example 7.7 proves this intuition wrong. In fact, _none_ of the remaining
definitions satisfying LP, AP, SP, and CP and listed in Table 6.1 satisfies
rMP or MP for $X=\mathbb{R}$:

######  Proposition 5.11.

For $X=\mathbb{R}$, all mode maps $\mathfrak{M}[Q_{1}\ldots Q_{K}]$ listed in
Table 6.1 satisfy neither MP nor rMP.

###### Proof.

We first show that Example 7.5 is a counterexample to rMP for all modes in
Table 6.1 that are not partial, i.e. $Q_{k}=\forall\mathsf{ns}$ (and not
$\exists\mathsf{ns}$) for some $k=1,\dots,K$. Making use of the implications
among some of the mode definitions that are discussed in Section 6 and
visualised in Figure 6.1, it is sufficient to show that, for
$A=(-\infty,-\nicefrac{{1}}{{2}})$ and $B=(\nicefrac{{1}}{{2}},\infty)$,

  1. (i)

$u_{A}=-1$ and $u_{B}=1$ are $\textswab{s:}$-modes of $\mu|_{A}$ and
$\mu|_{B}$, respectively.

  2. (ii)

$u_{A}$ and $u_{B}$ are neither w-modes nor wag-modes of $\mu|_{A\cup B}$.

Both statements follow directly from the derivations within Example 7.5.

Now we prove that Example 7.7 is a counterexample to rMP for all modes in
Table 6.1 that are partial, i.e. $Q_{k}=\exists\mathsf{ns}$ for some
$k=1,\dots,K$. Again, making use of the implications among some of the mode
definitions that are discussed in Section 6 and visualised in Figure 6.1, it
is sufficient to show that, for $A=(-\infty,-\nicefrac{{1}}{{2}})$ and
$B=(\nicefrac{{1}}{{2}},\infty)$,

  1. (i)

$u_{A}=-1$ and $u_{B}=1$ are $\textswab{ps:}$-modes of $\mu|_{A}$ and
$\mu|_{B}$, respectively.

  2. (ii)

$u_{A}$ and $u_{B}$ are neither wpag-modes nor wgap-modes of $\mu|_{A\cup B}$.

Both statements follow directly from the derivations within Example 7.7. ∎

As noted in the introduction, the failure of MP and rMP for weak modes is
intimately connected with the non-totality of the preorder (1.1) introduced by
Lambley and Sullivan (2023): the preorders for $\mu|_{A}$ and $\mu|_{B}$ may
each be total and admit a greatest element (weak mode), while these same
statements fail for $\mu|_{A\cup B}$. Proposition 5.11 shows that the same
issue persists for all the meaningful modes listed in Table 6.1.

##  6 A ‘periodic table’ of modes

Section 4 introduced a general template for mode definitions using small-ball
probabilities, with a combinatorially large number of ‘grammatically correct’
definitions (Appendix D); Section 5 undertook to eliminate those definitions
that are ‘meaningless’ in the sense that they fail to satisfy common-sense
desiderata such as LP, AP, SP, and CP (for $X=\mathbb{R}$, cf. Remark 4.4),
and to verify these axioms for the remaining definitions. Now, this section
seeks to establish the causal relations (implications and counterexamples)
among the surviving ‘meaningful’ modes and to present this information in a
systematic way: a ‘periodic table’ of modes.

###  6.1 Equivalences among meaningful modes

To reduce the complexity of the resulting ‘periodic table’ to a minimum, we
begin by giving alternative characterisations, i.e. equivalences, of several
mode types:

######  Proposition 6.1.

Let $\mu\in\mathcal{P}(X)$. Then

  1. (a)

$u\in X$ is a $[\exists\mathsf{as}\forall\mathsf{ns}]$-mode of $\mu$ if and
only if $u$ is an s:-mode ($[\forall\mathsf{ns}]$-mode) of $\mu$;

  2. (b)

the following notions of modes are equivalent:

    1. (i)

$[\forall\mathsf{ns}\forall\mathsf{cp}]$ (w),

    2. (ii)

$[\exists\mathsf{as}\forall\mathsf{ns}\forall\mathsf{cp}]$,

    3. (iii)

$[\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{ns}]$,

    4. (iv)

$[\forall\mathsf{cp}\exists\mathsf{cs}\forall\mathsf{ns}]$,

    5. (v)

$[\exists\mathsf{as}\forall\mathsf{cp}\exists\mathsf{cs}\forall\mathsf{ns}]$,

    6. (vi)

$[\forall\mathsf{cp}\exists\mathsf{as}\exists\mathsf{cs}\forall\mathsf{ns}]$;

  3. (c)

$u\in X$ is a $[\forall\mathsf{as}\exists\mathsf{ns}]$-mode of $\mu$ if and
only if $u$ is a ps:-mode ($[\exists\mathsf{ns}]$-mode) of $\mu$;

  4. (d)

$u\in X$ is a $[\forall\mathsf{as}\exists\mathsf{ns}\forall\mathsf{cp}]$-mode
of $\mu$ if and only if $u$ is a pw-mode
($[\exists\mathsf{ns}\forall\mathsf{cp}]$-mode) of $\mu$;

  5. (e)

the following notions of modes are equivalent:

    1. (i)

$[\forall\mathsf{cp}\exists\mathsf{ns}]$ (wp),

    2. (ii)

$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{ns}]$ (wap),

    3. (iii)

$[\forall\mathsf{as}\forall\mathsf{cp}\exists\mathsf{ns}]$,

    4. (iv)

$[\forall\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{ns}]$.

###### Proof.

The proof of (a) is similar to, but simpler than the one of (b), while (c) and
(d) are similar to, but simpler than (e). Therefore we only show (b) and (e).

For the proof of (b) note that the implications
(b)(i)$\Rightarrow$(b)(ii)$\Rightarrow$(b)(iii)$\Rightarrow$(b)(vi) and
(b)(i)$\Rightarrow$(b)(iv)$\Rightarrow$(b)(v)$\Rightarrow$(b)(vi) hold by
definition. Hence, it is sufficient to show the implication
(b)(vi)$\Rightarrow$(b)(i). For this purpose, assume that $u\in X$ is a
$[\forall\mathsf{cp}\exists\mathsf{as}\exists\mathsf{cs}\forall\mathsf{ns}]$-mode
and consider an arbitrary $\mathsf{ns}\ (r_{k}^{\ast})_{k\in\mathbb{N}}$ and
an arbitrary $\mathsf{cp}\ v\in X$. By assumption, there exist an
$\mathsf{as}\ (u_{n})_{n\in\mathbb{N}}$ of $u$ and a $\mathsf{cs}\
(v_{n})_{n\in\mathbb{N}}$ of $v$ such that, for each $\mathsf{ns}\
(r_{n})_{n\in\mathbb{N}}$,
$\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v_{n})\geqslant 1$. By
Lemma A.2 there exists, for each $k\in\mathbb{N}$, a radius
$0<s_{k}<r_{k}^{\ast}$ such that
$\mu(B_{s_{k}}(v))\geqslant\mu(B_{r_{k}^{\ast}}(v))(1-k^{-1})$, as well as,
since $u_{n}\to u$, $\tilde{N}_{k}\in\mathbb{N}$ such that, for each
$n\geqslant\tilde{N}_{k}$, $d(u_{n},u)\leqslant r_{k}^{\ast}-s_{k}$ and hence
$B_{s_{k}}(u_{n})\subseteq B_{r_{k}^{\ast}}(u)$,
$\mu(B_{s_{k}}(u_{n}))\leqslant\mu(B_{r_{k}^{\ast}}(u))$. Further, for each
$k\in\mathbb{N}$, Lemma A.3 guarantees the existence of
$N_{k}\geqslant\tilde{N}_{k}$ such that, for each $n\geqslant N_{k}$,
$\,\mathfrak{R}_{s_{k}}^{\mu}(v_{n},v)\geqslant 1-k^{-1}$. Without loss of
generality, we may assume $N_{k}$ to be strictly increasing in $k$. Now
denote, for $n\geqslant N_{1}$, $k(n)\coloneqq k$ whenever $N_{k}\leqslant
n<N_{k+1}$ and $r_{n}\coloneqq s_{k(n)}$. By construction,
$(r_{n})_{n\geqslant N_{1}}$ is a $\mathsf{ns}$ that satisfies

$\displaystyle\liminf_{k\to\infty}\,\mathfrak{R}_{r_{k}^{\ast}}^{\mu}(u,v)$
$\displaystyle=\liminf_{n\to\infty}\,\mathfrak{R}_{r_{k(n)}^{\ast}}^{\mu}(u,v)$
$\displaystyle\geqslant\liminf_{n\to\infty}\frac{\mu(B_{s_{k(n)}}(u_{n}))}{\mu(B_{s_{k(n)}}(v))}\,\big{(}1-\tfrac{1}{k(n)}\big{)}$
$\displaystyle=\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v_{n})\,\,\mathfrak{R}_{r_{n}}^{\mu}(v_{n},v)\,\big{(}1-\tfrac{1}{k(n)}\big{)}$
$\displaystyle\geqslant\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v_{n})\,\big{(}1-\tfrac{1}{k(n)}\big{)}^{2}$
$\displaystyle\geqslant 1,$

showing that $u$ is a w-mode and completing the proof of (b).

For the proof of (e) note that the implications
(e)(iv)$\Rightarrow$(e)(ii)$\Rightarrow$(e)(i) and
(e)(iv)$\Rightarrow$(e)(iii)$\Rightarrow$(e)(i) hold by definition. Hence, it
is sufficient to show the implication (e)(i)$\Rightarrow$(e)(iv). For this
purpose, assume that $u\in X$ is a wp-mode and consider arbitrary
$\mathsf{as}\ (u_{n})_{n\in\mathbb{N}}$, $\mathsf{cp}\ v\in X$ and
$\mathsf{cs}\ (v_{n})_{n\in\mathbb{N}}$. By Lemma A.2 there exists, for each
$k\in\mathbb{N}$, a radius $0<s_{k}<r_{k}^{\ast}$ such that
$\mu(B_{s_{k}}(u))\geqslant\mu(B_{r_{k}^{\ast}}(u))(1-k^{-1})$, as well as,
since $v_{n}\to v$, $\tilde{N}_{k}\in\mathbb{N}$ such that, for each
$n\geqslant\tilde{N}_{k}$, $d(v_{n},v)\leqslant r_{k}^{\ast}-s_{k}$ and hence
$B_{s_{k}}(v_{n})\subseteq B_{r_{k}^{\ast}}(v)$,
$\mu(B_{s_{k}}(v_{n}))\leqslant\mu(B_{r_{k}^{\ast}}(v))$. Further, for each
$k\in\mathbb{N}$, Lemma A.3 guarantees the existence of
$N_{k}\geqslant\tilde{N}_{k}$ such that, for each $n\geqslant N_{k}$,
$\,\mathfrak{R}_{s_{k}}^{\mu}(u_{n},u)\geqslant 1-\tfrac{1}{k}$.

Without loss of generality, we may assume $N_{k}$ to be strictly increasing in
$k$. Now choose, for $n\geqslant N_{1}$, $k(n)\coloneqq k$ whenever
$N_{k}\leqslant n<N_{k+1}$ and $r_{n}\coloneqq s_{k(n)}$. By construction,
$(r_{n})_{n\geqslant N_{1}}$ is a $\mathsf{ns}$ that satisfies

$\displaystyle\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(u_{n},v_{n})$
$\displaystyle=\liminf_{n\to\infty}\,\mathfrak{R}_{s_{k(n)}}^{\mu}(u_{n},u)\,\frac{\mu(B_{s_{k(n)}}(u))}{\mu(B_{r_{k(n)}^{\ast}}(v))}\,\frac{\mu(B_{r_{k(n)}^{\ast}}(v))}{\mu(B_{s_{k(n)}}(v_{n}))}$
$\displaystyle\geqslant\liminf_{n\to\infty}\big{(}1-\tfrac{1}{k(n)}\big{)}^{2}\,\,\mathfrak{R}_{r_{k(n)}^{\ast}}^{\mu}(u,v)$
$\displaystyle\geqslant 1,$

showing that $u$ is a
$[\forall\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{ns}]$-mode
and completing the proof of (b). ∎

###  6.2 A guide to the ‘periodic table’ of modes

Having disproved LP, AP, SP, and CP for a large number of mode definitions in
Section 5.1, we are left with 30 mode definitions which, as proved in Section
5.2, satisfy these properties (except for a few unresolved cases that could
only be derived under Conjecture 5.8; see Proposition 5.9). These 30
definitions are listed in Table 6.1. Due to the equivalences proved in
Propositions 3.8 and 6.1, these 30 definitions reduce to 17
inequivalent888Under Conjecture 5.8, three converse implications hold and
there are 14 inequivalent meaningful mode definitions. Our counterexamples
show that no further equivalences are possible. However, it is also
conceivable that some of these three equivalences might be proved
independently of the others (i.e. not using Conjecture 5.8) and future
research might propose axioms that would eliminate some of these 14 to 17
definitions for reasons that we have not considered. meaningful definitions of
a mode, as shown in Figure 6.1.

Table 6.1: All mode types satisfying Definition 4.2, omitting the
‘meaningless’ ones known to violate LP, AP, SP, or CP (for $X=\mathbb{R}$).
The remainder satisfy AP and LP (by Proposition 5.5) and violate MP as well as
rMP (for $X=\mathbb{R}$, by Proposition 5.11). Unresolved instances of SP and
CP are indicated with a ‘?’ (although these cases are resolved positively
under Conjecture 5.8 by Proposition 5.9), while ‘✓’ indicates that SP and CP
hold, as per Propositions 5.6 and 5.7.
mode type(s) |  comments |  SP |  CP
---|---|---|---
modes characterised as strong
$[\forall\mathsf{ns}]$ (s:) |  original definition of Dashti et al. (2013) |  ✓ |  ✓
$\Leftrightarrow[\exists\mathsf{as}\forall\mathsf{ns}]$ |  coincide by Proposition 6.1(a) |  ✓ |  ✓
$\Rightarrow[\forall\mathsf{ns}\exists\mathsf{as}]$ (gs:) |  introduced by Clason et al. (2019) |  ✓ |  ✓
$[\exists\mathsf{ns}]$ (ps:) |  $\Leftarrow$ s: |  ✓ |  ✓
$\Leftrightarrow[\forall\mathsf{as}\exists\mathsf{ns}]$ |  coincide by Proposition 6.1(c) |  ✓ |  ✓
$\Rightarrow[\exists\mathsf{ns}\exists\mathsf{as}]$ (pgs:) |  |  ✓ |  ✓
modes intermediate between strong and weak
$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}\forall\mathsf{ns}]$ |  $\Leftarrow$ s:; $\Rightarrow$ w and $\Rightarrow$ wag |  ✓ |  ✓
modes characterised as weak
$[\forall\mathsf{ns}\forall\mathsf{cp}]$ (w) |  global analogue (Ayanbayev et al., 2022) of
$E$-weak modes (Helin and Burger, 2015) |  ✓ |  ✓
$\Leftrightarrow[\forall\mathsf{cp}\exists\mathsf{cs}\forall\mathsf{ns}]$ |  coincide by Proposition 6.1(b) |  ✓ |  ✓
$\Leftrightarrow[\exists\mathsf{as}\forall\mathsf{ns}\forall\mathsf{cp}]$ |  coincide by Proposition 6.1(b) |  ✓ |  ✓
$\Leftrightarrow[\exists\mathsf{as}\forall\mathsf{cp}\exists\mathsf{cs}\forall\mathsf{ns}]$ |  coincide by Proposition 6.1(b) |  ✓ |  ✓
$\Leftrightarrow[\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{ns}]$ |  coincide by Proposition 6.1(b) |  ✓ |  ✓
$\Leftrightarrow[\forall\mathsf{cp}\exists\mathsf{as}\exists\mathsf{cs}\forall\mathsf{ns}]$ |  coincide by Proposition 6.1(b) |  ✓ |  ✓
modes characterised as weak and partial
$[\exists\mathsf{ns}\forall\mathsf{cp}]$ (pw) |  |  ✓ |  ✓
$\Leftrightarrow[\forall\mathsf{as}\exists\mathsf{ns}\forall\mathsf{cp}]$ |  coincide by Proposition 6.1(d) |  ✓ |  ✓
$\Rightarrow[\forall\mathsf{cp}\exists\mathsf{ns}]$ (wp) |  |  ✓ |  ✓
$\Leftrightarrow[\forall\mathsf{as}\forall\mathsf{cp}\exists\mathsf{ns}]$ |  coincide by Proposition 6.1(e) |  ✓ |  ✓
$\Leftrightarrow[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{ns}]$ (wap) |  coincide by Proposition 6.1(e) |  ✓ |  ✓
$\Leftrightarrow[\forall\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{ns}]$ |  coincide by Proposition 6.1(e) |  ✓ |  ✓
modes characterised as weak, generalised, and approximating
$[\forall\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}]$ (gwa) |  |  ? |  ✓
$\Leftrightarrow[\forall\mathsf{ns}\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{cs}]$ (wga) |  coincide by Proposition 3.8(a) |  ? |  ✓
$\Rightarrow[\forall\mathsf{ns}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}]$ (wag) |  reverse implication holds under Conjecture 5.8 |  ✓ |  ?
modes characterised as weak, partial, generalised, and approximating
$[\exists\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}]$ (pgwa) |  $\Leftarrow$ gwa |  ? |  ✓
$\Leftrightarrow[\exists\mathsf{ns}\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{cs}]$ (pwga) |  coincide by Proposition 3.8(b); $\Leftarrow$ wga |  ? |  ✓
$\Rightarrow[\exists\mathsf{ns}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}]$ (pwag) |  $\Leftarrow$ wag; $\Rightarrow$ pwga under Conjecture 5.8 |  ✓ |  ?
$\Rightarrow[\forall\mathsf{cp}\exists\mathsf{ns}\forall\mathsf{cs}\exists\mathsf{as}]$ (wpag) |  implies wpga under Conjecture 5.8 |  ✓ |  ?
$\Leftarrow[\forall\mathsf{cp}\exists\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cs}]$ (wpga) |  |  ? |  ✓
$\Leftarrow[\exists\mathsf{as}\forall\mathsf{cp}\exists\mathsf{ns}\forall\mathsf{cs}]$ (gwpa) |  |  ? |  ✓
$\Rightarrow[\exists\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{ns}]$ (gwap) |  |  ✓ |  ✓
$\Rightarrow[\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{cs}\exists\mathsf{ns}]$ (wgap) |  |  ✓ |  ✓

$[\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{cs}\exists\mathsf{ns}]$
(wgap)
$[\exists\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{ns}]$
(gwap)
$[\forall\mathsf{cp}\exists\mathsf{ns}\forall\mathsf{cs}\exists\mathsf{as}]$
(wpag)
$[\forall\mathsf{cp}\exists\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cs}]$
(wpga)
$[\exists\mathsf{as}\forall\mathsf{cp}\exists\mathsf{ns}\forall\mathsf{cs}]$
(gwpa)
$[\exists\mathsf{ns}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}]$
(pwag)
$[\exists\mathsf{ns}\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{cs}]$
(pwga) $\Leftrightarrow$
$[\exists\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}]$
(pgwa) $[\exists\mathsf{ns}\exists\mathsf{as}]$ (pgs:) $[\exists\mathsf{ns}]$
(ps:) $\Leftrightarrow$ $[\forall\mathsf{as}\exists\mathsf{ns}]$
$[\exists\mathsf{ns}\forall\mathsf{cp}]$ (pw) $\Leftrightarrow$
$[\forall\mathsf{as}\exists\mathsf{ns}\forall\mathsf{cp}]$
$[\forall\mathsf{ns}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}]$
(wag)
$[\forall\mathsf{ns}\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{cs}]$
(wga) $\Leftrightarrow$
$[\forall\mathsf{ns}\exists\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}]$
(gwa) $[\forall\mathsf{ns}\exists\mathsf{as}]$ (gs:) $[\forall\mathsf{ns}]$
(s:) $\Leftrightarrow$ $[\exists\mathsf{as}\forall\mathsf{ns}]$
$[\forall\mathsf{ns}\forall\mathsf{cp}]$ (w) $\Leftrightarrow$
$[\exists\mathsf{as}\forall\mathsf{ns}\forall\mathsf{cp}]$ $\Leftrightarrow$
$[\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{ns}]$ $\Leftrightarrow$
$[\forall\mathsf{cp}\exists\mathsf{cs}\forall\mathsf{ns}]$ $\Leftrightarrow$
$[\exists\mathsf{as}\forall\mathsf{cp}\exists\mathsf{cs}\forall\mathsf{ns}]$
$\Leftrightarrow$
$[\forall\mathsf{cp}\exists\mathsf{as}\exists\mathsf{cs}\forall\mathsf{ns}]$
$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}\forall\mathsf{ns}]$

$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{ns}]$ (wap)
$\Leftrightarrow$
$[\forall\mathsf{as}\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{ns}]$
$\Leftrightarrow$ $[\forall\mathsf{cp}\exists\mathsf{ns}]$ (wp)
$\Leftrightarrow$ $[\forall\mathsf{as}\forall\mathsf{cp}\exists\mathsf{ns}]$

Ex. 7.1$|$Ex. 7.2$|$$|$Ex. 7.5$|$Ex. 7.8Ex. 7.9$|$Ex. 7.10(a)$|$

Figure 6.1: Relations among the ‘meaningful’ mode definitions; cf. Table 6.1.
Unlabelled implications hold by definition; equivalences between mode
definitions hold by Propositions 3.8 and 6.1; dashed implications hold under
Conjecture 5.8. Each named counterexample explicitly constructs a point that
is a mode with respect to some definitions but not others, thus dividing the
system of modes into two parts, marked by a red line. Hence, each definition
from the one part fails to imply each definition from the other part, and the
direction of non-implication is marked by the crossed-out red arrow. In
particular, the converse of every black implication arrow that crosses one of
the red lines (against the direction of the corresponding red arrow) is false,
and a counterexample is provided by the label on the red arrow. Thus, apart
from the three ‘dashed’ implications that hold under Conjecture 5.8, no
further implications or equivalences are possible and so the number of
essentially distinct meaningful definitions of a mode is between 14 and 17.

The following remarks may serve as a guide to Table 6.1 and Figure 6.1. First
of all, the 17 definitions have been divided into six groups, which have been
colour-coded for easy recognition:

  * •

strong (blue): all mode types where the comparison point quantifier
$\forall\mathsf{cp}$ does not occur, i.e. the small ball probability
$\mu(B_{r}(u))$ of the potential mode $u$ (or the corresponding probability
$\mu(B_{r}(u_{n}))$ of an $\mathsf{as}$ $(u_{n})_{n\in\mathbb{N}}$ of $u$) is
always compared with the overall supremum $M_{r}^{\mu}\coloneqq\sup_{x\in
X}\mu(B_{r}(x))$ over such balls;

  * •

intermediate between strong and weak (violet): the singleton set consisting
only of the novel
$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}\forall\mathsf{ns}]$-modes;

  * •

weak (red): all mode definitions where the $\forall\mathsf{cp}$ quantifier
occurs, but none of the quantifiers $\exists\mathsf{ns}$ and
$\forall\mathsf{cs}$ do; on the other hand, the quantifiers
$\forall\mathsf{ns}$, $\forall\mathsf{as}$, $\exists\mathsf{as}$, and
$\exists\mathsf{cs}$ are allowed;

  * •

weak and partial (orange): mode definitions with the quantifiers
$\forall\mathsf{cp}$, $\exists\mathsf{ns}$, but without the quantifiers
$\forall\mathsf{ns}$ and $\exists\mathsf{as}$;

  * •

weak, generalised, and approximating (yellow): mode definitions with the four
quantifiers $\forall\mathsf{ns}$, $\forall\mathsf{cp}$, $\exists\mathsf{as}$
and $\forall\mathsf{cs}$ in various orders, but with $\forall\mathsf{ns}$
appearing before $\exists\mathsf{as}$ (note that $\textswab{g}$-modes that are
not $\textswab{a}$-modes were excluded by Proposition 5.4(a), but some
instances of the $\exists\mathsf{as}$ quantifier still survive in the weak
group, as long as they appear before the $\forall\mathsf{ns}$-quantifier);

  * •

weak, partial, generalised, and approximating (green): mode definitions with
the four quantifiers $\forall\mathsf{cp}$, $\exists\mathsf{ns}$,
$\exists\mathsf{as}$ and $\forall\mathsf{cs}$ in various orders.

Some of these mode definitions may seem rather strange. In particular, the
later the quantifiers $\forall\mathsf{ns}$ or $\exists\mathsf{ns}$ occur, the
less intuitively accessible the corresponding notion of a mode seems to be.
Furthermore, the quantifiers $\forall\mathsf{as}$ and $\exists\mathsf{cs}$ are
not the most natural ones, which is why they have not been studied in previous
works.

Arguably, the most exotic modes are
$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}\forall\mathsf{ns}]$-modes,
which are a slightly stronger notion than wag-modes, yet also sit strictly
between classical s:-modes and w-modes; these are also the _only_ meaningful
modes that strictly require the terminology of Definition 4.2 and cannot be
equivalently expressed using the notation $\textswab{s:}$, $\textswab{w}$,
$\textswab{p}$, $\textswab{g}$, and $\textswab{a}$ from Section 3.

However, no matter how odd some definitions might appear, we stress again
that, at this stage, there is no objective reason to prefer one over another,
since they all satisfy the properties LP, AP, SP, and CP (apart from the few
unresolved cases) and fail to satisfy MP and rMP.

##  7 Examples and counterexamples

This section provides examples and counterexamples to the concepts and
implications discussed in Sections 3 and 5. We begin with an example of a weak
mode that is not strong.999Note that such examples have already been
constructed by Lie and Sullivan (2018, Example 4.4) and Ayanbayev et al.
(2022, Example B.5), but ours is notably simpler. The idea behind the
construction of the measure $\mu_{w}$ is that it has exactly one singularity
$u$, which, naturally, dominates each other point in terms of small ball
probabilities $\mu(B_{r}(u))$ as $r\to 0$, making it a weak mode, but, for
each fixed radius $r>0$, there exists a point $v$ with larger mass
$\mu(B_{r}(v))$ of the corresponding ball centred at $v$, so that it is not a
strong mode.

Figure 7.1: Choice of the probability densities $\rho_{\alpha}$,
$\rho_{\beta}$ and $\rho_{w}$, $w=\nicefrac{{1}}{{10}}$, in Example 7.1. $u=0$
is an s:-mode of $\mu_{\alpha}$, while $\mu_{\beta}$ and $\mu_{w}$ have no
(partial generalised) strong modes. However, $u=0$ is a w-mode of $\mu_{w}$
for any value of $w\in(0,1)$.

######  Example 7.1 (w-mode $\not\Rightarrow$ pgs-mode; gwa-mode
$\not\Rightarrow$ pgs-mode;
$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}\forall\mathsf{ns}]$-mode
$\not\Rightarrow$ pgs-mode).

Consider the probability measures $\mu_{\alpha}$, $\mu_{\beta}$,
$\mu_{w}\in\mathcal{P}(\mathbb{R})$ with PDFs given by

$\displaystyle\rho_{\alpha}(x)$ $\displaystyle\coloneqq 2\big{(}\left\lvert
2x\right\rvert^{-1/3}-1\big{)}\,\chi_{[-\nicefrac{{1}}{{2}},\nicefrac{{1}}{{2}}]}(x),$
$\displaystyle\rho_{\beta}(x)$
$\displaystyle\coloneqq\frac{6}{\pi^{2}}\sum_{n\in\mathbb{N}}n\,t\big{(}n^{3}(x-n)\big{)},$
$\displaystyle
t(x)\coloneqq(1+2x)\,\chi_{[-\nicefrac{{1}}{{2}},0]}(x)+(1-2x)\,\chi_{[0,\nicefrac{{1}}{{2}}]}(x),$
$\displaystyle\rho_{w}(x)$ $\displaystyle\coloneqq
w\,\rho_{\alpha}(x)+(1-w)\,\rho_{\beta}(x),$

where $w\in(0,1)$ is a parameter. Figure 7.1 shows $\rho_{\alpha}$,
$\rho_{\beta}$, and $\rho_{w}$ for $w=\nicefrac{{1}}{{10}}$. Since $\rho_{w}$
has its only singularity at $u=0$ and is continuous everywhere else, $u=0$ is
the only w-mode of $\mu_{w}$:

$\liminf_{r\to 0}\,\mathfrak{R}_{r}^{\mu_{w}}(u,v)\geqslant\liminf_{r\to
0}\frac{3(2r)^{2/3}}{4r\rho_{w}(v)}=\infty\geqslant 1\qquad\text{for any
}v\neq u.$

Similarly, $u=0$ is a gwa-mode and a
$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}\forall\mathsf{ns}]$-mode
of $\mu_{w}$. However, for $w=\nicefrac{{1}}{{10}}$, $u=0$ is not a pgs-mode
of $\mu_{w}$: Defining $n(r)\coloneqq\lceil r^{-1/3}\rceil$, i.e.
$n(r)^{-3}\leqslant r\leqslant(n(r)-1)^{-3}$, we have, for each $\mathsf{ns}\
(r_{n})_{n\in\mathbb{N}}$ and each $\mathsf{as}\ (u_{n})_{n\in\mathbb{N}}$ of
$u=0$,

$\displaystyle\liminf_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu_{w}}(u_{n},\sup)$
$\displaystyle\leqslant\frac{w}{1-w}\limsup_{r\to
0}\frac{3(2r)^{2/3}-4r}{\mu_{\beta}(B_{n(r)^{-3}}(n(r)))}$
$\displaystyle\leqslant\frac{w}{1-w}\limsup_{r\to 0}\frac{3\cdot
2^{2/3}(n(r)-1)^{-2}}{\frac{6}{\pi^{2}}\,n(r)^{-2}}$
$\displaystyle=\frac{w}{1-w}\,\frac{\pi^{2}}{\sqrt[3]{2}}\approx 0.87<1.$

In summary, for $w=\nicefrac{{1}}{{10}}$, $\mu_{w}$ has a w-mode that is a
gwa-mode and a
$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}\forall\mathsf{ns}]$-mode
but not a pgs-mode.

The next example illustrates the difference between generalised and non-
generalised modes:

######  Example 7.2 (gs-mode $\not\Rightarrow$ wp-mode; similar to Clason et
al. 2019, Example 2.2).

Consider the probability measure $\mu\in\mathcal{P}(\mathbb{R})$ with pdf
$\rho$ given by

$\rho(x)\coloneqq\begin{cases}\lvert x\rvert&\text{for }x\in[-1,1],\\\
0&\text{otherwise.}\end{cases}$

Clearly, the only points that could be wp-mode are $u=\pm 1$. However, for any
$0<r<\nicefrac{{1}}{{3}}$,

$\mu(B_{r}(\pm 1))<r<\tfrac{4}{3}r=\mu(B_{r}(\tfrac{2}{3}))$

and hence $\mu$ has no wp-mode. On the other hand, for each $\mathsf{ns}\
(r_{n})_{n\in\mathbb{N}}$ (without loss of generality, we may assume $r_{n}<1$
for each $n\in\mathbb{N}$), we can choose the $\mathsf{as}\ u_{n}=1-r_{n}$ of
$u=1$ to obtain

$\mu(B_{r_{n}}(u_{n}))=2r_{n}(1-r_{n})=M_{r_{n}}^{\mu},$

proving that $u=1$ is a gs-mode of $\mu$ (a similar computation can be
performed for $u=-1$).

In the three examples below we will use the following terms and probability
measures. Our construction is similar to the more general one of Lambley and
Sullivan (2023).

Figure 7.2:  Quantities from Notation 7.3 for $b=\nicefrac{{13}}{{10}}$:
_left:_ choice of the values $a_{n}$, $n\in\mathbb{N}$; _middle:_ density
$\rho_{\textup{even}}$; _right:_ density $\rho_{\textup{odd}}$.

######  Notation 7.3.

Let $b>1$ and, for $n\in\mathbb{Z}$, let $a_{n}\coloneqq b^{-2n}$,
$r_{n}=a_{2n}=b^{-4n}$, $\tilde{r}_{n}=a_{2n+1}=b^{-4n-2}$ and
$C=\frac{b^{2}+1}{2b}>1$. Further, define the finite measures (_not_
probability measures) $\mu_{\textup{even}}$ and $\mu_{\textup{odd}}$ on
$\mathbb{R}$ with Lebesgue densities

$\displaystyle\rho_{\textup{even}}(x)$
$\displaystyle\coloneqq\begin{cases}\sup\bigl{\\{}b^{2n}\,\big{|}\,n\in\mathbb{Z},\,\lvert
x\rvert^{1/2}\leqslant b^{-2n}\bigr{\\}}&\text{for }x\in[-1,1],\\\
0&\text{otherwise},\end{cases}$ $\displaystyle\rho_{\textup{odd}}(x)$
$\displaystyle\coloneqq\begin{cases}\sup\bigl{\\{}b^{2n+1}\,\big{|}\,n\in\mathbb{Z},\,\lvert
x\rvert^{1/2}\leqslant b^{-2n-1}\bigr{\\}}&\text{for }x\in[-1,1],\\\
0&\text{otherwise},\end{cases}$

illustrated in Figure 7.2. The measures $\mu_{\textup{even}}$ and
$\mu_{\textup{odd}}$ are finite since $\rho_{\textup{even}}$ and
$\rho_{\textup{odd}}$ are bounded by the function $f(x)=\lvert x\rvert^{-1/2}$
(with $f(0)\coloneqq\infty$) which is Lebesgue integrable on $[-1,1]$.

######  Proposition 7.4.

Using Notation 7.3, we have, for any $0<r<1$,

$\frac{\mu_{\textup{even}}(B_{r_{n}}(0))}{\mu_{\textup{odd}}(B_{r_{n}}(0))}=C,\qquad\frac{\mu_{\textup{even}}(B_{\tilde{r}_{n}}(0))}{\mu_{\textup{odd}}(B_{\tilde{r}_{n}}(0))}=C^{-1},\qquad\frac{\mu_{\textup{even}}(B_{r}(0))}{\mu_{\textup{odd}}(B_{r}(0))}\in\left[C^{-1},C\right].$

In particular, $\liminf_{r\to
0}\frac{\mu_{\textup{even}}(B_{r}(0))}{\mu_{\textup{odd}}(B_{r}(0))}=C^{-1}$
and $\limsup_{r\to
0}\frac{\mu_{\textup{even}}(B_{r}(0))}{\mu_{\textup{odd}}(B_{r}(0))}=C$.

###### Proof.

For any $n\in\mathbb{N}$ and $r\in[\tilde{r}_{n},r_{n}]$ we have:

$\displaystyle\mu_{\textup{even}}(B_{r}(0))$
$\displaystyle=2rb^{2n}+\sum_{j\geqslant
n+1}2a_{2j}(b^{2j}-b^{2j-2})=2rb^{2n}+2b^{-2n-2},$
$\displaystyle\mu_{\textup{odd}}(B_{r}(0))$
$\displaystyle=2rb^{2n-1}+\sum_{j\geqslant
n}2a_{2j+1}(b^{2j+1}-b^{2j-1})=2rb^{2n-1}+b^{-2n-1}.$

By Lemma A.4 the ratio
$R(r)\coloneqq\frac{\mu_{\textup{even}}(B_{r}(0))}{\mu_{\textup{odd}}(B_{r}(0))}$
is strictly increasing in $r$ on each interval $[\tilde{r}_{n},r_{n}]$,
$n\in\mathbb{N}$, and that its values on the boundary are
$R(\tilde{r}_{n})=C^{-1}<1$ and $R(r_{n})=C>1$. A similar calculation shows
that $R$ is strictly decreasing on every interval $[r_{n+1},\tilde{r}_{n}]$,
$n\in\mathbb{N}$. ∎

Figure 7.3:  Choice of the density $\rho$ in Example 7.5 for
$b=\nicefrac{{13}}{{10}}$ (before normalisation). As the radius decreases,
$r\to 0$, the two points $u=+1$ and $v=-1$ ‘take turns’ in dominating the
$r$-ball probability at the corresponding point; the ratio
$\,\mathfrak{R}_{r}^{\mu}(-1,1)$ oscillates alternates between values either
side of unity so that its limits superior and inferior are either side of
unity. Hence, $u,v$ are neither gw-mode nor wag-modes of $\mu$, while they are
ps-mode of $\mu$.

The following example illustrates the difference between partial and non-
partial modes by considering the density
$\rho(x)\propto\rho_{\textup{even}}(x+1)+\rho_{\textup{odd}}(x-1)$,
illustrated in Figure 7.3. As $r\to 0$, the points $\pm 1$ take turns in being
dominant in terms of their radius-$r$ ball probabilities. Therefore, the
points $\pm 1$ cannot be non-partial modes (wg-modes), while, since partial
modes allow for dominance only along a certain subsequence $r_{n}\to 0$, both
are ps-mode. This example also illustrates how previous definitions of modes
failed to satisfy the merging properties MP and rMP, while partial modes give
hope for a remedy of this shortcoming. However, as Example 7.7 will show,
partial modes also fail to fulfil MP and rMP.

######  Example 7.5 (ps-mode $\not\Rightarrow$ s-mode; ps-mode
$\not\Rightarrow$ gw-mode; ps-mode $\not\Rightarrow$ wag-mode; certain non-
partial modes satisfy neither MP nor rMP).

Using Notation 7.3, consider the probability measure
$\mu\in\mathcal{P}(\mathbb{R})$ with pdf
$\rho(x)\propto\rho_{\textup{even}}(x+1)+\rho_{\textup{odd}}(x-1)$,
illustrated in Figure 7.3. By Proposition 7.4,

$\,\mathfrak{R}_{r_{n}}^{\mu}(-1,+1)=C>1,\qquad\,\mathfrak{R}_{\tilde{r}_{n}}^{\mu}(-1,+1)=C^{-1}<1.$

Hence, both $u_{A}=-1$ and $u_{B}=1$ are neither s-mode nor gw-mode nor wag-
modes nor $[\forall\mathsf{cp}\exists\mathsf{as}\forall\mathsf{ns}]$-modes of
$\mu$ (note that the constant $\mathsf{as}\ u_{n}=u_{A}$ of $u_{A}$ is optimal
in terms of small ball probabilities, and similarly for $u_{B}$; and that it
is sufficient to consider the two singularity points $u_{A}$ and $u_{B}$ since
they clearly dominate every other point). However, both are ps:-modes of
$\mu$, since, for the sequences $r_{n}$ (respectively $\tilde{r}_{n}$) we
obtain

$\lim_{n\to\infty}\,\mathfrak{R}_{r_{n}}^{\mu}(-1,\sup)=1,\qquad\lim_{n\to\infty}\,\mathfrak{R}_{\tilde{r}_{n}}^{\mu}(1,\sup)=1.$

Further, if we choose $A=(-\infty,-\nicefrac{{1}}{{2}})$ and
$B=(\nicefrac{{1}}{{2}},\infty)$, then $u_{A}$ and $u_{B}$ are
$\textswab{s:}$-modes of $\mu|_{A}$ and $\mu|_{B}$, respectively, while, as we
have just seen, $\mu|_{A\cup B}$ has no w-modes, gw-modes or wag-modes, which
disproves both MP and rMP for all relevant non-partial modes; see Proposition
5.11.

######  Example 7.6 (counterexample to (Helin and Burger, 2015, Lemma 3); cf.
Remark 3.5).

Using Notation 7.3, consider the probability measure $\mu$ on $\mathbb{R}$
with pdf $\rho(x)\propto\rho_{\textup{even}}(x+1)+C\rho_{\textup{odd}}(x-1)$.
By Proposition 7.4,

$\liminf_{r\to 0}\,\mathfrak{R}_{r}^{\mu}(-1,1)=C^{-2},\qquad\limsup_{r\to
0}\,\mathfrak{R}_{r}^{\mu}(-1,1)=1.$

Hence, $u_{B}=1$ is a strong mode of $\mu$ (note again that it is sufficient
to consider the two singularity points $u_{A}=-1$ and $u_{B}=1$), but not an
$\mathbb{R}$-weak mode in the sense of Helin and Burger (2015, Definition 4),
since the limit $\lim_{r\to 0}\,\mathfrak{R}_{r}^{\mu}(-1,1)$ fails to exist,
providing a counterexample to Helin and Burger (2015, Lemma 3). It is,
however, a (global) weak mode in the sense of our definition of weak modes
(Definition 3.4), which ‘saves’ the implication that every s:-mode is a
w-mode, cf. Remark 3.5.

Figure 7.4:  Choice of the density $\rho$ in Example 7.7 for
$b=\nicefrac{{9}}{{5}}$ and $\beta=10$ (before normalisation). The parts in
blue correspond to (restricted, scaled, shifted) versions of
$\rho_{\textup{even}}$, while the parts in red correspond to (restricted,
scaled, shifted) versions of $\rho_{\textup{odd}}$. While $u=-1$ dominates
$u_{k}=-k-2$ for any $k\in\mathbb{N}$ with respect to the $\mathsf{ns}\
(r_{n})_{n\in\mathbb{N}}$ and is thereby a partial strong mode of $\mu|_{A}$
with $A=(-\infty,\nicefrac{{1}}{{2}})$, it is dominated by $v_{k}=k+2$ for
each $k\in\mathbb{N}$ with respect to _every_ $\mathsf{ns}\ (r_{n}^{\prime})$.
It is therefore not a partial mode of $\mu$ for all definitions of partial
modes in this paper (i.e. definitions listed in Table 6.1 that include the
$\exists\mathsf{ns}$ quantifier). A similar argument works for $v=1$,
$(\tilde{r}_{n})_{n\in\mathbb{N}}$ and $\mu|_{B}$ with
$B=(-\nicefrac{{1}}{{2}},\infty)$. Note that no $u_{k}$ can be a partial mode
of $\mu$, since it is strictly dominated by its neighbour $u_{k+1}$ with
respect to every $\mathsf{ns}\ (r_{n}^{\prime})_{n\in\mathbb{N}}$, and
similarly for $v_{k}$. Thus, $\mu$ has no partial modes, showing that all
definitions of partial modes violate both MP and rMP.

As illustrated by Example 7.5, partial modes give hope to remedy the
shortcoming of conventional definitions of modes to satisfy the merging
properties MP and rMP. However, the following slightly more sophisticated
example shows that this is not the case:

######  Example 7.7 (certain partial modes satisfy neither MP nor rMP).

Using Notation 7.3, let $\beta>1$ such that $C-\beta^{-1}>1$, let
$I_{k}\coloneqq(-2^{-k},2^{-k})$, $u_{k}=-k-2$ and $v_{k}=k+2$ for
$k\in\mathbb{N}$. Define the probability measure
$\mu\in\mathcal{P}(\mathbb{R})$ by the pdf101010We restrict to the intervals
$I_{k}$ merely to make the right-hand side Lebesgue integrable, which holds
since the values
$\int_{I_{k}}\rho_{\textup{even}}\leqslant\int_{-2^{-k}}^{2^{-k}}\lvert
x\rvert^{-1/2}\mathrm{d}x=4\cdot 2^{-k/2}$ are summable over $k$ (similarly
for $\rho_{\textup{odd}}$). This has no effect on the ratios of small ball
probabilities since, for each $k\in\mathbb{N}$ and sufficiently small $r>0$,
$B_{r}(0)\cap I_{k}=B_{r}(0)$.

$\rho(x)\propto\rho_{\textup{even}}(x+1)+\rho_{\textup{odd}}(x-1)+\sum_{k=1}^{\infty}(C-\beta^{-k})\left(\rho_{\textup{even}}\big{|}_{I_{k}}(x-v_{k})+\rho_{\textup{odd}}\big{|}_{I_{k}}(x-u_{k})\right).$

Note that, in the following, it is sufficient to consider only the singularity
points of $\rho$ for generalised (partial) strong and generalised (partial)
weak modes, since no other points have a chance of being modes. Further, it is
sufficient to consider constant $\mathsf{as}\ x_{n}=x$ of $x$, since these are
optimal in terms of small ball probabilities for all singularities of $\rho$.

Now let $A=(-\infty,-\nicefrac{{1}}{{2}})$ and
$B=(\nicefrac{{1}}{{2}},\infty)$. Then $u_{A}=-1$ is a ps-mode of $\mu_{A}$
since, for any $k\in\mathbb{N}$ and sufficiently large $n\in\mathbb{N}$,
Proposition 7.4 implies

$\,\mathfrak{R}_{r_{n}}^{\mu|_{A}}(u_{A},\sup)=\min\left(1,\frac{\mu(B_{r_{n}}(u_{A}))}{\sup_{k\in\mathbb{N}}\mu(B_{r_{n}}(u_{k}))}\right)=\min\left(1,\frac{\mu_{\textup{even}}(B_{r_{n}}(0))}{\mu_{\textup{odd}}(B_{r_{n}}(0))\sup_{k\in\mathbb{N}}(C-\beta^{-k})}\right)=1.$

However, each $u_{k}$ is strictly dominated by its neighbour $u_{k+1}$ in
terms of small ball probabilities for each $\mathsf{ns}\
(r_{n}^{\prime})_{n\in\mathbb{N}}$, since, for sufficiently small $r>0$,

$\,\mathfrak{R}_{r}^{\mu|_{A}}(u_{k+1},u_{k})=\frac{(C-\beta^{-k-1})\mu_{\textup{odd}}(B_{r}(0))}{(C-\beta^{-k})\mu_{\textup{odd}}(B_{r}(0))}=\frac{C-\beta^{-k-1}}{C-\beta^{-k}}>1.$

Hence, for each $k\in\mathbb{N}$, $u_{k}$ is not even a wapg-mode of $\mu_{A}$
or $\mu|_{A\cup B}$ (and hence neither a wpag-mode nor a wgap-mode). Similar
statements for $\mu_{B}$ and $u_{B}=1$ can be derived in the same way.
Further, $u_{A}=-1$ is strictly dominated by each $v_{k}$, $k\in\mathbb{N}$,
for each $\mathsf{ns}\ (r_{n}^{\prime})_{n\in\mathbb{N}}$, since, for any
$0<r<\nicefrac{{1}}{{2}}$,

$\,\mathfrak{R}_{r}^{\mu|_{A}}(v_{k},u)=\frac{(C-\beta^{-k})\mu_{\textup{even}}(B_{r}(0))}{\mu_{\textup{even}}(B_{r}(0))}=C-\beta^{-k}>1.$

A similar argument holds for $u_{B}=1$. Hence, $u_{A}$ and $u_{B}$ are the
only ps-mode of $\mu_{A}$ and $\mu_{B}$, respectively, while $\mu|_{A\cup B}$
has no wpag-mode or wgap-mode, violating MP and rMP for a large class of
partial modes, cf. Proposition 5.11.

The following example is very similar to Example 7.5 — but with three
singularities instead of just two, and a fourth that is a ‘mean’ of the others
— and shows that the order of the quantifiers w and p is significant. Since
the line of argument is hopefully becoming familiar, we do not spell out all
the details.

Figure 7.5:  Illustration of Example 7.8. _Top:_ The Lebesgue density $\rho$
of $\mu$ for $b=\nicefrac{{6}}{{5}}$. _Bottom:_ The ‘radial cumulative
distribution function’ $\mu(B_{r}(x))$ and ratios
$\,\mathfrak{R}_{r}^{\mu}(x,0)$ as functions of $r$ for several critical
values of $x$ and $b=\nicefrac{{3}}{{2}}$.
As $r\to 0$, the singularities at $x=2$, $4$, and $6$ take turns in dominating
the other ones. As a result, the ‘mean’ singularity in $u=0$ dominates each of
the other singularities for certain null sequences, but there is no
$\mathsf{ns}$ for which it dominates the other points simultaneously.

######  Example 7.8 (wp-mode $\not\Rightarrow$ pw-mode; gwpa-mode
$\not\Rightarrow$ pwag-mode).

Let $b>1$ and $\mu_{k}$, $k=1,2,3$, be finite measures on $\mathbb{R}$ given
by their Lebesgue densities

$\displaystyle\rho_{k}(x)$
$\displaystyle\coloneqq\begin{cases}\sup\bigl{\\{}b^{3n+k}\,\big{|}\,n\in\mathbb{Z},\,\lvert
x\rvert^{1/2}\leqslant b^{-3n-k}\bigr{\\}}&\text{for }x\in[-1,1],\\\
0&\text{otherwise}.\end{cases}$ Consider the measure $\mu$ given by its
Lebesgue density $\displaystyle\rho(x)$
$\displaystyle\coloneqq\sum_{k=1}^{3}\tfrac{1}{3}\rho_{k}(x)+\rho_{k}(x-2k).$

For simplicity, we will work with the unnormalised density $\rho$, illustrated
in Figure 7.5, instead of the corresponding normalised pdf, which clearly has
no influence on the small ball probability ratios. We will argue that $u=0$ is
a wp-mode but not a pw-mode of $\mu$. The reason is that, for each
$\mathsf{cp}\ v\in\mathbb{R}$, there exists a $\mathsf{ns}$ for which $u=0$
dominates $v$, but there is no $\mathsf{ns}$ that works for all $\mathsf{cp}$
simultaneously. For the same reason, $u=0$ is a gwpa-mode, but not a pwag-mode
of $\mu$.

Analogously to the calculations in the proof of Proposition 7.4, we derive,
for any $k=1,2,3$ and $b^{-3(n+1)-k}<r^{1/2}\leqslant b^{-3n-k}$,

$\mu(B_{r}(2k))=2rb^{3n+k}+\sum_{j\geqslant
n+1}2b^{-6j-2k}\big{(}b^{3j+k}-b^{3(j-1)+k}\big{)}=2rb^{3n+k}+2b^{-3n-3-k},$

showing that $u=0$ dominates $u^{(k)}=2k$ for the $\mathsf{ns}\
r_{n}^{(k)}=b^{-3n-k-1}$, $k=1,2,3$, respectively, as illustrated in Figure
7.5 (bottom right). However, no $\mathsf{ns}$ works for all the three
$u^{(k)}$ simultaneously. Consequently, $u=0$ is a wp-mode, a gwpa-mode and an
$[\forall\mathsf{as}\forall\mathsf{cp}\exists\mathsf{ns}]$-mode (note that the
constant $\mathsf{as}$ of all singularities are clearly ‘optimal’), but
neither a pw-mode nor a pwag-mode nor a
$[\forall\mathsf{as}\exists\mathsf{ns}\forall\mathsf{cp}]$-mode.

In order to show that the quantifiers w and g are not interchangeable, the
next example modifies the previous one by making the singularities ‘one-sided’
(cf. Figure 7.6) such that approximating sequences other than $u_{n}=u$ and
$v_{n}=v$ are actually significant. Again, since the line of argument is
similar to previous ones, we do not spell out all the details.

Figure 7.6: Illustration of the Lebesgue density $\rho$ in Example 7.9.

######  Example 7.9 (wpga-mode $\not\Rightarrow$ gwap-mode; wpga-mode
$\not\Rightarrow$ pwag-mode).

Let $1<b\leqslant\nicefrac{{3}}{{2}}$. Further, define the finite measures
$\mu_{k}$, $k=1,2,3$, on $\mathbb{R}$ by their Lebesgue densities

$\displaystyle\rho_{k}(x)$
$\displaystyle\coloneqq\begin{cases}\sup\bigl{\\{}b^{3n+k}\,\big{|}\,n\in\mathbb{Z},\,\lvert
x\rvert^{1/2}\leqslant b^{-3n-k}\bigr{\\}}&\text{for }x\in[0,1],\\\
0&\text{otherwise},\end{cases}$ as well as the measure $\mu$ by its density
$\displaystyle\rho(x)$
$\displaystyle\coloneqq\sum_{k=1}^{3}\tfrac{1}{3}\rho_{k}(x)+\rho_{k}(x-2k).$

For simplicity, we will work with the unnormalised density $\rho$, illustrated
in Figure 7.6, instead of the corresponding normalised pdf, which clearly has
no influence on the small ball probability ratios. We argue that $u=0$ is a
wpga-mode but neither a gwap-mode, nor a pwag-mode of $\mu$. The reason is
that, for each $\mathsf{cp}\ v\in\mathbb{R}$, there exists a $\mathsf{ns}\
(r_{n})_{n\in\mathbb{N}}$ and an $\mathsf{as}\ (u_{n})_{n\in\mathbb{N}}$ of
$u$, namely $u_{n}\coloneqq r_{n}$, such that $(u_{n})_{n\in\mathbb{N}}$
dominates each $\mathsf{cs}\ (v_{n})_{n\in\mathbb{N}}$ of $v$ (the critical
one being $v+r_{n}$), but there is no $\mathsf{ns}$ of $u$ (with corresponding
$\mathsf{as}$ $u_{n}\coloneqq r_{n}$) that works for all $\mathsf{cp}$
simultaneously. The calculations are similar111111However, one has to be
slightly careful not to choose $b$ too large, otherwise the constant
$\mathsf{as}\ u_{n}=0$ will also dominate each $\mathsf{cp}$ for the
corresponding $\mathsf{ns}$. to Example 7.8 and are therefore left to the
reader.

Figure 7.7:  Illustration of the densities $\rho_{1}$ and $\rho_{2}$ in
Example 7.10.

######  Example 7.10 (w $\not\Rightarrow$ wpag; violation of SP and CP).

Consider the probability measures
$\mu_{0},\mu_{1},\mu_{2}\in\mathcal{P}(\mathbb{R})$ with probability densities

$\displaystyle\rho_{0}(x)$ $\displaystyle=\begin{cases}\tfrac{1}{2}\lvert
x\rvert^{-1/2}&\text{if }x\in(0,1),\\\ 0&\text{otherwise},\end{cases}$
$\displaystyle\rho_{1}(x)$
$\displaystyle=\alpha\rho_{0}(x)+\alpha\rho_{0}(-x)+(1-2\alpha)\rho_{0}(x-2),$
$\displaystyle 0\leqslant\alpha\leqslant\nicefrac{{1}}{{2}},$
$\displaystyle\rho_{2}(x)$
$\displaystyle=\beta\rho_{0}(x)+(1-\beta)\rho_{0}(x-2),$
$\displaystyle\beta=0.49,$

respectively. $\rho_{1}$ and $\rho_{2}$ are illustrated in Figure 7.7. Note
that, for $r<\nicefrac{{1}}{{2}}$ and $-r\leqslant t\leqslant r$,
$\mu_{0}(B_{r}(t))=\sqrt{r+t}$, in particular,

$\mu_{0}(B_{r}(0))=\sqrt{r},\qquad\mu_{0}(B_{r}(r))=\sqrt{2r},\qquad\mu_{0}(B_{r}(-r/2))=\sqrt{r/2}.$
(7.1)

Hence it is crucial which of the points $u$ and $v$ in the definition of modes
is/are allowed an approximating sequence:

  1. (a)

For the measure $\mu_{1}$ with $\alpha=\nicefrac{{1}}{{4}}$ the points $u=0$
and $v=2$ clearly dominate every other point in terms of small ball
probabilities (as they are the only singularities), so it is sufficient to
compare these two points. Since $\,\mathfrak{R}_{r}^{\mu_{1}}(0,2)=1$ for each
$r>0$, both are w-mode of $\mu_{1}$. However, for $v=2$ and for each
$\mathsf{ns}\ (r_{n})_{n\in\mathbb{N}}$ (without loss of generality we assume
$r_{n}<1$ for each $n\in\mathbb{N}$) and each $\mathsf{as}\
(u_{n})_{n\in\mathbb{N}}$ of $u=0$ (note that the constant $\mathsf{as}\
u_{n}=0$ of $u=0$ is always ‘optimal’), the $\mathsf{cs}\ v_{n}=2+r_{n}$ of
$v=2$ satisfies
$\,\mathfrak{R}_{r_{n}}^{\mu_{1}}(u_{n},v_{n})\leqslant\nicefrac{{1}}{{\sqrt{2}}}<1$.
Hence, $u=0$ is a w-mode, but not a wpag-mode of $\mu_{1}$.

  2. (b)

For the measure $\mu_{0}$ and certain mode types in Definition 4.2, $u=0$
dominates every other $\mathsf{cp}\ v\neq u$ but fails to dominate itself,
excluding such definitions by SP, as listed in Proposition 5.3.

  3. (c)

If CP holds and $u=0$ is a mode of $\mu_{0}$, then $u=2$ must be the only mode
of $\mu_{2}$. This excludes a large number of mode definitions from Definition
4.2, listed in Proposition 5.4.

##  8 Conclusion

Defining a mode for a probability measure without a continuous Lebesgue PDF,
e.g. in infinite-dimensional Hilbert and Banach spaces or other metric spaces,
is neither trivial nor unambiguous, and still a topic of current research.
This manuscript introduced a structured way of defining modes that
incorporates all previous definitions as well as over 100 alternatives that
are at least _grammatically correct_. In order to filter out those definitions
that are actually _meaningful_ , we introduced several common-sense axioms
that modes ought to satisfy — such as the correct identification of modes for
measures that possess atoms or a continuous Lebesgue PDF, as well as proper
behaviour under copying, shifting and taking averages of probability measures
on normed spaces. Excluding all definitions that violate these axioms for
$X=\mathbb{R}$ — except for the challenging ‘merging property’ (MP) — leaves
17 distinct meaningful definitions for a mode, some rather exotic, but none
objectively inferior to another at this point.

We established causal relations among these definitions and presented a large
number of illuminating examples and counterexamples, which hopefully will
shape the future development of the theory on modes and MAP estimators. In
particular, we have realised a new ‘exotic’ mode, the
$[\forall\mathsf{cp}\forall\mathsf{cs}\exists\mathsf{as}\forall\mathsf{ns}]$-mode,
located strictly between the strong mode of Dashti et al. (2013) and the weak
mode of Helin and Burger (2015) and Ayanbayev et al. (2022). Note that the
probability measures in all our examples could be constructed on
$X=\mathbb{R}$ and possess a (possibly discontinuous) PDF.

One of our proposed axioms, the MP, highlights a crucial shortcoming of all
extant definitions of modes: for open sets $A,B\subseteq X$ it can happen that
both $\mu|_{A}$ and $\mu|_{B}$ possess a mode while $\mu|_{A\cup B}$ has no
modes, which runs counter to our intuition about modes. Our own attempt to fix
this issue led to the definition of ‘partial modes’, which, unfortunately,
also do not satisfy the MP. In fact, none of the 17 meaningful definitions
that we identified satisfies the MP, leaving the fully satisfactory definition
of modes and MAP estimators as an open problem that deserves further research.

##  Appendix A Supporting technical results

######  Lemma A.1.

Let $(X,d)$ be a separable metric space, $\mu\in\mathcal{P}(X)$,
$u\in\operatorname{supp}(\mu)$ and let $(\Delta_{n})_{n\in\mathbb{N}}$ be any
null sequence. Then there exist null sequences $(r_{n})_{n\in\mathbb{N}}$,
$(\tilde{r}_{n})_{n\in\mathbb{N}}$ and $(\varepsilon_{n})_{n\in\mathbb{N}}$
such that

$\mu(B_{r_{n}-\Delta_{n}}(u))\geqslant(1-\varepsilon_{n})\,\mu(B_{r_{n}}(u)),\qquad\mu(B_{\tilde{r}_{n}+\Delta_{n}}(u))\leqslant(1+\varepsilon_{n})\,\mu(B_{\tilde{r}_{n}}(u)).$

###### Proof.

We will first show that there exist a null sequences
$(s_{k})_{k\in\mathbb{N}}$ and $(\tilde{s}_{k})_{k\in\mathbb{N}}$ and a
strictly increasing sequence $(n_{k})_{k\in\mathbb{N}}$ such that, for each
$n\geqslant n_{k}$,

$\mu\bigl{(}B_{s_{k}}(u)\setminus B_{s_{k}-\Delta_{n}}(u)\bigr{)}\leqslant
k^{-1}\mu(B_{s_{k}}(u)),\quad\mu\bigl{(}B_{\tilde{s}_{k}+\Delta_{n}}(u)\setminus
B_{\tilde{s}_{k}}(u)\bigr{)}\leqslant k^{-1}\mu(B_{\tilde{s}_{k}}(u)).$ (A.1)

Since $\mu$ is a finite measure, this holds trivially for $k=1$ with $s_{1}=1$
and $n_{1}=1$ and sufficiently large $\tilde{s}_{1}>0$, while for $k\geqslant
2$ we can construct $s_{k}$ and $n_{k}$ recursively: Set $\hat{s}=s_{k-1}/2$,
$\check{s}=s_{k-1}/4$, $Q=\mu(B_{\hat{s}}(u))/\mu(B_{\check{s}}(u))\geqslant
1$, $J=\lceil{Qk}\rceil$ and $\delta=(\hat{s}-\check{s})/J>0$. Since the rings
$A_{j}\coloneqq B_{\check{s}+j\delta}(u)\setminus
B_{\check{s}+(j-1)\delta}(u)\subseteq B_{\hat{s}}(u)$, $j=1,\dots,J$, are
disjoint,

$\sum_{j=1}^{J}\mu(A_{j})\leqslant\mu(B_{\hat{s}}(u))=Q\,\mu(B_{\check{s}}(u)).$

Hence, there exists $j^{\ast}\in\\{1,\dots,J\\}$ such that
$\mu(A_{j^{\ast}})\leqslant J^{-1}Q\,\mu(B_{\check{s}}(u))$. Choose
$s_{k}\coloneqq\check{s}+j^{\ast}\delta$ and
$\tilde{s}_{k}\coloneqq\check{s}+(j^{\ast}-1)\delta$. Further, since $u_{n}\to
u$ as $n\to\infty$, there exists $n_{k}\geqslant n_{k-1}$ such that, for each
$n\geqslant n_{k}$, $\Delta_{n}\leqslant\delta$. Since $J^{-1}Q\leqslant
k^{-1}$ and $B_{s_{k}}(u)\setminus B_{s_{k}-\Delta_{n}}(u)\subseteq
A_{j^{\ast}}$ as well as $B_{\tilde{s}_{k}+\Delta_{n}}(u)\setminus
B_{\tilde{s}_{k}}(u)\subseteq A_{j^{\ast}}$ for $n\geqslant n_{k}$, this
proves (A.1), (note that $(s_{k})_{k\in\mathbb{N}}$ and
$(\tilde{s}_{k})_{k\in\mathbb{N}}$ are null sequences since $s_{k}\leqslant
s_{k-1}/2$, $\tilde{s}_{k}\leqslant s_{k-1}/2$).

Now, choosing $r_{n}=s_{k}$, $\tilde{r}_{n}=\tilde{s}_{k}$ and
$\varepsilon_{n}=k^{-1}$ whenever $n_{k}\leqslant n<n_{k+1}$ for some (unique)
$k\in\mathbb{N}$, the above construction yields, for each $n\in\mathbb{N}$,

$\displaystyle\mu\bigl{(}B_{r_{n}}(u)\setminus
B_{r_{n}-\Delta_{n}}(u)\bigr{)}$
$\displaystyle\leqslant\varepsilon_{n}\mu(B_{r_{n}}(u))\text{ and}$
$\displaystyle\mu\bigl{(}B_{\tilde{r}_{n}+\Delta_{n}}(u)\setminus
B_{\tilde{r}_{n}}(u)\bigr{)}$
$\displaystyle\leqslant\varepsilon_{n}\mu(B_{\tilde{r}_{n}}(u)).$

Since

$\displaystyle B_{r_{n}}(u)$
$\displaystyle=B_{r_{n}-\Delta_{n}}(u)\uplus\bigl{(}B_{r_{n}}(u)\setminus
B_{r_{n}-\Delta_{n}}(u)\bigr{)}\text{ and}$ $\displaystyle
B_{\tilde{r}_{n}+\Delta_{n}}(u)$
$\displaystyle=B_{\tilde{r}_{n}}(u)\uplus\bigl{(}B_{\tilde{r}_{n}+\Delta_{n}}(u)\setminus
B_{\tilde{r}_{n}}(u)\bigr{)},$

this proves the claim. ∎

######  Lemma A.2.

Let $\mu\in\mathcal{P}(X)$. Then, for each $x\in X$ and $s>0$,

$\lim_{r\nearrow s}\mu(B_{r}(x))=\mu(B_{s}(x)),\qquad\lim_{r\nearrow
s}M_{r}^{\mu}=M_{s}^{\mu}.$

###### Proof.

Since $B_{r}(x)$ is monotonically increasing in $r$ with
$B_{s}(x)=\bigcup_{r<s}B_{r}(x)$, the first identity follows directly from the
continuity of measures. Now assume that the second identity does not hold,
i.e. there exists $\varepsilon>0$ such that, for each $r<s$,
$M_{r}^{\mu}\leqslant M_{s}^{\mu}-\varepsilon$. By definition of the supremum,
there exists $x_{\ast}\in X$ such that
$\mu(B_{s}(x_{\ast}))>M_{s}^{\mu}-\varepsilon$. Hence, using the first
identity, we obtain the contradiction

$M_{s}^{\mu}-\varepsilon\geqslant\limsup_{r\nearrow
s}M_{r}^{\mu}\geqslant\lim_{r\nearrow
s}\mu(B_{r}(x_{\ast}))=\mu(B_{s}(x_{\ast}))>M_{s}^{\mu}-\varepsilon.\qed$

######  Lemma A.3.

Let $\mu\in\mathcal{P}(X)$. Then, for each $r>0$, the map
$x\mapsto\mu(B_{r}(x))$ is lower semicontinuous on $X$, i.e.
$\liminf_{k\to\infty}\mu(B_{r}(x_{k}))\geqslant\mu(B_{r}(x))$ for any
convergent sequence $(x_{k})_{k\in\mathbb{N}}$ in $X$ with limit $x\in X$.

###### Proof.

See e.g. Lambley and Sullivan (2023, SM1.1). ∎

######  Lemma A.4.

Let $a,b,c,d>0$ and $f\colon(0,\infty)\to\mathbb{R}$ be given by
$f(x)\coloneqq\frac{ax+b}{cx+d}$. Then $f$ is strictly monotonically
increasing if and only if $ad>bc$.

###### Proof.

Note that $f$ is well defined, as its domain has been chosen to exclude the
singularity at $x=-\tfrac{d}{c}<0$. The derivative of $f$,

$f^{\prime}(x)=\frac{a(cx+d)-c(ax+b)}{(cx+d)^{2}}=\frac{ad-bc}{(cx+d)^{2}},$

is strictly positive for arbitrary $x\in(0,\infty)$ if and only if $ad-bc>0$.
∎

##  Appendix B Further plausible axioms for modes

Here we briefly sketch out some additional axioms or properties that one might
require a ‘meaningful’ definition of a mode to satisfy. The first two have a
geometrical flavour, asking that modes behave well under simple linear
transformations of the parameter space.

######  Definition B.1.

Let $X$ be a separable normed space. We say that a mode map
$\mathfrak{M}\colon\mathcal{P}(X)\to 2^{X}$ satisfies

  1. (a)

the homothety invariance property (HIP) if, for any $\mu\in\mathcal{P}(X)$,
$b\in X$, and $s>0$,

$\mathfrak{M}(\mu(s^{-1}(\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\displaystyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\textstyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\scriptstyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\scriptscriptstyle\bullet\;$}}}}}-b)))=b+s\,\mathfrak{M}(\mu);$

  2. (b)

the affine invariance property121212This property can be generalised to any
normed space $X$, in which case one would have to determine the class of
admissible operators $A\colon X\to X$, e.g. bounded operators with bounded
inverse. (AIP) if $X=\mathbb{R}^{d}$ and, for any $\mu\in\mathcal{P}(X)$,
$b\in X$, and invertible $A\in\mathbb{R}^{d\times d}$,

$\mathfrak{M}(\mu(A^{-1}(\mathchoice{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\displaystyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\textstyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\scriptstyle\bullet\;$}}}}}{\mathbin{\vbox{\hbox{\scalebox{0.5}{$\;\scriptscriptstyle\bullet\;$}}}}}-b)))=b+A\,\mathfrak{M}(\mu);$

HIP holds trivially for all definitions in Table 6.1, since they rely on small
ball probabilities. Therefore, in order to avoid any distractions, we did not
state these properties in Definition 5.1. AIP, on the other hand, cannot be
expected to hold for modes defined via small ball probabilities, since these
notions depend crucially on the chosen metric, which is modified non-trivially
by scaling with an invertible matrix, in particular under rotation (Lie and
Sullivan, 2018, Example 5.6). For this reason we did not study this property
here.

Another attractive property arises naturally in the context of Bayesian
inverse problems:

######  Definition B.2.

Let $X$ be a separable metric space. We say that a mode map
$\mathfrak{M}\colon\mathcal{P}(X)\to 2^{X}$ satisfies the reweighting property
(RP) if, for any $\mu,\nu\in\mathcal{P}(X)$ with $\nu\ll\mu$, the existence of
a mode of $\mu$ implies the existence of a mode of $\nu$, i.e.
$\mathfrak{M}(\mu)\neq\varnothing\implies\mathfrak{M}(\nu)\neq\varnothing$.

RP states that — for posteriors that are absolutely continuous with respect to
the prior — the existence of a prior mode implies the existence of a posterior
mode, i.e. of a MAP estimator; see e.g. Dashti et al. (2013), Klebanov and
Wacker (2023), and Lambley (2023) for treatments of Gaussian priors. To make
this an effective definition, one would also impose further restrictions on
the Radon–Nikodym derivative of $\nu$ with respect to $\mu$ which are typical
in the context of Bayesian inference; see e.g. (Stuart, 2010, Section 2,
Assumption 2.6). A thorough study of mode maps satisfying RP is beyond the
scope of this paper.

Finally, one could consider a formal restriction that $\mathfrak{M}(\mu)$
depends on $\mu$ _only_ through small ball probabilities $\mu(B_{r}(x))$ in
the limit as $r\to 0$:

######  Definition B.3.

Let $X$ be a separable metric space. We say that a mode map
$\mathfrak{M}\colon\mathcal{P}(X)\to 2^{X}$ satisfies the small balls property
(SBP) if, whenever $\mu,\nu\in\mathcal{P}(X)$ admit $R>0$ such that

$\mu(B_{r}(x))=\nu(B_{r}(x))\quad\text{for all $x\in X$ and $0<r<R$,}$