# What is the plausibility of probability?(revised 2003, 2015)

Stefan Arnborg
Royal Institute of Technology KTH
<EMAIL_ADDRESS>
Gunnar Sjödin
Swedish Institute of Computer Science
<EMAIL_ADDRESS>
###### Abstract

We present and examine a result related to uncertainty reasoning, namely that
a certain plausibility space of Cox’s type can be uniquely embedded in a
minimal ordered field. This, although a purely mathematical result, can be
claimed to imply that every rational method to reason with uncertainty must be
based on sets of extended probability distributions, where extended
probability is standard probability extended with infinitesimals.

This claim must be supported by some argumentation of non-mathematical type,
however, since pure mathematics does not tell us anything about the world. We
propose one such argumentation, and relate it to results from the literature
of uncertainty and statistics.

In an added retrospective section we discuss some developments in the area
regarding countable additivity, partially ordered domains and robustness, and
philosophical stances on the Cox/Jaynes approach since 2003. We also show that
the most general partially ordered plausibility calculus embeddable in a ring
can be represented as a set of extended probability distributions or, in
algebraic terms, is a subdirect sum of ordered fields. In other words, the
robust Bayesian approach is universal. This result is exemplified by relating
Dempster-Shafer’s evidence theory to robust Bayesian analysis.

Keywords. belief, plausibility, probability, Bayesianism

##  1 Introduction

We consider plausibility spaces of the type defined by Cox[13], with auxiliary
functions $F$, $S$ and $G$ used for computing plausibilities of conjunctions,
complements and disjunctions, respectively. The domain of plausibility values
with the auxiliary functions and some ordering relation $\leq$ is called a
plausibility space111We use the term space instead of measure or algebra in
order to avoid confusion with terminology in related work., where $a\leq b$
means that an event with plausibility value $a$ is equally or less plausible
than one with the value $b$. A system of conditional propositions or events
with plausibility values is a plausibility model 222Model is one of several
possible terms; it suggests the analoguous use of models in logic.. Standard
probability is one possible plausibility space, where the domain is the real
numbers in $[0,1]$ and the functions $F$ and $G$ are multiplication and
addition, respectively. Extended probability is standard probability extended
with infinitesimal probabilities.

We will motivate a definition, prove two theorems and justify a claim.
Definition 1 below of proper ordered plausibility spaces and models lists a
number of properties we assume for the domain of plausibility values and the
combination functions $F$ and $G$ used to find plausibilities of conjunctions
and disjunctions when we have the plausibilities of their operands. These
properties are similar to those assumed by Aczél and Cox. In particular, $F$
and $G$ satisfy the algebraic laws of the operators $\times$ and $+$ of a ring
and have certain monotonicity properties.

Theorem 1 says that a proper ordered plausibility space can be embedded in an
ordered field where the field operators $\times$ and $+$ are extensions of $F$
and $G$. An ordered field is a ring where certain monotonicity and solvability
properties hold, the best known examples being the fields of reals ${\bf R}$
and rationals ${\bf Q}$ . But there are more complex ordered fields that may
be relevant and which we will discuss.

In order to motivate Definition 1, we investigate several assumptions, some
old and some new, which state desirable properties of a plausibility space. We
argue for the acceptance of a number of these assumptions, which we call
Sufficiency, Monotonicity, Propositional limit, Refinability and Closedness.

Theorem 2 says that a plausibility space satisfying the above assumptions is a
proper ordered plausibility space as defined in Definition 1. Our claim is:

###### Claim 1

Theorems 1 and 2 imply that the most general rational way to deal with
uncertainty, under the assumptions made, is the Extended Bayes’ method, where
uncertainty is completely described with an extended probability distribution.
With the additional Robustness assumption, uncertainty is completely described
with a set of such distributions.

There are many results of the type above in the 20th century literature on
probability and uncertainty. Nevertheless, there is no consensus among
researchers in uncertainty management that Claim 1 is even approximately
valid. The Claim is of obvious interest as a foundational issue. But interest
is not confined to the ivory tower: Designers of future complex systems that
need new types of human-agent and agent-agent interactions struggle with this
question, and fielded solutions have already unraveled difficult compatibility
problems, e.g., when advanced system components using different, sometimes
incompatible, unknown or irrational ways to describe uncertainty are put
together to form systems of systems. If Claim 1 above can be accepted, this
task becomes easier, since questions can center around requirements on useful
probability models rather than comparing disparate schemes of uncertainty
management.

The auxiliary argumentation required to support our claims is similar to
inference principles in statistics, like the Sufficiency, Likelihood and
Conditionality principles[39]. Such principles have only a pragmatic validity,
meaning that their acceptability may be questioned in new applications
different from those conceived when they were first accepted. We will state a
number of assumptions made in derivations of this kind, and they are indicated
in bold face. Those that will eventually be retained in this study are marked
with an asterisk (*).

In section 2 we review the basic features of plausibility spaces and state a
number of common assumptions used to analyze them. In section 3 we review the
two basic methods used to derive probability as canonical uncertainty measure,
coherence arguments and Cox consistency based argument. In section 4 we
discuss a number of fundamental principles that are also required but which
are often hastily glossed over: these principles have the common feature that
they say that inferences possible should not depend in an arbitrary manner on
how our problem is embedded in a larger context. This discussion leads to
Definition 1. The consequences of Theorem 1 are discussed in section 5. In
section 6 we discuss extended probability. Theorems are proved in the
appendix. The scientific contribution claimed is the concept of embeddability
of the auxiliary functions in an ordered field, and its proofs. These are
somewhat more complex than we hoped, mainly because the function $G$ is a
partial function on the domain of plausibility values.

There are two papers that analyze a similar question and lead to different but
related and compatible conclusions, namely Kraft et al[27] and Hardy[22].

##  2 Symbols and Plausibilities.

The methods of definition and induction are attributed, by Aristotle, to
Socrates[46], and were developed further by Aristotle and the Stoic
philosophers. Although present English translations of Aristotle contain words
like symbol, utility and probability, it is clear that Aristotle was not very
interested in mathematics and does not give quantitative models of decision
problems. So his terms do not reflect a detailed and precise understanding of
modern Bayesian decision theory. In particular, the modern concept of
probability was certainly not known at the time. Nevertheless, Aristotle is
the first to give a surviving qualitative version of the Bayesian decision
making principle in his Nichomachean ethics[24]: “ Find out what you think is
a good life, and consider the probabilities of your possible actions to
achieve this. Then follow the course of action which with highest
probabibility results in a good life.”

Two thousand years later, Thomas Bayes was the first to apply the newly
invented probability calculus to an inference problem. In his posthumous essay
he computes the posterior probability for the probability of heads in tossing
a coin, with respect to observed outcomes and under the assumption that tosses
are independent and identically distributed, and that the prior probability of
getting a head is uniformly distributed between 0 and 1. Somewhat later,
Laplace gave a more satisfactory analysis of the Bayesian method.

Our modern plausibility models assume a set of events, or statements, or
possible world sets $A$, $B$, $C$, etc., which in an application will
correspond to conditions existing in the world or in the minds of humans and
agents. This step is completely standard. Many theories of plausibility go on
to define conditional event plausibilities, which in many cases are written
using the probability inspired notation $A|B$ \- the plausibility of $A$ given
that we know $B$ to be true333$A|B$ is thus the plausibility, not the
conditional event. It is sometimes written $pl(A|B)$. This is the model in
which foundational studies are usually made, and which we use here.

There are essentially two different methods that were used to analyse possible
plausibility spaces for consistency. The more common is based on a gambling
analogy: plausibilities are assumed analog to betting odds, and a situation
where a gambler can pick bets on a set of related events to hedge completely
the risk of loosing money is called incoherent. The plausibility assignment
leading to such a situation is considered incoherent, and under various
assumptions several arguments exist that end up in concluding that probability
or a set of probability distributions is the only coherent plausibility space
[41, 28, 15, 45]. We will use here the alternative approach devised by Cox[13]
which avoids the betting scenario and instead analyses how plausibilities of
combined events can be defined, and what properties the plausibility
combination functions must have in order to avoid that an events plausibility
will depend on which of several possible derivations is used.

Cox[13] assumes that the domain of plausibility values is an interval of real
values, which without loss of generality can be assumed to be $[0,1]$, with 0
for falsity and 1 for truth. He furthermore finds that two functions $F$ and
$S$ must exist such that $A\land B|C=F(A|B\land C,B|C)$ and $\lnot
A|C=S(A|C)$. With a number of – sometimes implicit – regularity assumptions,
he shows that the domain of plausibility values can be rescaled so that $F$ is
tranformed to multiplication, and $S$ to the function $S$ such that
$S(x)=1-x$. In other words, if his assumptions are accepted, plausibility
models must be equivalent to probability models. We can state an assumption
made implicitly by Cox here (the function $G$ was introduced by Aczél[1] in a
related investigation):

Sufficiency assumption*: The plausibility value of a proposition is a
sufficient characterization of the statement with respect to uncertainty and
propositional connectives, i. e., there are functions $F$ and $S$, and a
partial function $G$ such that $A\land B|C=F(A|BC,B|C)$,
$\overline{A}|C=S(A|C)$ and $A\lor B|C=G(A|C,B-A|C)$.

There has been some concern with alternative interpretations of symbols used.
In rough and fuzzy set theory[37, 51], the meanings of symbols are taken to be
ambiguous, in the sense that even with full information we cannot definitely
say, e.g., whether or not $A$ obtains or if $x$ is a member of set $X$. The
ambiguity problem has in many articles been claimed to be an obstacle to
probabilistic uncertainty management[49]. An alternative way is to see
ambiguity as an aspect that should go into the structure of a probabilistic
model. So instead of introducing $A$ as an objective fuzzy set, we can
introduce the judgments made by various agents and decision makers as
$A|C_{i}$ for the judgment made e.g., by agent $i$ based on the information
and background factors available to agent $i$. This makes fuzziness and
plausibility orthogonal concepts, and both can be present in an application.
Those who focus on impreciseness assume that this is the most prominent
characteristic of their applications, while those focusing on plausibility do
not think so.

Which is the domain $D$ of plausibility values? Fine[17, Ch 1] lists a number
of theories, which correspond to different order structures of the domain of
plausibility values. In short, the domain can be discrete (with true/false as
the case equivalent to propositional logic), ordered, continuous or partially
ordered. All applications of plausibility in real systems aim at decision
making, and for this reason we must eventually be able to say that one event
is more plausible than another one. For this reason it is essential to have
some order structure in the domain, and we rule out completely unordered
domains like the complex numbers. This means that the relation ’more plausible
than’ is transitive. This is not necessarily the case in practical decision
making but because of psychological effects like framing, unstable preferences
or computational limitations of the brain. Our study aims at prescriptive
theories so this problem will be ignored: the domain of plausibility values is
always (at least) partially ordered and usually linearly ordered.

There are several uncertainty management methods that make use of partially
ordered domains of plausibility values, typically in the form of intervals of
real numbers.

Robustness assumption*: A plausibility space with a partial order relation on
the domain of plausibility values, describing the relation ’more plausible
than’, is definable using indexed sets of plausibility spaces each having a
linearly ordered domain of plausibility values. In such a space, an event
$e_{1}$ is more plausible than $e_{2}$ if and only if it is more plausible in
each constituent indexed space.

We have not yet a compelling argument for the robustness assumption444see
however the restrospective section 8.2 – existing proposals have rather large
gaps. A possible approach is to develop a general theory of partially ordered
plausibility spaces. Until this has been done, we assume robustness and
consequently analyze models with a linearly ordered domain $D$ of plausibility
values.

Monotonicity assumption*:  The domain of plausibility is ordered, $S$ is
decreasing555Decreasing is a stronger condition than non-increasing. The
assumption could also have been called strict monotonicity. Despite the subtle
distinction, the weaker assumption would lead to completely different
conclusions[14] since non-strict monotonicity does not entail cancellation
laws., and $F$ and $G$ are increasing in each argument (in the case of $F$ if
the other argument is non-$\bot$).

By considering the limiting case of propositional logic we also find a number
of constraints based on the requirement that reasoning with entirely false or
true events should follow the rules of propositional logic. We omit a number
of rules which will follow from the algebraic rules of auxiliary functions $F$
and $G$ that we will soon adopt:

Propositional limit assumption*: $G(x,\bot)=x$, $F(\bot,x)=\bot$,
$F(x,\top)=x$, and $S(\bot)=\top$.

An immediate consequence of Monotonicity and Propositional limit is
$F(x,y)\leq\min(x,y)$ and $G(x,y)\geq\max(x,y)$.

##  3 Coherence, or Dutch Book avoidance

We have now stated the common assumptions of most existing theories of
plausibility[17, 15, 13, 41]. But more is required before probability appears
as inevitable in a form we recognize. What is required is some means to derive
constraints on sets of plausibilities of different conditional statements.
Without such constraints we can easily define plausibility spaces that are not
equivalent to probability. There are basically two ways to proceed. One
followed by de Finetti is to construct a gambling situation where different
plausibilities are tied together in composite bets offered by a bookmaker
according to his beliefs. A bookmaker who offers a set of bets among which a
gambler can choose a combination which gives positive payoff in every
situation seems to have made his bets from a globally incoherent set of
beliefs. Several papers show, with important differences in detail, that every
coherent belief set is equivalent to a probability model[15, 13, 41, 29]. The
coherence concept can also be applied to some domains not totally but
partially ordered, and Goodman, Ngyuen and Rogers[19] show that some
plausibilty measures not coherent under the total order assumption, among
others the standard version of DS-theory, are coherent under the latter
assumption.

Another way to connect different plausibilities is via propositional logic
connectives, and this gives us some constraints on the auxiliary functions
$F$, $S$ and $G$ used in Cox’s approach. As an example, the rule of
associativity for conjunction enforces $F$ to be associative for certain
arguments, namely, from $ABC|D=F(AB|CD,C|D)=F(A|BCD,BC|D)$ and $x=A|BCD$,
$y=B|CD$ and $z=C|D$, we can derive

$F(x,F(y,z))=F(F(x,y),z).$

In other words, under some circumstances associativity is inherited by $F$
from the associativity of conjunction. If we want to show that every good
plausibility space must be rescalable to probability, we must prove that $F$
and $G$ satisfy certain algebraic laws that are satisfied by $\cdot$ and $+$
and invariant under rescaling - the laws of associativity, commutativity and
distributivity satisfied by the operators of a ring. As an example, a model is
discussed by Halpern[20] that has only a few events. Thus the domain of
plausibility values is finite and associativity and differentiability of $F$
does not seem inevitable. In order to claim that the auxiliary function must
be associative, we must either claim that no sensible person would deny it,
which is difficult, or find some type of argument that goes outside the finite
example. In [13], it is plainly assumed (as pointed out in [47]) that the
auxiliary function $F$ has certain regularity properties:

Associativity and differentiability assumption: The function $F$ is defined on
$[0,1]^{2}$, and is associative and twice continously differentiable; the
auxiliary function $S$ is continuously differentiable and $S(S(x))=x$

The differentiability part is implicit – as was common when Cox wrote his
paper – but is a standard assumption made to justify switching the order of
differentiation which occurs in the derivations of [13], and also in [25].

##  4  Embeddability, Denseness, Refinability.

The argumentation reviewed in section 3 must be complemented by some arguments
that bind together plausibilities of events that are not connected by
propositional identities. In Cox’s work, this is effected by regularity
assumptions: the auxiliary functions $F$ and $S$ are assumed to be defined on
an interval of real numbers, and obey the associativity and differentiability
assumption on this interval. Implicit in this assumption seems to be the idea
that the functions $F$ and $S$ are universal, and every plausibility problem
should be solvable with the same functions $F$ and $S$. In the criticisms of
Cox’s work, this assumption has either been assumed not to exist[20, 21] or
been ignored[36]. The statistics based derivations of probability as universal
uncertainty measure often mention similar assumptions, but surprisingly often
these are somewhat glossed over, the main exceptions being [9, 17, 41, 49]. We
make a sketchy review of some of the assumptions that have been proposed. In
the prevision-based analyses, the first methods were based on assuming that an
arbitrarily fine partitioning of the events space exists; this allows one to
deduce the existence of a probability measure based on a relation $\leq$ on
events with the meaning $A\leq B$ if $A$ is equally or less plausible than
$B$.

Uniform partition assumption[15]: For every event $B$, and every number $n$,
there is a partition of $B$ into $n$ equally plausible events.

Almost uniform partition assumption[41, Ch 3]: For every event $B$, and every
number $n$, there is a partition of $B$ into $n$ events, such that for
$r=1,\ldots,n-1$, the union of any $r$ elements of the partition is less
plausible than the union of $r+1$ members.

Standard events and precise measurements assumptions[9, Ch. 2.3, Ax. 4,5]: For
every conditional event there exists a standard event with the same
plausibility, and for every number $r\in[0,1]$ there is a standard event with
probability $r$.

Embeddability assumption[49]: Inference should not depend on how the events
participating in calculations are embedded in a larger event plausibility
system.

An alternative assumption used to derive Cox’s result is given by Paris[36]:

Denseness assumption[36]: For all real values $x$, $y$, $z$ and each
$\epsilon>0$, there are events $A$, $B$, $C$ such that $|(A|BCD)-x|$,
$|(B|CD)-y|$ and $|(C|D)-z|$ are all less than $\epsilon$.

This assumption might be more motivated than Cox’s original assumptions,
although it does not seem to be weaker. In response to [20], we developed very
weak assumptions binding the different parts of a system of events together[4,
5].

Refinability assumption*[4]: In a plausibility model with a conditional event
of plausibility $p$, it must be possible to introduce a new subcase $B$ of a
non-false event $A$ with plausibility value $p$ given to $B|A$. This means
that it should also be possible to define a new subcase of an event and get a
new model which is equivalent to the original one as long as no reference is
made to the new event. If two new subcases $B$ and $B^{\prime}$ of an event
$A$ are defined in this way, they can be specified to be information
independent, i.e., $B|B^{\prime}A=B|A$, $B^{\prime}|BA=B^{\prime}|A$. For two
plausibility values $x$, $y$ such that $x<S(y)$, it should be possible to
define two new subcases $C$, $C^{\prime}$ of any non-false event $A$ such that
$x=C|A$, $y=C^{\prime}|A$ and $C\land C^{\prime}|A=\bot$.

These augmentations of the model are called refinements. A refinement just
introduces an already existing plausibility value in ’another part’ of the
model. This means that there is no assumption of a ’dense domain’ or that a
given new plausibility value can be introduced. The motivation for
refinability is that the same cognitive or other process that resulted in a
particular plausibility value for one event can always be mirrored in another
part of the model to produce the same plausibility value for another event. If
this process introduces an inconsistency, then we feel that there is a basic
shortcoming of the model, and we should not accept it. We claim that this
assumption is weaker than those referenced above in this section, and also
more motivated. This is the weakest condition we have found that ensures that
the auxiliary functions $F$, $G$ and $S$ have enough algebraic properties –
associativity, symmetry(i.e., commutativity for corresponding binary
operators) and distributivity properties of + and $\cdot$ in a ring – to
ensure that the main theorem follows. Somewhat surprisingly, Cox’s result on
rescalability of the plausibility space to standard probability follows from
refinability and the assumptions in section 2 for models that have a finite
number of plausibility values and an ordered plausibility space[4]. Even more
surprisingly, the same result cannot be proved for infinite plausibility
spaces (non-provability follows from a counter-example[5]). Two responses to
this are possible: either more assumptions are required, or the result would
not be right. We choose the second alternative. The reason is that we want to
extend probability to extended probability, where infinitesimal probability
values are allowed. This idea goes back to Adams[2] and has been found to give
a very basic uncertainty management that seems to incorporate many uncertainty
calculi as special cases (several are analyzed using extended but not
necessarily Bayesian probability in [7]). The combination of robust and
extended probability has been analyzed by Wilson[50]. Our motivation for
accepting extended probability is the existence of infinite plausibility
models that are refinable and embedded in the real numbers, but whose
plausibility spaces are not rescalable to standard probability[5].

In order to derive extended probability as a canonical plausibility space, we
must assume that models can be refined to the limit:

Closedness assumption*[5]: The functions $F$, $S$ and $G$ have the following
additional properties:

$F:D\times D\rightarrow D$, $S:D\rightarrow D$, $E=\\{(x,y)\in D\times D:x\leq
S(y)\\}$ and $G:E\rightarrow D$.

In other words, the functions $F$ and $S$ are total while $G(x,y)$ is defined
when $x\leq S(y)$. This requirement is similar (but not equivalent) to
Kolmogorov’s insistence that a probability space is defined on a
$\sigma$-algebra. Refinability would imply that $F$ and $G$ must be
associative and symmetric, and also that $F$ must distribute over $G$ on the
domain of plausibilities (since $G$ is a partial function, we must moderate
its laws by only requiring that both sides of its associativity and symmetry
equations are equal when one of them has a defined value). These algebraic
properties will simply be inherited from the corresponding properties of
$\land$ and $\lor$. We can now see that Definition 1 of a proper ordered
plausibility space is relevant in the sense that we must work with such spaces
if we accept the general framework of section 2, and if we also accept the
refinability and closedness assumptions. If we accept the Robustness
assumption we can concentrate on ordered spaces.

###### Definition 1

A proper plausibility space is a seven-tuple $(D,F,G,S,\leq,\bot,\top)$, where
$D$ is a partially ordered domain with smallest value $\bot$ and largest value
$\top$, $F:D\times D\rightarrow D$, $S:D\rightarrow D$, $E=\\{(x,y)\in D\times
D:x<S(y)\\}$ and $G:E\rightarrow D$. Moreover, $F$ and $G$ are symmetric and
associative, $F$ distributes over $G$, $F$ and $G$ are increasing in their
arguments and $S$ is decreasing. Additionally, $G(x,\bot)=x$,
$F(\bot,x)=\bot$, $F(x,\top)=x$, $S(S(x))=x$, and $S(\bot)=\top$. A proper
ordered plausibility space is a proper plausibility space where $\leq$ is a
total ordering.

###### Theorem 1

A proper ordered plausibility space can be uniquely embedded in a minimal
ordered field where multiplication and addition are extensions of $F$ and $G$,
respectively.

Proof. See appendix.

###### Theorem 2

Assume that a plausibility model and its plausibility space satisfies the
assumptions of Sufficiency, Monotonicity, Propositional limit, Refinability
and Closedness. Then the plausibility space is a proper plausibility space.

Proof. See appendix.

We can now see that Theorem 2 leads us to concentrate on proper plausibility
spaces, and that with the Robustness assumption Claim 1 should follow from
Theorem 1.

##  5 Common sense assumptions entail extended robust probability.

The best known ordered fields are ${\bf Q}$ and ${\bf R}$. However, there are
ordered fields that contain more than the real numbers. One example is the
field of rational functions $R(\epsilon)$ in one variable $\epsilon$. These
functions can be added to and multiplied with each other, and if they are
ranked by the lexicographical ordering of their values and derivatives of
order 1, 2, … at $\epsilon=0$, we get an ordered field. This field contains
all real numbers (as constant functions) but does not admit lowest upper and
greatest lower bounds, and thus cannot be embedded in the field of reals. The
variable $\epsilon$ can be regarded as an infinitesimal, positive but smaller
than every positive real number. There are many ordered fields that are
superfields of the reals, and there is even a unique maximal ordered field No,
described by Conway[12]. This field consists of reals, infinite numbers and
infinitesimals. There is an infinitesimal element (a non-zero element smaller
than every real number) for every infinite ordinal and transfinite number,
namely its inverse. The field No has an extremely complex structure and cannot
be easily represented in a computer. The concepts of infinitesimals and
infinite numbers in non-standard analysis[40] are closely related to extended
probability. However, non-standard analysis has a set-model-theoretic basis
whereas our basis is classical algebra, and it is not completely obvious that
the two concepts are exactly identical.

The above leads us to a belief structure where belief is an extended
probability value. Many arguments have been forwarded for belief to be modeled
not by an ordered set, but a partially ordered set like a set of intervals.
Because of the subtle properties of belief, it is difficult to resolve what
the right thing is in a convincing way. The preceding analysis started out
with the assumption that belief values are totally ordered, and cannot get to
other conclusions. However, assume that the possible plausibility values form
a partially ordered set, and accept the Robustness assumption of section 2.
This partially ordered set can then be regarded as a set of different contexts
(e.g., different unfused expert assessments) of linearly ordered plausibility
models. For each linearly ordered domain belonging to an indexed member ,
Theorem 1 is applicable, and thus a plausibility model can be fully
characterized by a set of extended probability distributions, one for each
index. One way to fuse such sets of distributions is to form a weighted
average of the distributions, but it is not clear which principles such fusion
should be based on.

##  6 Pragmatics

If we design a methodology where users are permitted to define epistemic
states as sets of extended probability distributions we will soon find us in a
situation where these users are encouraged to define things they cannot
possibly understand. Extended Robust Bayes allows us to construct extremely
complex descriptions of systems, where instead of points we must work with
polytopes in high-dimensional or even infinite-dimensional spaces. Even simple
things cannot easily be done with such models[3]. The most interesting aspect
of the above analysis is that it indicates a possible maximal generality in
uncertainty management, but in every particular application this generality
must probably be pruned. Is the full generality really required? There are
strong claims in the literature that it is not. There are also strong claims
that conventional Bayesianism, where only one probability distribution is used
and where probabilities are standard, is inadequate. So, a pragmatic approach
would be to investigate claims made, and see how reasonable they are. In this
investigation there seems no definite need to distinguish human from machine
decision making – most requirements we find are obviously applicable to human
and machine alike, particularly when we want machines to interact with humans
in terms of beliefs and intentions.

###  6.1 Extended Probability

Is extended probability needed or is standard probability adequate? Extended
probability is not completely unavoidable, in the sense that every finite
extended probability model is equivalent to a standard probability model[4].
But in some cases extended probability seems useful as a pragmatic
simplification of a modeling problem and obtaining natural problem
descriptions. It seems clear that there is a phase in cognitive assessments
where qualitative and order-of-magnitude reasoning is done[26], followed by a
quantitative phase where more quantitative reasoning occurs. In AI reasoning
research, many non-probabilistic methods have been proposed, and it seems as
if many of them can be described in terms of extended probability[7, 8]. Non-
standard analysis approaches have been advocated as an alternative to measure-
theoretic ones in stochastic processes. Although no immediate reaction
followed from the user communities, this is still a promising direction[34]. A
number of assumptions have been proposed in the literature for arriving at the
inevitability of standard probability and thus the exclusion of extended
probability:

Real valued assumption: Assume (with, e.g, Cox) that plausibility is real
valued.

Archimedean assumption: Fine[17] assumes that for every non-zero probability
$e$, with $ne=G((n-1)e,e)$ and $1\cdot e=e$, there is an $N$ such that
$Ne>S(e)$. This assumption is introduced to step from comparative to standard
probability.

Separability assumption: Arnborg and Sjödin[5] define a separable model as one
in which, for every $x<y$ and $c$, there are $n$, $m$ such that
$x^{n}<c^{m}<y^{n}$, with $x^{n}=F(x,x^{n-1})$ and $x^{1}=x$. This is
introduced as a weaker assumption than continuity of the auxiliary function
$F$ introduced in [1].

With the hindsight given by Theorem 1, these assumptions appear only as
alternative ways to say that we do not accept infinitesimal probabilities.
None of these assumptions are very compelling except possibly the first, if
plausibility values are used in an application strictly using expected utility
decision making. In this case plausibilities separated by an infinitesimal
amount cannot be distinguished and can be considered equivalent.

##  7 Summary and Conclusions

Many of the objections to Bayesian methods come from the sometimes very
complex analytical models preferred by theoretical statisticians, and many
practitioners have seen alternative methods like neural networks as a way to
by-pass statistics. Unfortunately (or fortunately) this is to a large extent
an illusion. There is no principled reason why models used could not be based
on other types of models like neural networks, linguistic coding or case based
reasoning, and indeed efforts have been made to view these techniques as
special types of Bayesian models[23, 38]. As detailed above, there is no
simple way around the normative claims of the Robust Extended Bayes’ method.
So if one uses an alternative method, it will sooner or later have to be
evaluated against the standard scale of rationality. Bayesian methods contain
a large amount of freedom in the sense that there are no ’correct’ models or
model sets on which to base conclusions about real world or inner world
phenomena, but models must be chosen and tested against requirements of
applications. This makes the Robust Extended Bayesian method itself more or
less impossible to falsify - but so are other very basic methods like
arithmetic.

Is it now possible to answer the question in the title of this paper? Even a
subjectivistic probabilist would hesitate, but mainly because the question is
a little ambiguous. Our answer is that extended robust probability is
completely plausible as a universal uncertainty or belief measure until an
example is given where some of the starred assumptions above can be
demonstrated dubious.

## Acknowledgments

The reviewers of previous versions of this contribution have influenced it
significantly, by pointing out questionable statements (some now deleted,
others elaborated) and several relevant related papers in many disciplines.

SA is indebted to David Draper for discussions of the importance of and
approach to including countable additivity in the Cox/Jaynes framework, which
also made me finalize this unpublished manuscript and add a retrospective
section below (after the bibliography).

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##  8 Retrospect

Various versions of this manuscript were accidentally indexed by search
engines and are cited by other papers. This is the last 2003 version, with
very few changes made in 2015: typos removed and a few too short arguments
elaborated. Nothing reflecting developments since 2003 has been added, except
in this retrospective section. Several aspects of possible polishing of
Cox/Jaynes’s development have been published, for a survey see the
introductory sections of [48]. Apparently, little effort has been spent on
analysing extended probability and partially ordered plausibility domains, but
there are a few results to review. I will only mention those results easily
discussed in our algebraic framework. Some papers argue that we have made
above quite a number of assumptions compared to other papers in the same area.
But this is, on closer reading, only an effect of our need to be explicit
about assumptions and to discuss several alternative assumption sets. Counting
assumptions as axioms is a bit misleading since we actually want compelling
axioms, not minimum number of them. The shortest axiom set would be our Claim
above…

###  8.1 Countable additivity

In the paper by Terenin and Draper[48], a new aspect is taken up, namely
countable additivity of probabilities which has several equivalent
definitions, the one coupled to its name is that for all events C and all sets
of mutually disjoint events $\\{E_{i}\\}$, we must have
$\Sigma_{i}P(E_{i}|C)=P(\bigcap_{i}E_{i}|C)$. Yosida and Hewitt, however,
define countable additivity as the condition that for every event $C$ and
sequence $(A_{i})$ such that $A_{i+1}\subset A_{i}$ and
$\bigcap_{i}A_{i}=\emptyset$, we also have $\lim_{i}P(A_{i}|C)=0$, and they
claim it trivially equivalent to the former definition. The latter seems more
understandable than the first, and could possibly serve as a ’compelling
assumption’ in the Cox/Jaynes framework, as is approximately the case in [48].

Although one normally, in applications where the question arises, just assumes
or postulates that this assumption is fulfilled (as among others Kolmogorov
did), it actually does not have to be true for infinite sets of events (but it
must be true when $\\{E_{i}\\}$ is a finite set, a property called finite
additivity), this being a consequence of somewhat counter-intuitive measure-
theoretic considerations. A simple example of a probability distribution
violating countable additivity is obtained by starting with a uniform
distribution over integers $1$ to $N$ and letting $N$ go to $\infty$. Then
every probability of an integer goes to $0$ but all the sums are $1$. The
limit of the sums is thus $1$ while the sum of the limits is $0$. One common
way to handle this is to claim that the limit, an improper prior, is not a
probability distribution, despite the fact that unnormalized improper priors
are sometimes (actually, quite often) used ’as if’ they can be normalized. If
the likelihood is enough concentrated, the posterior obtained by applying
Bayes rule can be normalized and thus gives an inference, although it is clear
that there are some problems with this approach. An improper prior, however,
contains more significant information than the normalized finitely additive
probability distribution, as can be seen in that the standard improper priors
(uniform, Jeffrey’s, etc) are identical as probability distributions: they all
have density zero (except, for Jeffrey’s prior, at zero). However, in the
framework of extended probability they are proper distributions with
infinitesimal densities. The question whether probabilities in infinite spaces
should be countably or just finitely additive has been debated a lot, while in
most applications one just assumes, with Kolmogorov, countable additivity. The
Cox/Jaynes framework is explicitly based on what Jaynes refers to as common
sense, but the question of countable/finite additivity has not previously
(before Terenin and Draper) been analysed within this framework. Both Cox and
Jaynes most likely regarded this as a non-issue, and for non-parametric
Bayesian inference it is clear that current practice is based on the
assumption of countably additive probability. Considering the current
situation, we have here a case were researchers have not quite agreed about
what common sense requires. From an engineering perspective the most difficult
to swallow property of spaces not fulfilling countable additivity is known as
non-conglomerability (de Finetti[15]). Non-conglomerability can be defined in
terms of plausibilities: For an infinite partition $\\{E_{i}\\}$ of $E$ and an
event $C$, it is not necessarily the case that $\inf_{i}C|E_{i}\leq
C|E\leq\sup_{i}C|E_{i}$. Although my own reaction to this, both immediately
and after considerable reflecting over it, is that non-conglomerability is
absurd and must be rejected, the analyses of consequences of non-
conglomerability are sometimes just mentioning that the concept invalidates
some familiar inference methods. For example, it is not permitted to conclude
that $C|E$ is a weighted average of the $C|E_{i}$ for an infinite partition
$\\{E_{i}\\}$ of $E$, in other words among the $C|E_{i}$ we will not
necessarily find both numbers not less than and numbers not greater than
$C|E$. Is my reaction to this a sound reaction or just unwillingness to change
habits?

If non-conglomerability is rejected and conglomerability is accepted as a
common sense assumption, we can only end up with a system of plausibilities
that can be rescaled to a system of probabilities that has to be
conglomerable. Although it is relatively easy to see that a countably additive
probability space is conglomerable, the reverse is also true but not so easy
to see: a conglomerable probability space is countably additive. This was
proved in [42], so conglomerability entails countable additivity, at least for
real valued probability systems. This is an alternative axiom to that
presented in [48], maybe also more compelling. What happens in extended
probability has to be investigated, since the existing derivation relies in
many places on standard probability, where, for example, least upper bounds
exist.

###  8.2 Partially ordered domains and robustness

Concerning partially ordered plausibility domains, little progress has been
made. However, Jörg Zimmermann in his thesis [52] shows that a partially
ordered plausibility space is embeddable in a partially ordered ring, and
shows, assuming that this ring has a greatest ordered subfield called backbone
(and satisfies additional technical constraints), that any plausibility value
$r$ can be uniquely decomposed into $s+a\cdot t$, where $s$ and $t$ are from
the backbone and $a$ is interactive, which means that $a$ is incomparable to
elements between 0 and 1. He also shows that DS-theory is interpretable in
this way and clearly robust probability also fits the model. In robust
probability the plausibility domain is a vector (or indexed set) of
probabilities and the backbone elements have all components equal. The domain
is a ring, not a field, since it has zero divisors. The interpretation of the
decomposition here is that $s$ is an estimator of the probability (actually
the min probability), $t$ is a measure of the ’uncertainty of the probability’
(maybe $s+t/2$ is a better estimator of the probability), and $a$ gives a kind
of ’profile’ characterizing the deviation of the uncertainty from the
estimated probability. This is quite nice, but the assignment of the estimate
to every plausibility in the model does not itself give a coherent probability
assignment. It is also not clear that the mentioned subfield always exists and
thus a general understanding of partially ordered plausibility has not yet
been obtained, in particular we do not know what weight the robustness
assumption carries. Zimmermann, in his ring theorem, clearly makes an
assumption slightly stronger than we have done, in the axiom
$\mathbf{And}_{2}$. This entails the cancellability law for + of a ring, and
we did (probably) not make enough assumptions to get cancellability. His
assumption is clever and plausible, how compelling it is I cannot yet quite
see. We could prove cancellability for $G$ only for totally ordered
plausibility domains. Without some type of cancellability assumption it is
difficult to characterize the possible plausibility domains, and in particular
to develop the algebraic approach. Even with the assumption it is difficult to
get a real grip on what the most general such domains mean in practical terms.
Ring structures are quite flexible. There is a fair number of in-depth
treatments of partially ordered rings, the more recent motivated by its
relevance to real algebraic geometry, but as far as I know they have not yet
been seriously applied to the philosophical study of plausibility. It seems as
if Zimmermann’s assumption ($\mathbf{And}_{2}$) is a good starting point, and
by his ring theorem the plausibility domain is then a partially ordered ring.
He calls rings suitable for plausibility calculus $c$-rings. Now, the $c$-ring
does not have to contain a field (the backbone) under assumptions made, since
the ring of integers is a counterexample, and powers (repeated products) of
this ring are others. But if there is a (non-integer) rational number (a
rational number is a ring element $f$ such that the $n$-fold sum of $f$:s
equals $m$ for some integers $n$ and $m$) in the $c$-ring it must also contain
lots of rational numbers since the ring is closed under addition, subtraction
and multiplication and this subring of rationals seems a good approximation to
a field, the backbone. Moreover, any ordered field can be added to any
partially ordered ring, creating a $c$-ring with a backbone from a $c$-ring
without one. So the decomposition theorem of Zimmermann holds for all
$c$-rings, even if some $c$-rings may not have interesting decompositions. A
second question raised by the referenced thesis is the structure of $c$-rings.
Our robustness assumption can also be formulated as a conjecture:

Conjecture: Every $c$-ring is embeddable in a product of ordered fields.

Considering the intense studies made of rings and fields, one would expect the
literature to contain the verdict on this conjecture. And it seems to do, but
the standard algebraic operating procedures makes it more appropriate to say
that every c-ring is a subdirect sum of fields:

####  8.2.1 Representing rings as subdirect sums

The problem of representing a complicated ring as a product of simpler rings
has been thoroughly studied in [10] and [32, 31, 33]. A product ring
$P=S\times T$ of two rings $S$ and $T$ has a carrier that is the Cartesian
product of those of the factors $S$ and $T$, and the operations are defined
component-wise, thus, for example, if $S$ contains $s_{1}$ and $s_{2}$ and $T$
contains $t_{1}$ and $t_{2}$, then $P$ contains $(s_{i},t_{j})$ for $i=1,2$
and $j=1,2$. Moreover, $(s_{1},t_{1})\cdot(s_{2},t_{2})=(s_{1}\cdot
t_{1},s_{2}\cdot t_{2})$ and
$(s_{1},t_{1})+(s_{2},t_{2})=(s_{1}+t_{1},s_{2}+t_{2})$. Two observations: for
some, probably historical, reason, our main tool is actually a product but is
called subdirect sum. Another important thing to note is that the product
construction always introduces zero divisors, so the product of two integral
domains is never itself an integral domain: $(s,0)\cdot(0,t)=(0,0)$ regardless
of $s$ and $t$, and $(0,0)$ is the zero element of the product $P$ since it is
the unit for $+$. Therefore, a cancellation law for $\cdot$ never holds in a
product or subdirect sum. By a subdirect sum S of rings $\\{S_{i}\\}_{i\in I}$
(where the index set $I$ is not necessarily finite or countable) we mean a
ring isomorphic to a subring $P$ of $\prod_{i\in I}S_{i}$, and such that the
projection of $P$ on component $i$ is exactly $S_{i}$ (i.e., the homomorphism
is onto). Moreover, we require the representation to be non-trivial, i.e.,
none of the subrings $S_{i}$ is allowed to be isomorphic to $P$. If such a
non-trivial representation exists, the ring is called reducible. If no such
representation exists, the ring is irreducible. Note that an irreducible ring
is not the same as a reduced ring; the former can not be split into a
subdirect sum whereas a reduced ring is a ring that has no nilpotent elements.

Theorem: [10] Every reducible ring is a subdirect sum of irreducible rings.

What makes this relatively simple theorem interesting is the connection to
fields :

Theorem[33, Theorem 3.14] Every subdirectly irreducible ring with no nilpotent
elements is a field.

This is as close as we can get using the literature on rings to deriving the
robustness assumption (In [32, Theorem 4], the corresponding statement was
that every reduced (in the sense of absence of nilpotent elements) and
reducible ring is a subdirect sum of integral domains, a slightly weaker
result). The question now is: is there some good argument to reject factors in
the subdirect sum with nilpotent elements, which would recover the robustness
conjecture? Yes, there is. If a factor in a subdirect sum $P$ has a nilpotent
element then $P$ also has a nilpotent element and there is a nilpotent element
in the plausibility domain: If $S_{i}$ has the nilpotent element $a$, then the
product has the nilpotent element $(0,\ldots,0,a,0,\ldots,0)$ with only one
non-zero component. If there is a nilpotent plausibility value then there is
also $a\neq 0$ such that $a^{2}=0$. Use the refinement assumption to define
two new subevents $B_{1}$ and $B_{2}$ of $A$ such that $B_{1}|A=B_{2}|A=a$ and
such that they are information independent: $B_{2}|B_{1}A=a$. Now $B_{1}$ and
$B_{2}$ are both possible ($a\neq 0$) in the context of $A$, but their
combination in the context $A$ has plausibility $a^{2}=0$ and is impossible.
This contradicts the assumption that the events are information independent:
since $B_{2}$ is possible in the context $A$ and information independent of
$B_{1}$, it should also be possible in the context $AB_{1}$. We conclude that
the ring is reduced, i.e., lacks nilpotent elements. Since the decomposition
of a reduced ring will result in a subdirect sum of fields, and these will be
partially ordered by assumption, we can use the fact([43]) that a partially
ordered field can always be totally ordered and thus is isomorphic to one of
the fields between $Q$ and No, to recover the robustness conjecture.

We have just informally proved:

Robustness Theorem: A plausibility calculus embeddable in a partially ordered
ring is, under the refinability assumption, equivalent to a subring of a power
$\mathbf{No}^{X}$, for some set $X$. The domain of this calculus is the set of
maps from $X$ to subfields of $\mathbf{No}$ and the operations $\cdot$ and $+$
are defined component-wise using the corresponding field operations.

The question remains of course how this set of probability assignments should
be interpreted. Since the domain is a direct sum of ordered fields, there is
only one partial order to choose, where $a\geq b$ if and only if $a_{i}\geq
b_{i}$ for all $i\in X$.

The assumption is that event $A$ is more plausible than $B$ if each
distribution in the index set says that $A$ is more plausible than $B$. Our
assumptions do not let us conclude that the lower and upper envelopes have the
specific meaning of lower and upper limits of an unknown probability
distribution, although this interpretation is done in robust Bayesian
inference. However, the domain values having all components equal represent
obviously precise probabilities (as well as the backbone of [52]) and an event
$B$ with imprecise probability (not all components equal) is more plausible
than an event $A$ with precise plausibility if every component of $B$ is more
plausible than $A$, and thus an upper and by symmetry lower plausibility is
indicated for the maximum and minimum components plausibilities. The robust
Bayesian interpretation of domain values is thus justified, and Zimmermann’s
decomposition $p=s+a\cdot t$ contains indeed the lower and upper probabilities
of $p$, namely $s$ and $s+t$. There is a small complication in that sets of
extended probabilities are sets of hyperreal numbers which do not necessarily
have tight lower and upper bounds. The standard way to define infemum and
supremum of such sets is to use the standard part of the numbers, although we
have not investigated the appropriateness of this method in this particular
application.

####  8.2.2 Robustness examples

Since the robust Bayesian representation seems universal one can ask why there
are other uncertainty management schemes than robust Bayesian analysis, or
rather how these fare with respect to the Robustness theorem. The robustness
theorem says that a problem where uncertainty is present should be modeled
with a set of conditional probability assignments, and that these have the
interpretation of possible probability distributions. This seems in agreement
with how robust Bayesian analysis is performed and interpreted. Zimmerman
considers also other uncertainty management schemes, and observes that the
system of lower probabilities violates his ring theorem. This is because lower
and upper probabilities are insufficient for recovering a set of
distributions: in the lower and upper envelopes of a set of distributions the
latter are mixed up and there is no way to recover the set of distributions
from which the upper and lower probabilities were generated. This has the
consequence that the calculus cannot be embedded in a ring and there is no
useful definition of conditional plausibility. This appears as a weakness or
at least inconvenience of the method.

A third method, probably the most serious competitor besides Bayes, is
Dempster-Shafer evidence theory, DS-theory[16, 44]. Here, an assessment is
described as a ’body of evidence’ or mass assignment, and this is a random set
over the frame of discernment $\Omega$ which in turn is a partition of the
universe of discourse. A generative model of the outcome given a body of
evidence is that an outcome of the random set is generated and then one member
of the resulting set is chosen arbitrarily. If the random set has non-zero
probability only for singleton sets there is no arbitrary choice and we have a
precise Bayesian probability model. Zimmerman observes that a body of evidence
can be translated to a set of distributions and this is a good beginning.
However, the DS-theory also has the peculiarity that bodies of evidence that
are independent (of which apparently no precise characterization has been
given) should be combined with random set intersection (conditioned on being
nonempty), the Dempster’s rule. Dempster’s rule is different from the robust
Bayesian combination rule. A very useful example of the difference was
presented by Gelman[18]: it involves a match between a boxer and a wrestler,
the outcome of which is entirely uncertain, and a coin flip which is a
standard random event with probability 0.5 for each outcome. Here the events
are: $B$: boxer wins; $\overline{B}$: wrestler wins; $C$: heads up;
$\overline{C}$: tails up. The frame of discernment is
$\Omega=\\{BC,B\overline{C},\overline{B}C,\overline{B}\overline{C}\\}$. The
total uncertainty of $B$ is expressed with the vacuous mass assignment
$[m_{1}(\\{BC,B\overline{C},\overline{B}C,\overline{B}\overline{C}\\})=1]$, a
random set whose outcome is always $\Omega$. The information on the coin flip
is expressed by the body of evidence
$[m_{2}(\\{BC,\overline{B}C\\})=0.5;m_{2}\\{B\overline{C},\overline{B}\overline{C}\\})=0.5)]$,
a random set whose outcome is $\\{BC,\overline{B}C\\}$ or
$\\{B\overline{C},\overline{B}\overline{C}\\}$, each with probability 0.5.
Gelman now goes on and adds the evidence that an impartial and trustworthy
person reports that the two events had identical outcome (either $BC$ or
$\overline{B}\overline{C}$ happened) This can be encoded as the body of
evidence $[m_{3}(\\{BC,\overline{B}\overline{C}\\})=1]$. Combining, using
Depster’s rule, the bodies of evidence $m_{1}$, $m_{2}$ and $m_{3}$, gives the
random set $[m(\\{BC\\})=0.5;m(\\{\overline{B}\overline{C}\\})=0.5)$, in other
words the outcome is that of the standard coin flip. On the other hand, if the
bodies of evidence are translated to sets of distributions and combined using
Laplace’s combination (component-wise multiplication followed by
normalization), the outcome is completely uncertain between $BC$ and
$\overline{B}\overline{C}$, in other words
$[m_{r}(\\{BC,\overline{B}\overline{C}\\})=1]$, the random set outcome is
always $\\{BC,\overline{B}\overline{C}\\}$ . There is a significant practical
difference between these two results: In the first case any bet on the outcome
at better than even odds is known to be advantageous, in the second case no
bets or odds are known to be advantageous. And it is not difficult to see, by
considering probabilities for $B$ close to 0 and 1, that in this simple
example the robust Bayesian method is the appropriate one. Gelman is
dissatisfied with both results, but the robust Bayesian method actually gives
the right result in this simple example, the difficulty seems to be in the
problem formulation. Dempster’s rule gives no imprecision when an imprecise
body of evidence is combined with a precise one (which, as mentioned, is
characterized by being a singleton random set). This is a trivial consequence
of the random set intersection in Dempster’s rule: the intersection of a
singleton random set with any random set is itself a singleton or empty.

It is possible to bring in DS-theory under the robust Bayesian methodology,
but only by modelling the random set operations directly – they are after all
entirely kosher probability models. This would entail introduction of new
symbols for the subsets of $\Omega$. These symbols are new and do not
correspond to unions or intersections but to the random sets involved, so for
example the symbol $E=\\{BC\\}$ gives rise to the assignments $E|m_{1}=0$,
$E|m_{2}=0$, $E|m_{3}=0$ and $E|m=0.5$, whereas the correct answer based on
$[m_{r}(\\{BC,\overline{B}\overline{C}\\})=1]$ would have $E|m_{r}=0$ but
$F|m_{r}=1$ where $F=\\{BC,\overline{B}\overline{C}\\}$. This gives a precise
Bayesian model where only the last step, translating the final random set to a
convex set of probabilities, uses the robust Bayesian framework. The problem
is thus not that DS-theory does not fit into the Cox/Jaynes framework, but
that the implied probability model sometimes (like in Gelman’s case) gives
wrong answers for simple problems, as has been pointed out in [6, 11]. The
discussion of the justification of Dempster’s rule in [16] seems fair, but the
problem is that the implied probability model is only known to be appropriate
for problems where imprecision can be attributed to private frames of
reference.

###  8.3 Philosophy

A number of philosophical papers on the justifications of the line of
investigations based on Cox’s and de Finetti’s work have also appeared. An
article by John D Norton[35] groups assumptions in several groups, gives
counterarguments to some of them, and suggests that each group is checked and
a subset is chosen according to the needs of each specific application. This
seems a reasonable idea, but it requires a lot of effort to find out what
these subsets actually mean in terms of the ensuing uncertainty calculi, and
my experience is that finding the specific needs in an application area is
difficult, since as Norton points out the game is really to find convincing
arguments for an already made decision. In practice it seems that, currently,
a number of methods are developed in separate communities and one of their
major tasks (given that the claim to universality is too difficult as they
usually know by now) is to find the applications that confirm the method. The
conclusion of Norton is quite plausible while his specific examples in the
argumentation are sometimes difficult to follow and agree with, like the lack
of distinction between descriptive (Cialdini, Ellsberg, Kahneman, Tversky,
Klein) and prescriptive studies. As an example, the non-associativity of the
’more plausible than’ relation occurs only in descriptive studies (reflecting
framing, cognitive barriers, unstable preferences or approximate computations)
and there is no real argument against it in a prescriptive system. When and if
a difference is found it seems that three lines of investigation are possible:
(i) explaining the difference, for example by the wiring of the brain, the
social context of the decision making, computational or cognitive limitations,
or a real problem with the prescription; (ii) explaining what is wrong in
practical decision making and trying to fix it by adjusting the education
system; (iii) finding what is wrong with the prescriptive theories and
adapting them to the real needs or finding a new prescriptive theory of some
generality if not universality. It seems to me that accepting (iii) as the
only appropriate way is not so convincing. The development of Bayesian
analysis is however an example of the third option: from being harshly
criticised most of the time before 1980, it has been tuned by an enormous
effort by Bayesians addressing both fundamental and computational questions as
well as developing sophisticated model families suitable for areas such as
health sciences, bioinformatics, language translation, language and species
evolution, robotics and vision, web analytics, signals and systems and
finance, in several of which it has supplanted earlier methodologies. There is
by now a toolbox that, by its diversity and the versatility of supporting
computer codes, makes the Bayesian framework rather hard to replace. It is
also not the case that spectacular developments have been achieved all over
the field and certainly some Bayesian efforts have faded away, but enough
remains to make a convincing case for robust Bayesian analysis as the starting
effort in an area, and in case the application area is resisting, see what
modifications of the method can be made before looking for completely
different alternatives. Such modifications have been made, and I find it
difficult to see many of them as contrived or exotic.

In summary, it seems that despite progress the last decade, there is more to
find out about plausibility calculi.

##  Appendix A Proof of Theorem 1

Recall the definition of an ordered field: it is a structure
$R=(D,<,\cdot,+,1,0)$, where $D$ is a domain ordered by $<$, $\cdot$ and $+$
are total functions from $D^{2}$ to $D$, satisfying the properties of
associativity and symmetry, and where $\cdot$ distributes over $+$. Moreover,
it has an additive inverse, i.e.,, the equation $a+x=b$ can be solved for $x$,
and the equation $a\cdot x=b$ can be solved for $x$ if $a\neq 0$. The element
1 of $D$ is a unit for $\cdot$ and 0 is a unit for $+$. The function $+$ is
increasing and $\cdot$ is increasing for arguments larger (by $<$) than 0.
Fields are special cases of rings and integral domains. An ordered integral
domain satisfies the rules of a field except that we do not require
solvability for $x$ of $a\cdot x=b$.

We will show how an ordered plausibility space is embedded in an ordered
field. We have thus the space defined by the ordered domain $D$ with smallest
and largest elements now called 0 and 1, respectively, and they will always be
embedded as 0 and 1 of the field, respectively. Because of the significant
number of algebraic formulas used here, we also use infix notation $+$ and
$\cdot$ for $G$ and $F$, as well as for their extensions.

The function $S:D\rightarrow D$ maps its argument $x$ to a solution \- the
only one because of the monotonicity assumption - $y$ of the equation $x+y=1$.
The function $\cdot$ is defined on $D\times D$, and $+$ on $\\{(x,y):(x,y)\in
D\times D\land x\leq S(y)\\}$. We now extend $D$ while extending the
definitions of $\cdot$ and $+$. We know that $\cdot$ is associative and
symmetric, and that $+$ is symmetric. If $(a+b)+c$ and $a+(b+c)$ are both
defined, they are equal. Likewise, if $a+b$ is defined, $c\cdot(a+b)=c\cdot
a+c\cdot b$, and if $a+b$ is defined and $c<b$, then $a+c$ is also defined.

The strict monotonicity assumption leads to cancellation properties:

###### Lemma 1

If $a+b=a+c$ then $b=c$, if $a\cdot b=a\cdot c$ and $a\neq 0$ then $b=c$, if
$a+b\leq a+c$ then $b\leq c$, and if $a\cdot b\leq a\cdot c$ and $a\neq 0$
then $b\leq c$.

If there are no elements in $D$ except 0 and 1, the embedding is trivial, so
we assume an element $e$ with $0<e<1$ in $D$. We do the embedding in three
steps, using the standard technique of defining an extension as the quotient
of a set of pairs by an equivalence relation, and indicating which element of
the extension that corresponds to each element of the original domain. We use
the notation $[a]_{\sim}$ for the equivalence class of $\sim$ containing $a$.
It is easy, in each embedding step, to define the functions $\cdot$ and $+$ on
the extensions and verify that they are indeed functions and extensions, that
their laws are preserved, as well as to verify that no two elements of the old
domain become equivalent in the new domain. Details of this verification are
shown here, in the form of a series of Lemmas, whose proofs are sometimes
terse and sometimes omitted.

###### Lemma 2

If $a_{1}+a_{2}$ is defined and $a_{1}\geq b_{1}$ and $a_{2}\geq b_{2}$, then
$b_{1}+b_{2}$ is defined.

Proof: Obviously, $b_{1}\leq a_{1}\leq S(a_{2})\leq S(b_{2})$.

###### Lemma 3

If $e\neq 0,1$ and $f=\min(e,S(e))$, then $f\cdot a+f\cdot b$ is defined.

Proof: Assume (no loss of generality because of symmetry of $+$) that $a\leq
b$.

  1. 1.

$f=e\leq S(e)$: In this case, $f\cdot a\leq f\cdot b\leq S(f)\cdot b$, and
$f\cdot b+S(f)\cdot b$ is defined, thus by Lemma 1 $f\cdot a+f\cdot b$ is also
defined.

  2. 2.

$e>S(e)=f$: Since $e\cdot b+S(e)\cdot b$ is defined and $e\cdot b\geq f\cdot
a$, thus by Lemma 2 $f\cdot a+f\cdot b$ is also defined.

###### Lemma 4

For every sequence $(a_{i})_{1}^{n}$ there is a non-zero $c_{n}$ depending
only on $n$ such that $c_{n}\cdot a_{1}+c_{n}\cdot a_{2}\ldots c_{n}\cdot
a_{n}$ is defined.

Proof: For any non-trivial plausibility value $e$, choose $c=\min(e,S(e))$ and
$c_{n}=c^{\lceil\log n\rceil}$. Use Lemma 3 inductively on half sequences.

The first embedding step introduces non-negative rationals and is similar to
the standard quotient construction for integral domains. Let
$D^{+}=D-\\{\bot\\}$ and $D^{(1)}=(D\times D^{+})/\sim$, where, for $a,c\in D$
and $b,d\in D^{+}$, $(a,b)\sim(c,d)$ iff $a\cdot d=b\cdot c$. This makes
$\sim$ an equivalence relation. Use notation $[a,b]$ for $[(a,b)]_{\sim}$. An
element $d\in D$ is identified with $[d,1]\in D^{(1)}$.

Define $<$, $\cdot$ and $+$ as total functions on $D^{(1)}$ by $[a,b]<[c,d]$
iff $a\cdot d<c\cdot b$, $[a,b]\cdot[c,d]=[a\cdot c,b\cdot d]$, and
$[a,b]+[c,d]=[e\cdot a\cdot d+e\cdot c\cdot b,e\cdot b\cdot d]$, for some $e$
such that the expressions are defined (see Lemma 4). The rational number $2/3$
is identified with $[x+x,x+x+x]$ for some $0<x<c_{3}$, and the other non-
negative rationals are similarly defined. In this extension the rules for a
field are satisfied, except that we have not yet an additive inverse or
negative values:

###### Lemma 5

The relation $\sim$ is an equivalence relation and if $[d,1]=[d^{\prime},1]$,
then $d=d^{\prime}$.

Proof: The relation $\sim$ is obviously reflexive and symmetric. It is also
associative, since if $(a,b)\sim(c,d)$ and $(c,d)\sim(e,f)$ with $b,d,f\neq 0$
then $a\cdot d=b\cdot c$ and $c\cdot f=d\cdot e$, hence $c\neq 0$, $a\cdot
f\cdot(c\cdot d)=e\cdot b\cdot(c\cdot d)$ and by the cancellation Lemma,
$(a,b)\sim(e,f)$.

###### Lemma 6

The relation $<$ is a total order on $D^{(1)}$.

Proof: Omitted.

###### Lemma 7

The relations $\cdot$ and $+$ are total functions on $D^{(1)}\times D^{(1)}$.

Proof: Consider $+$: If $[a,b]+[c,d]=[f,g]$ and
$[a^{\prime},b^{\prime}]+[c,d]=[f^{\prime},g^{\prime}]$ with
$[f,g]=[f^{\prime},g^{\prime}]$, then $[f,g]=[e\cdot a\cdot d+e\cdot c\cdot
b,e\cdot b\cdot d]$ and $[f^{\prime},g^{\prime}]=[e\cdot a^{\prime}\cdot
d+e\cdot c\cdot b^{\prime},e\cdot b^{\prime}\cdot d]$, The equivalence
condition $[f,g]=[f^{\prime},g^{\prime}]$ translates to $(e\cdot a\cdot
d+e\cdot c\cdot b)\cdot e\cdot b^{\prime}\cdot d=(e\cdot a^{\prime}\cdot
d+e\cdot c\cdot b^{\prime})\cdot e\cdot b\cdot d$, in other words (by
cancellation) $(a\cdot d+c\cdot b)\cdot b^{\prime}=(a^{\prime}\cdot d+c\cdot
b^{\prime})\cdot b$ so $[a,b]=[a^{\prime},b^{\prime}]$. In other words the
type of $+$ is indeed $D^{(1)}\times D^{(1)}\rightarrow D^{(1)}$. The other
cases are similar but easier.

###### Lemma 8

The functions $\cdot$ and $+$ are associative and symmetric, and $\cdot$
distributes over $+$ on $D^{(1)}\times D^{(1)}$.

Proof: Consider associativity for $+$: If $([a,b]+[c,d])+[f,g]=[h,i]$ and
$[a,b]+([c,d]+[f,g])=[h^{\prime},i^{\prime}]$, then $[h,i]=[e\cdot(e\cdot
a\cdot d+e\cdot c\cdot b)\cdot g+e\cdot f\cdot e\cdot b\cdot d,e\cdot e\cdot
b\cdot d\cdot g]$ and $[h^{\prime},i^{\prime}]=[e\cdot(e\cdot c\cdot g+e\cdot
d\cdot f)\cdot b+e\cdot a\cdot d\cdot g\cdot e,e\cdot b\cdot e\cdot d\cdot
g]$. These two are equal. The other cases are similar.

Our next embedding step introduces subtraction and negative values: Let
$D^{(2)}=(D^{(1)}\times(D^{(1)})/\approx$, where, for $a,b,c,d\in D^{(1)}$,
$(a,b)\approx(c,d)$ iff $a+d=b+c$. Use notation $[[a,b]]$ for
$[(a,b)]_{\approx}$. An element $d\in D^{(1)}$ is identified with $[[d,0]]\in
D^{(2)}$. Define $<$, $\cdot$ and $+$ in this extension by $[[a,b]]<[[c,d]]$
iff $a+d<c+b$, $[[a,b]]\cdot[[c,d]]=[[a\cdot c+b\cdot d,a\cdot d+b\cdot c]]$,
and $[[a,b]]+[[c,d]])=[[a+c,b+d]]$.

###### Lemma 9

The relation $\approx$ is an equivalence relation and if
$[[d,0]]=[[d^{\prime},0]]$, then $d=d^{\prime}$.

###### Lemma 10

The relation $<$ is a total order on $D^{(2)}$ and extends $<$ on $D^{(1)}$.

###### Lemma 11

The relations $\cdot$ and $+$ are total functions on $D^{(2)}\times D^{(2)}$
and extend $\cdot$ and $+$ on $D^{(1)}\times D^{(1)}$.

###### Lemma 12

The functions $\cdot$ and $+$ are associative, symmetric and $\cdot$
distributes over $+$ on $D^{(2)}\times D^{(2)}$.

###### Lemma 13

The structure $R=(D^{(2)},<,\cdot,+,1,0)$ is an ordered integral domain.

###### Lemma 14

([30, Ch V.2, Theorem 6]) Every ordered integral domain can be embedded in an
ordered field

The structure $R$ can thus be embedded in an ordered field. This field embeds
the plausibility space. This finishes the proof of Theorem 1.

##  Appendix B Proof of Theorem 2

Let the plausibility space be $(D,\cdot,+,S,\leq,\bot,\top)$. Most of the
properties of a proper ordered plausibility space are immediate consequences
of our stated assumptions. The non-trivial part is to show that the stated
laws for $F$, $G$ and $S$ must hold in a consistently refinable model. This
follows from a sequence of Lemmas, each showing how an algebraic law follows
from the corresponding law of propositional logic. We only state them and
prove some of them:

###### Lemma 15

For all $x\in D$, $S(S(x))=x$.

Proof: If $x=A|B$, then $x=A|B=\overline{\overline{A}}|B=S(S(x))$.

###### Lemma 16

For all $x,y,z\in D$, $(x\cdot y)\cdot z)=x\cdot(y\cdot z))$.

Proof: If we have worked out a model where $a\cdot(b\cdot c)\neq(a\cdot
b)\cdot c$ for some plausibilities $a$, $b$ and $c$ that occur in the model,
then we take an arbitrary statement $S$ (not false) and refine with
$A^{\prime}$, $B^{\prime}$ and $C^{\prime}$: $A^{\prime}\rightarrow
B^{\prime}$, $B^{\prime}\rightarrow C^{\prime}$ and $C^{\prime}\rightarrow S$,
$A^{\prime}|B^{\prime}=a$, $B^{\prime}|C^{\prime}=b$ and $C^{\prime}|S=c$. Now
the value $A^{\prime}B^{\prime}C^{\prime}|S$ can be computed in two ways
giving different results, as
$(A^{\prime}B^{\prime})C^{\prime}|S=F(A^{\prime}B^{\prime}|SC^{\prime},C^{\prime}|S)=F(F(A^{\prime}|B^{\prime},B^{\prime}|C^{\prime}),c)=(a\cdot
b)\cdot c$ and as
$A^{\prime}(B^{\prime}C^{\prime})|S=F(A^{\prime}|SB^{\prime}C^{\prime},B^{\prime}C^{\prime}|S)=F(A^{\prime}|B^{\prime},F(B^{\prime}|C^{\prime},C^{\prime}|S))=a\cdot(b\cdot
c)$.

###### Lemma 17

For all $x,y\in D$, $x\cdot y=y\cdot x$.

Proof: If $x=A|D$ and $y=B|E$, introduce by refinement $B^{\prime}$ such that
$D\rightarrow B^{\prime}$, $B^{\prime}|D=y$ and $B^{\prime}$ is independent of
$A$. Now $AB^{\prime}|D=x\cdot y$ and $AB^{\prime}|D=B^{\prime}A|D=y\cdot x$,
so $x\cdot y=y\cdot x$.

###### Lemma 18

For all $x,y\in D$, $x+y=y+x$, in the sense that if one side is defined, then
the other side is defined and equal.

Proof: If $x=A|D$ and $y=B|E$ and $x\leq S(y)$, introduce by refinement
$B^{\prime}$ such that $D\rightarrow B^{\prime}$, $B^{\prime}|D=y$ and
$AB=\bot$. Now $A\lor B|D=x+y$ and $A\lor B|D=B\lor A|D=y+x$, so $x+y=y+x$.

###### Lemma 19

For all $x,y,z\in D$, $(x+y)+z=x+(y+z)$, in the sense that if one side is
defined, then the other side is defined and equal.

Proof: By refinement and associativity of $\lor$.

###### Lemma 20

For all $x,y,z\in D$, $x\cdot z+y\cdot z=(y+x)\cdot z$, in the sense that if
$x+y$ is defined, then both sides are defined and equal.

Proof: Assume $x$, $y$ and $z$ are non-trivial plausibilities of the model and
$x\leq S(y)$, because otherwise the Lemma is obvious. Introduce for non-$\bot$
$D$ by refinement $A\rightarrow D$, $B\rightarrow D$ and $C\rightarrow D$ such
that $A\land B=\bot$ and $A$ and $B$ are independent of $C$, and $x=A|D$,
$y=B|D$ and $z=C|D$. Now, $(A\lor B)\land C|D=(x+y)\cdot z$ and $(A\lor
B)\land C|D=(A\land C)\lor(B\land C)|D=x\cdot z+y\cdot z$ and the Lemma
follows.