Submitted to Statistical Science

Judicious Judgment Meets
Unsettling Updating:
Dilation, Sure Loss, and
Simpson’s Paradox
arXiv:1712.08946v1 [] 24 Dec 2017

Ruobin Gong and Xiao-Li Meng
Harvard University

Abstract. Statistical learning using imprecise probabilities is gaining
more attention because it presents an alternative strategy for reducing
irreplicable findings by freeing the user from the task of making up
unwarranted high-resolution assumptions. However, model updating as a mathematical operation is inherently exact, hence updating
imprecise models requires the user’s judgment in choosing among
competing updating rules. These rules often lead to incompatible inferences, and can exhibit unsettling phenomena like dilation, contraction and sure loss, which cannot occur with the Bayes rule and precise
probabilities. We revisit a number of famous “paradoxes”, including the three prisoners/Monty Hall problem, revealing that a logical
fallacy arises from a set of marginally plausible yet jointly incommensurable assumptions when updating the underlying imprecise
model. We establish an equivalence between Simpson’s paradox and
an implicit adoption of a pair of aggregation rules that induce sure
loss. We also explore behavioral discrepancies between the generalized
Bayes rule, Dempster’s rule and the Geometric rule as alternative posterior updating rules for Choquet capacities of order 2. We show that
both the generalized Bayes rule and Geometric rule are incapable of
updating without prior information regardless of how strong the information in our data is, and that Dempster’s rule and the Geometric
rule can mathematically contradict each other with respect to dilation
and contraction. Our findings show that unsettling updates reflect a
collision between the rules’ assumptions and the inexactness allowed
by the model itself, highlighting the invaluable role of judicious judgment in handling low-resolution information, and the care we must
take when applying learning rules to update imprecise probabilities.
MSC 2010 subject classifications: Primary 62A01; secondary 62C86.
Key words and phrases: Belief functions, Choquet capacities, Contraction, Dempster’s rule, Imprecise probability, Monty Hall problem.
1 Oxford Street, Cambridge, MA, 02138 (e-mail:
rgong@fas.harvard.edu; meng@stat.harvard.edu)
∗ Supported

in part by the John Templeton Foundation Grant 52366.

1
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

2

GONG AND MENG

1. THERE IS NO FREE LUNCH...
Statistical learning is a process through which models perform updates in
light of new information, according to a pre-specified set of operation rules.
As new observations arrive, a good statistical model revises and adapts its
existing uncertainty quantification to what has just been learned. If a model a
priori judges the probability of an event A to be P( A), after learning event B
happened, it may update the posterior probability according to the Bayes rule:
P( A | B) = P( A) ·

P( B | A)
.
P( B)

Exactly one of three things will happen: P( A | B) > P( A), P( A | B) < P( A),
or P( A | B) = P( A). Moreover, P ( A | B) > P ( A) if and only if P ( A | Bc ) <
P ( A), that is, if B expresses positive support for A, its complement must express negative support. The comparison of prior and posterior probabilities
of A encapsulates its association with the observed evidence B, a fundamental
characterization of the contribution made by a piece of statistical information.
Nevertheless, there exists modeling situations in which associations do not
comply with our well-founded intuition. We sketch a series of such examples,
well-known from probability classrooms to real life statistical inference, which
will serve as the basis of our analysis throughout the paper. Many of them,
known as paradoxes, bear multiple solutions that have long been the center
of dispute and explication in the literature. What makes all of them thoughtprovoking is the apparent change from prior to posterior judgments of an event
of interest that most will find counterintuitive. That, as we will see, is a consequence of the ambiguity in the probabilistic specification of the model itself,
ambiguity that cannot be meaningfully resolved by any automated rule.
1.1 Statistical paradox or imprecise probability?
Example 1 (Treatment efficacy before and after randomization; Section 2.2.1). Patients Oreta and Tang are participating in a clinical trial, in which one of them
will receive treatment I, and the other treatment II, with equal probability. Let
A denote the event that Oreta will improve more from this trial than Tang (assuming no ties), and let B denote the event that Tang is assigned to treatment I.
Before the treatment is assigned, we clearly have P ( A) = 1/2 because the situation is fully symmetric (in the absence of any other information). However,
after the assignment is observed, we seem to have no good idea of the value of
either P ( A | B) or P ( A | Bc ), other than they are both bounded within [0, 1].
Example 1 showcases a severe form of “confusion” expressed by the model
as the prior probability updates to posterior probability in light of any new
information. The precise prior judgments P ( A) = 1/2 and P ( Ac ) = 1/2 are
both bound to suffer a loss of precision by the sheer act of conditioning on any
event in B = { B, Bc }. A central topic of this paper is the dilation phenomenon,
revealed by Good (1974) and investigated in depth by Seidenfeld & Wasserman
(1993); Herron et al. (1994, 1997). A formal definition is given in Section 3.1.
Example 2 (The boxer, the wrestler, and the coin flip (Gelman, 2006); Sections 3.1
& 6.1). The greatest boxer and the greatest wrestler are scheduled to fight.
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

3

Who will defeat the other? Let Y = 1 if the boxer wins; Y = 0 if the wrestler
wins. Also, let X = 1 if a toss of a fair coin yields heads; X = 0 if tails. A
witness at both the fighting match and the coin flip tells us that X = Y. Given
this, what is the boxer’s chance of winning, P (Y = 1 | X = Y )?
Example 3 (Three prisoners (Diaconis, 1978; Diaconis & Zabell, 1983); Sections
3.2 & 6.2). Three death row inmates A, B, and C are told, on the night before
their execution, that one of them has been chosen at random to receive parole,
but it won’t be announced until the next morning. Desperately hoping to learn
immediately, prisoner A says to the guard: “Since at least one of B and C will
be executed, you’ll give away no information about my own chance by giving
the name of just one of either B or C who is going to be executed.” Convinced
of this argument, the guard truthfully says, “B will be executed.” Given this information, how should A judge his living prospect, P ( A lives | guard says B)?
Example 4 (Simpson’s paradox (Simpson, 1951; Blyth, 1972); Section 4). We
would like to evaluate the effectiveness of a novel treatment (experimental)
compared to its standard counterpart (control). Let Z = 1 denote assignment
of the experimental treatment, 0 the control treatment, and let Y = 1 denote
the event of a recovery, 0 otherwise. Let U ∈ {1, 2, ..., K } be a covariate of the
patients, a K-level categorical indicator variable. One could imagine K to be
very large, to the extent that the univariate U creates sufficiently individualized
strata among the patient population.
Suppose we learn from reliable clinical studies that the experimental treatment works better than the control for all K subtypes of patients. That is,
(1.1)
pk ≡ P (Y = 1 | Z = 1, U = k ) > qk ≡ P (Y = 1 | Z = 0, U = k) , k = 1, ..., K.
Nevertheless, field studies consisting of feedback reports from clinics and hospitals seem to suggest otherwise; that on an overall basis, the control treatment
cures more patients than the experimental treatment. That is,
(1.2)

p̄obs ≡ Pobs (Y = 1 | Z = 1) < q̄obs ≡ Pobs (Y = 1 | Z = 0) .

How do we resolve the apparent conflict between the conditional inference in
(1.1) and the marginal inference in (1.2)?
The above examples will be examined in detail in Sections 3 through 5. All of
them, despite being disguised with cunning descriptions, share the characteristic of being an imprecise model. Their narratives imply the existence of a joint
distribution, yet only a subset of marginal information is precisely specified.
For instance, in Example 1, while the treatment assignment (B) is known to
be fair prior to randomization, the improvement event A is not measurable with
respect to the B margin, effectively posing a Fréchet class of joint distributions
on the { A, B} space. The only statements we can make about P( A | Z ) are the
trivial bounds 0 ≤ P ( A | Z ) ≤ 1, whether Z = B or Z = Bc , leading to the
dilation phenomenon. For Example 2, the coin margin X is fully known a priori,
but the relationship between the fighters Y and the coin X, crucial for quantifying the event { X = Y }, is unspecified. In Example 3, the guard’s tendency to
report B over C is unspecified in the case that A was granted parole, yet A’s
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

4

GONG AND MENG

survival probability depends critically on this reporting tendency. In all of these
examples, the water gets muddied due to an unspecified but necessary piece of
relational knowledge, which in turn imposes on the modeler a choice among a
multiplicity of updating rules, each supplying a distinct set of assumptions to
complement this ambiguity.
1.2 What do we try to accomplish in this paper?
Unsettling phenomena to be discussed in this paper reflect unusual ways
through which more information can seemingly “harm” our existing knowledge of the state of matters. These phenomena are not foreign to statisticians,
but are seen as anomalies and paradoxes, far from everyday model building. In
fact, whenever there is a fully and precisely specified probability model, none
of these phenomena would occur. Wouldn’t we all be safer then by staying
away from any imprecise model? Quite the contrary, we argue. Imprecise models are unavoidable even in basic statistical modeling, and sometimes they are
disguised as precise models only to trick us into blindness. Simpson’s paradox,
re-examined in Section 4, is one of such cases. Without acknowledging the imprecise nature of modeling, one is ill-suited to make judicious choices among
the updating rules and treatments of evidence.
We aim to investigate these perceived anomalies as they occur during the
updating of imprecise models, and their implications on the choice of updating rules. Imprecise models in statistical modeling are ubiquitous and can be
easily induced from precise models through the introduction of external variables. When model imprecision is present, a choice among updating rules is a
necessity, and it reflects the modeler’s judgment on how statistical evidence at
hand should be used. With the recent surge of interest in imprecise probabilitybased and related statistical frameworks including generalized Fiducial inference (Hannig, 2009), confidence distribution (Hannig & Xie, 2012; Schweder
& Hjort, 2016) and inferential models (Martin & Liu, 2015), we are compelled
to bring attention to the non-negligible choice of combining and conditioning
rules for statistical evidence.
The remainder of this paper starts with an introduction to some formal notations of imprecise probabilities in Section 2.1, particularly of Choquet capacities of order 2 as well as belief functions, a versatile special case which can
also be formulated as a precise model for imprecise states, that is, set-valued
random variables. Three main updating rules are introduced in Section 2.2, all
of which are applicable to Choquet capacities of order 2. Section 3 defines dilation, contraction and sure loss as phenomena that happen during imprecise
model updating, and Section 4 shows how Simpson’s paradox is a consequence
of an ill-chosen updating rule by establishing its equivalence to a sure loss in
aggregation. It also shows how imprecise models can be easily induced from
precise ones. Section 5 compares and contrasts the behavior of the updating
rules, especially as they exhibit dilation and sure loss, and illustrates them with
an additional example. When do the updating rules differ, and how? We believe
these questions will shed light on the means through which information could
contribute to imprecise statistical models, a topic we will discuss in Section 6,
among others.

imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

5

2. IMPRECISE PROBABILITIES AND THEIR UPDATING RULES
This section introduces some formal concepts and notation governing imprecise probability needed within the scope of this paper. Readers who are familiar
with such notion may skip to Section 3.
2.1 Coherent lower and upper probabilities
Definition 2.1 (Coherent lower and upper probabilities). Let Ω be a separable
and completely metrizable space and B (Ω) be its Borel σ-algebra. The lower
and upper probabilities of a set of probability measures Π on Ω are set functions
P ( A) = inf P ( A) and P ( A) = sup P ( A) ,
P∈Π

P∈Π

for all A ∈ B (Ω).

Respectively, P and P are said to be coherent lower and upper probabilities if Π
is closed and convex (Walley, 2000).
Note that P and P are conjugate in the sense that P ( A) = 1 − P ( Ac ); thus one
is sufficient for characterizing the other. We may refer to either P or P individually with the understanding of their one-to-one relationship. Next we introduce
Choquet capacities, an important class of imprecise probabilities widely used
in robust statistics (Huber & Strassen, 1973; Wasserman, 1990).
Definition 2.2 (Choquet capacities of order k). Suppose a coherent lower probability P is such that { P; P ≥ P} is weakly compact. P is a Choquet capacity
of order k, or k-monotone capacity, if for every Borel-measurable collection of
{ A, A1 , ..., Ak } such that Ai ⊂ A for all i = 1, ..., k, we have
(2.1)

P ( A) ≥

∑

(−1)| I |−1 P (∩i∈ I Ai )

∅6= I ⊂{1,...,k }

where |S| denotes the number of elements in the set S. Its conjugate capacity
function P is a called a k-alternating capacity, because it satisfies for every Borelmeasurable collection of { A, A1 , ..., Ak } such that A ⊂ Ai for all i = 1, ..., k,
(2.2)

P ( A) ≤

∑

(−1)| I |−1 P (∪i∈ I Ai ) .

∅6= I ⊂{1,...,k }

If a Choquet capacity is (k + 1)-monotone, it is k-monotone as well: the
smaller the k, the broader the class. Choquet capacities of order 2 are a special
case of coherent lower probability. They satisfy P ( A ∪ B) ≥ P ( A) + P ( B) −
P ( A ∩ B) for all A, B ∈ B (Ω). A most special case of Choquet capacities consists of belief functions (Shafer, 1979).
Definition 2.3 (Belief function). P is called a belief function if it is a Choquet
capacity of order ∞, i.e., if (2.1) holds for every k.
It can be easily verified that precise probabilities are a special case of belief
functions, and in turn Choquet capacities of any order, thus all of the above
are more general constructs than precise probability functions. That being said,
they constitute a small class of imprecise probabilities imaginable. Belief functions in particular have their own specializations and limitations when it comes
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

6

GONG AND MENG

to characterizing imprecise knowledge in probability specifications. Pearl (1990)
noted that belief functions are often incapable of characterizing imprecise probabilities expressed in conditional forms, a category in which Examples 1 and 4
fall, thus neither models are belief function-representable. However, unique to
belief function is its intuitive interpretation as a random set object that realizes
itself as subsets of Ω.
Definition 2.4 (Mass function of a belief function). If P is a belief function,
its associated mass function is the non-negative set function m : P (Ω) → [0, 1]
such that
(2.3)

m ( A) =

∑

(−1)| A− B| P ( B) ,

for all A ∈ B (Ω)

B⊆ A

where A − B = A ∩ Bc , and the notation ∑ B⊆ A should be understood as taking
the sum under the additional constraint that B ∈ B (Ω) (a convention for the
rest of this article, whenever appropriate). Here m satisfies (1) m (∅) = 0, (2)
∑ B⊆Ω m ( B) = 1, and (3) P ( A) = ∑ B⊆ A m ( B) and is unique to P.
Formula (2.3) is called the Möbius transform of P (Yager & Liu, 2008). A mass
function m induces a precise probability distribution on the subsets of Ω, as the
distribution of a random set R. In Section 3 we will invoke the mass function
representation of belief functions in Examples 2, 3, and 5 (to be discussed in
Section 5.5).
2.2 Updating rules for coherent lower and upper probabilities
To update a set of probabilities Π given a set B ∈ B (Ω) is to replace the
set function P with a version of the conditional set function P• (· | B). The definition of P• is precisely the job of the updating rule. We emphasize that to say
an event is “given” does not necessarily mean it is “observed”. In hypothetical contemplations we often employ conditional statements about all events in a
partition, for example B = { B, Bc }, even if logically we cannot observe B and Bc
simultaneously. Therefore, the phrase “given” should be understood as imposing a mathematical constraint generated by B. When Π contains a single, precise
statistical model, the Bayes rule entirely dictates how we use the information
supplied by B. But when Π is imprecise and does not possess sharp knowledge
about B, i.e., P ( B) < P ( B) (Dempster, 1967), the updating rule itself becomes
an imprecise matter. As a consequence, there exists multiple reasonable ways
to use the information in B, e.g, “supported by B” and “not contradicted by B”
generate two different constraints. This raises both flexibility and confusion in
defining the updating rules. Here we supply the formal definitions of three viable updating rules for coherent lower and upper probabilities: the generalized
Bayes rule, Dempster’s rule, and the Geometric rule. Important differences and
relationships exist among these rules, as we shall present in Section 5.
2.2.1 Generalized Bayes rule. Recall Example 1. Using the notation in 2.1, we
rewrite the imprecise model in terms of its prior upper and lower probabilities
of event A, which are precisely one half: P( A) = P( A) = 0.5. The question
is: what are the upper and lower probabilities of A given the treatment assignments in B = { B, Bc }? For example, a version of the answer supplied here
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

7

is
PB ( A | B) = 0 , PB ( A | B) = 1,

and

PB ( A | Bc ) = 0 , PB ( A | Bc ) = 1.

The expressions PB and PB , where the subscript B is for Bayes, signify the use
of the generalized Bayes rule, as defined below.
Definition 2.5 (Generalized Bayes rule). Let Π be a closed, convex set of
probability measures on Ω. The conditional lower and upper probabilities according to the generalized Bayes rule are set functions PB and PB such that, for
A, B ∈ B (Ω),
(2.4)
(2.5)

P ( A ∩ B)
,
P∈Π
P ( B)
P ( A ∩ B)
PB ( A | B) = sup
.
P ( B)
P∈Π
PB ( A | B ) =

inf

That is, the conditional lower and upper probabilities are respectively the
minimal and maximal Bayesian conditional probability among elements of Π.
In their definition of the generalized Bayes rule, Seidenfeld & Wasserman (1993)
worked with the requirement that P ( B) > 0, which guarantees P ( B) > 0 for
all P ∈ Π; hence the ratios in (2.4) and (2.5) are always well-defined.
The generalized Bayes rule is a most widely employed updating rule for
coherent lower and upper probabilities (Walley, 1991), and is notable for its
dilation phenomenon. In Example 1, as a consequence of employing the rule,
the conclusion appears puzzling: Tang will surely receive one of the two treatments, and one would expect that, in the worst case scenario, learning about
the treatment assignment is completely useless, i.e., having no effect on our a
priori assessment of P( A). But how could it be that the knowledge of something
can do more harm than being useless?
To gain a better understanding of the behavior of the generalized Bayes rule,
we introduce two alternative updating rules for sets of probabilities as a means
of comparison. Both Dempster’s rule of conditioning and the Geometric rule
were originally proposed for use with the special case of belief functions, however their expressions compose intriguing counterparts to the generalized Bayes
rule. Section 5 is dedicated to a comparison among the trio of rules.
2.2.2 Dempster’s rule. Dempster’s rule of conditioning is central to the DempsterShafer theory of belief functions (Dempster, 1967; Shafer, 1976). The conditioning operation is a special case of Dempster’s rule of combination, equivalent to
combining one belief function with another that puts 100% mass on one particular subset, B ∈ B (Ω), on which we wish to condition. Specifically, let P be
a belief function such that P ( B) > 0, and m be its associated mass function
given by (2.3). Let P0 be a separate belief function such that its associated mass
function m0 ( B) = 1. The conditional belief function PD (· | B) is defined as
PD ( A | B) = P ( A) ⊕ P0 ( B) ,

for all A ∈ B (Ω) ,

where the combination operator “⊕” is defined in Shafer (1976) to imply that
the mass function associated with PD (· | B) is
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

8

GONG AND MENG

mD ( A | B) =

(2.6)

∑C ∩ B= A m (C )
,
∑C0 ∩ B6=∅ m (C 0 )

for all A ∈ B (Ω) .

Consequently, Dempster’s rule of conditioning yields the following form.
Definition 2.6 (Dempster’s rule of conditioning). For P a belief function over
B (Ω) and Π the set of probability measures compatible with P, the lower and
upper probabilities according to Dempster’s rule of conditioning are set functions
PD and PD such that for A, B ∈ B (Ω) with P ( B) > 0,
PD ( A | B) = 1 − PD ( Ac | B) ,
supP∈Π P ( A ∩ B)
PD ( A | B) =
.
supP∈Π P ( B)

(2.7)
(2.8)

Hence PD ( A | B) differs from PB ( A | B) of (2.5) by taking the ratio of the
suprema, instead of the supremum of the ratio P ( A ∩ B) /P ( B).
An operational view of (2.8) is helpful for understanding exactly what information is retained by Dempster’s rule (Gong & Meng, 2018). Denote by R
the set-valued random variable representing the distribution as dictated by the
mass function corresponding to P. Dempster’s rule of conditioning on set B is
akin to taking a “B-shaped cookie cutter” to all realizations of R, i.e., retaining
all intersection sets that R has with B given it is non-empty, discarding the rest
while renormalizing the retained sets such that their mass function mD(·| B) as
in (2.6) normalizes to one.
2.2.3 The Geometric rule. The Geometric rule was proposed by Suppes & Zanotti (1977) as an intended alternative to Dempster’s rule.
Definition 2.7 (The Geometric rule). Let P be a belief function as in Definition 2.6. The conditional lower and upper probabilities according to the
Geometric rule are set functions PG and PG such that for A, B ∈ B (Ω) with
P ( B) > 0,
infP∈Π P ( A ∩ B)
,
infP∈Π P ( B)

(2.9)

PG ( A | B) =

(2.10)

PG ( A | B) = 1 − PG ( Ac | B) .

Mathematically, the Geometric rule appears to be a natural dual to Dempster’s rule, by replacing the latter’s suprema for upper probability as in (2.8)
with the infima for lower probability, as in (2.9). Operationally, the Geometric
rule differs from Dempster’s rule by retaining all mass-bearing sets of R that
are contained within B, discarding the rest while renormalizing the resulting
mass function. Section 5 further describes some relationships between the two
rules. In his review of Shafer (1976), Diaconis (1978) discussed a paradoxical
conclusion for the three prisoners example (reproduced here as Example 3) using Dempster’s rule, and inquired about the option of the Geometric rule as an
alternative rule of updating. As we will show in Section 3.2, the Geometric rule
does no better job than Dempster’s rule for this paradox, as in fact both exhibit
precisely a sure loss phenomenon.
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

9

More updating rules for belief functions exist beyond Dempster’s and the
Geometric rule, including the disjunctive rule by Smets (1993) based on set
union operations, the open-world conjunctive rule which is the unnormalized
version of Dempster’s rule as employed in the transferable belief models, as
well as others, e.g., Yager (1987); Kohlas (1991); Kruse & Schwecke (1990). Smets
(1991) provided a broad overview of an array of updating rules.
2.2.4 Applicability to Choquet capacities. The generalized Bayes rule was designed to work with coherent sets of probabilities, thus by default applicable
to special cases of coherent probabilities such as Choquet capacities of order 2.
Wasserman & Kadane (1990) showed that, when applied to prior sets of probabilities that are Choquet capacities of order 2, the posterior sets of probabilities
by the generalized Bayes rule remain in the class. Specifically, Fagin & Halpern
(1991) showed that, when they are applied to belief functions, the conditional
probabilities will remain a belief function with the expression

PB ( A | B ) = P ( A ∩ B ) / P ( A ∩ B ) + P ( A c ∩ B ) ,
(2.11)

(2.12)
PB ( A | B ) = P ( A ∩ B ) / P ( A ∩ B ) + P ( A c ∩ B ) .
The formula of the mass functions induced by the Möbius transform corresponding to PB (· | B) was given in Jaffray (1992).
On the other hand, can Dempster’s rule and the Geometric rule be applied
to sets of probabilities more general than belief functions? Below we show that
they can for Choquet capacities of order k where k ≥ 2, and the corresponding
conditional lower probability will remain Choquet capacities of order k.
Theorem 2.1. Let P be a k-monotone Choquet capacity on B (Ω), and event B
such that the set functions PD (· | B) in (2.7) and PG (· | B) in (2.9) are well-defined.
Then, PD (· | B) and PG (· | B) are both k-monotone.
Proof. To say P is k-monotone implies for all Borel-measurable collections
{ A1 , ..., Ak },


P ∪ik=1 Ai ≥

k

∑ P ( Ai ) − ∑ P

i =1




Ai ∩ A j + · · · + (−1)k+1 P ∩ik=1 Ai

i< j

or, equivalently, P is k-alternating:


P ∩ik=1 Ai ≤

k




k +1
k
P
A
−
P
A
∪
A
+
·
·
·
+
1
P
∪
A
(
)
(−
)
i
i
j
∑
∑
i =1 i .

i =1

i< j

imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

10

GONG AND MENG

For Dempster’s rule, we have
PD



∩ik=1 Ai

|B



=
≤

P




∩ik=1 Ai ∩ B
P ∩ik=1 ( Ai ∩ B)
=
P ( B)
P ( B)
"

1
·
P ( B)

k

∑ P ( Ai ∩ B ) − ∑ P

( Ai ∩ B ) ∪ A j ∩ B



+···

i< j

i =1


i
+ (−1)k+1 P ∪ik=1 ( Ai ∩ B)
k

=

∑ PD ( Ai | B) − ∑ PD

i =1


Ai ∪ A j | B + · · ·

i< j




+ (−1)k+1 PD ∪ik=1 Ai | B .
Similarly, for the Geometric rule,
PG



∪ik=1 Ai

|B



=
≥

P




P ∪ik=1 ( Ai ∩ B)
∪ik=1 Ai ∩ B
=
P ( B)
P ( B)
"

1
·
P ( B)

k

∑ P ( Ai ∩ B ) − ∑ P

i =1


Ai ∩ A j ∩ B + · · ·

i< j



+ (−1)k+1 P ∩ik=1 Ai ∩ B

i

k

=

∑ PG ( Ai | B) − ∑ PG
i< j

i =1

+ (−1)


Ai ∩ A j | B + · · ·

k +1




PG ∩ik=1 Ai | B .

Hence k-monotonicity is preserved by both Dempster’s and the Geometric rules
of updating when applied to k-monotone Choquet capacities.
3. THE UNSETTLING UPDATES IN IMPRECISE PROBABILITIES
Because an imprecise model permits, and indeed requires, a choice of updating rule, it may exhibit updates that has troubling interpretations, notably
dilation, contraction and sure loss. This section supplies a detailed look at these
phenomena. We emphasize that the subscript “•” used in the definitions below is crucial because, given the same imprecise model specification, a phenomenon can be induced by one rule but not by another. The choice among
updating rules is inseparable from the choice of assumption of a missing information mechanism, and it would be wrong to think that an observable event,
as a mathematical constraint, is taken literatim in imprecise probability conditioning. The operational interpretations of Dempster’s rule and the Geometric
rule presented in Section 3 highlight the different uses, by different rules, of the
information in the same event being conditioned upon.
3.1 Dilation and contraction
Definition 3.1 (Dilation). Let A ∈ B (Ω) and B be a Borel measurable
partition of Ω. Let Π be a closed, convex set of probability measures defined
on Ω, P its lower probability function, and P• the conditional lower probability
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

11

function supplied by the updating rule “•”. We say that B strictly dilates A
under the •-rule if
(3.1)

sup P• ( A | B) < P ( A) ≤ P ( A) < inf P• ( A | B) .
B∈B

B∈B

If either (but not both) outer inequality is allowed to hold with equality, we
simply say B dilates A under the said updating rule.
Dilation means that the conditional upper and lower probability interval of
an event A contains that of the unconditional interval, regardless of which B in
the space of possibilities B is observed. Inference for A, as expressed by the imprecise probabilities under the chosen updating rule, will become strictly less
precise regardless of what has been learned. This is commonly perceived as
unsettling, because one would expect that learning, at least in some situations,
ought to help the model deliver sharper inference, reflected in a tighter probability interval. But when dilation happens, it seems that as we learn, knowledge
does not accumulate and quite the contrary, diminishes surely.
If dilation is something one finds unsettling, the opposing notion, contraction, should be nothing less. Contraction happens when the posterior upper
and lower probability interval becomes strictly contained within that of the
prior, regardless of what is being learned. If a tighter probability interval symbolizes more knowledge, when contraction happens, it is as if some knowledge
is created out of thin air. How could it be that whatever is learned, we could
always eliminate a fixed set of values of probability that were a priori considered possible? If we could have eliminated them by a pure thought experiment
that can never fail, why wouldn’t we have eliminated them a priori? Formally,
contraction is defined as follows.
Definition 3.2 (Contraction). Let A, B and P• be the same as in Definition
3.1. We say that B strictly contracts A under the •-rule if
(3.2)

P ( A) < inf P• ( A | B) ≤ sup P• ( A | B) < P ( A) .
B∈B

B∈B

If either (but not both) outer inequality is allowed to hold with equality, we
simply say B contracts A under the said updating rule.
We now illustrate these two unsettling updating phenomena using Example
2, although we defer the discussion of their interpretations to Section 6.
Example 2 cont. (The boxer, the wrestler, and the coin flip).
model, we know precisely that the coin is fair:
(3.3)

By the setup of the

P( X = 0) = P( X = 1) = 1/2.

However, no information is available about either fighter’s chance of winning.
That is, if we assume the probability of a boxer’s win P(Y = 1) = π, π is
allowed to vary between [0, 1]. Then according to the imprecise model,
(3.4)

P (Y = 1 ) = 0 ,

P (Y = 1 ) = 1

imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

12

GONG AND MENG

and similarly so for the wrestler’s win: P(Y = 0) = 0, P(Y = 0) = 1. The
known probabilistic margins specify a belief function, as displayed in Table 1.
Table 1
Example 2 (boxer and wrestler): mass function representation of the belief function model
coin lands head, either fighter wins

coin lands tails, either fighter wins

( X, Y ) ∈ {1} × {0, 1}

( X, Y ) ∈ {0} × {0, 1}

0.5

0.5

m (·)

When told X = Y, how should the model at hand be revised? Two aspects
are worth noting:
i) Posterior inference for the fighters. As Gelman (2006) noted, Dempster’s rule
contracts the boxer’s chance of winning, because
PD (Y = 1 | X = Y ) = 1/2,

PD (Y = 1 | X = Y ) = 1/2,

PD (Y = 1 | X 6= Y ) = 1/2,

PD (Y = 1 | X 6= Y ) = 1/2

which are strictly contained within the vacuous prior probability interval as in
(3.4). The calculations given the two alternative conditions X = Y and X 6= Y
are identical due to symmetry of the setup. In contrast, the generalized Bayes
rule cannot contract vacuous prior interval, in this example and in general (see
Theorem 5.8).
ii) Posterior inference for the coin. Intriguingly, the generalized Bayes rule dilates
the precise a priori information (3.3) on the coin’s chance of coming up heads,
because
PB ( X = 1 | X = Y ) = 0,

PB ( X = 1 | X = Y ) = 1,

PB ( X = 1 | X 6= Y ) = 0,

PB ( X = 1 | X 6= Y ) = 1.

In contrast, Dempster’s intervals remain identical to that of the prior interval under either X = Y or X 6= Y. Notice that in this example, P ( X = Y ) =
P ( X 6= Y ) = 0 hence the Geometric rule is not applicable. The generalized
Bayes rule in the sense of Seidenfeld & Wasserman (see Definition 2.5) is not
applicable either; however since the the model is a belief function, we use Fagin
& Halpern’s results as given in (2.11) and (2.12) to obtain the above expressions. This is equivalent to minimizing and maximizing over the restricted sets
of probabilities { P : P ≥ P & P ( X = Y ) > 0} and { P : P ≥ P & P ( X 6= Y ) > 0}
respectively, thus avoiding ill-defined probability ratios.
3.2 Sure loss
The next type of updating anomaly is even more unsettling, as it is usually
regarded as an infringement on the logical coherence of probabilistic reasoning.
Definition 3.3 (Sure loss). Let A, B , P, and P• be the same as in Definition
3.1. We say that B incurs sure loss in A under the •-rule if either
(3.5)

inf P• ( A | B) > P ( A) ,

B∈B

imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

13

or
(3.6)

sup P• ( A | B) < P ( A) .
B∈B

Sure loss describes a universal and uni-directional displacement of probability judgment before and after conditioning on any event from a subalgebra. That is, after learning anything, the event in question becomes altogether
more (or less) likely than before. The terminology “sure loss” stems from the
Bayesian decision-theoretic context, where probabilities are seen to profess personal preferences contingent on which one is willing to make bets. If B incurs
sure loss in A, the beholder of P and P• as her personal prior and posterior
imprecise probabilities respectively, can be made to commit a compound bet
with a guaranteed negative payoff.
To see this, let s, t be two numbers such that infB∈B P• ( A | B) > s > t >
P ( A), that is, we assume sure loss in the form of (3.5). Since t > P ( A), I shall
accept a bet for which I pay 1 − t, get 1 back if A did not occur, and nothing if it
did, because my expected payoff is P( Ac ) − (1 − t) = t − P( A) ≥ t − P( A) > 0.
On the other hand, since P• ( A | B) > s for all B, contingent on any B I shall
also accept bets for which I pay s, get 1 back if A did occur and nothing if it did
not, because regardless of which B occurs, my expected payoff P( A | B) − s ≥
infB∈B P• ( A | B) − s > 0. It therefore seems perfectly logical for me to take both
bets, as both are expected to have positive return. However, if I do take both
bets, then the compound bet is the one with guaranteed payoff of only 1, less
than what I have paid for 1 − t + s because s > t. Therefore, endorsing P• as
the updating rule means one is willing to accept a finite collection of bets and
be certain to lose money, a trademark incoherent behavior.
Note that if B incurs sure loss in A in the form of (3.5), it equivalently incurs
sure loss in Ac as well in the form of (3.6), though perhaps the term sure gain
would be more appropriate – in Émile Borel’s words, the former the “imbecile”
and the latter the “thief”. Whenever a distinction is necessary, we will use the
term sure gain in addition to sure loss to highlight the directionality of displacements of posterior probability intervals compared to that of the prior, and
will otherwise follow the pessimistic convention (which seems to be a hallmark
of statistical or probabilistic terms, such as “risk”, “regret”, “regression”) of the
literature and use “sure loss” to refer to both situations if non-ambiguous.
We emphasize again that both dilation and sure loss, as concepts describing
the change from prior to posterior sets of probabilities, are contingent upon the
updating rule. Even with the same imprecise probability model P, the same
partition B and the same event A, it can well be the case that B dilates A
under one rule and induces sure loss in A under the other. Example 3 below
is a situation in which all three rules behave very differently, and Section 5 is
dedicated to a characterization of their differential behavior.
We are now ready to take a careful look at the three prisoners paradox.
Example 3 cont. (Three prisoners). What do we have about the probabilistic
model behind the three prisoners’? Since exactly one of the three prisoners will
receive parole randomly, the prior probabilities of living for each of them are
all exact:
P ( A lives) = P ( B lives) = P (C lives) = 1/3.
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

14

GONG AND MENG

Furthermore, since the guard cannot lie, he has no choice on who to report if
the inquirer A does not receive parole. That is,
P(guard says C | B lives) = P(guard says B | C lives) = 1.
The above probability specification can be expressed as a belief function model,
with mass distribution dictated by the known model margins as represented in
Table 2:
Table 2
Example 3 (three prisoners): mass function representation of the belief function model

m (·)

A lives, guard says {B, C}

B lives, guard says C

C lives, guard says B

1/3

1/3

1/3

We see from the specification that what remains unknown is, in case A indeed receives parole, the propensity of the guard reporting either B or C as
dead had he the freedom to choose:
(3.7)

δB = P(guard says B | A lives) ∈ [0, 1].

As a consequence, the posterior probability of A living is
(3.8)

P( A lives | guard says B) = δB /(1 + δB ).

This extra degree of freedom δB fully characterizes the set of probabilities implied by the model.
There is a long literature documenting the variety of modes of reasoning
to this problem, e.g., Mosteller (1965) and Morgan et al. (1991) which invoked
a similar construction as the δB above, in explicating the reasons why many
are seemingly intuitive yet riddled with logical fallacies. Four types of “popular” answers are reproduced below, reflecting different ways of treating the
unknown value δB . What’s interesting is that, as we will see, three of these
answers correspond to those given by the three conditioning rules respectively.
i) The indifferentist: assumption of ignorability One of the most commonly made
assumptions is that the guard has no preference one way or the other about
who to report when given the freedom, that is, δB = 1/2, thus
P( A lives | guard says B, δB = 1/2) = 1/3.
That is to say, prisoner A would not have benefitted from the knowledge that B
is going to be executed, precisely as he claimed to the guard to begin with. The
assumption of guard’s indifference is equivalent to the ignorability assumption
commonly employed in the treatment of missing and coarse data. Despite being
intuitive, the assumption is not backed by the model description per se. Neither
the posited imprecise model nor the data as reported by the guard can supply
any logical evidence to support the ignorability assumption. Therefore, the assertion that ignorability is “intuitive” is a judgment that can be as unreasonable
as any other seemingly less intuitive ones, such as the ones below.

imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

15

ii) The optimist: Dempster’s rule Applying Dempster’s rule, we have
PD ( A lives | guard says B) = 1/2 ,

PD ( A lives | guard says B) = 1/2.

Thus prisoner A felt happier now that his chance of survival increased from
1/3 to 1/2. This happiness is gained from assuming the optimistic scenario of
δB = 1, that is, the guard chose a reporting mechanism that has the highest
likelihood given A lives. However, one realizes that the guard could have only
reported either B or C, both fully symmetrical in the prior. Had the guard
said C would be executed, A would again apply Dempster’s rule, thus grow
happier following the same logic by effectively assuming δC = P(guard says C |
A lives) = 1. Under the assumption that the guard cannot lie and cannot refuse
to answer, δB + δC = 1, thus δB and δC cannot be 1 simultaneously. Hence the
reasoning that whatever the guard says, the probability of A living will go up
from 1/3 to 1/2, which is equivalent to assuming the impossible δB = δC = 1,
is a direct consequence of a logical fallacy.
iii) The pessimist: the Geometric rule Applying the Geometric rule, we have
PG ( A lives | guard says B) = PG ( A lives | guard says B) = 0
and, by symmetry,
PG ( A lives | guard says C ) = PG ( A lives | guard says C ) = 0.
This answer is perhaps the most striking among all, directly pointing at the
absurdity of the assumptions behind the updating rule within this context.
Upon hearing anything, prisoner A will deny himself of any hope of living,
effectively assuming δB = 0 if guard says B and δC = 0 if guard says C, two
assumptions that are incommensurable with each other because δB + δC = 1,
much in the same way as the previous case with Dempster’s rule.
iv) The conservatist: generalized Bayes rule The solution suggested by Diaconis
(1978), and indeed supplied by the generalized Bayes rule, is
(3.9) PB ( A lives | guard says B) = 0 ,

PB ( A lives | guard says B) = 1/2.

This answer is a direct consequence of (3.8). As δB varies within [0, 1] without
any further assumption, one is bound to concur with (3.9). The caveat to it,
however, is that again due to prior symmetry of B and C, the generalized Bayes
rule will also yield
PB ( A lives | guard says C ) = 0 ,

PB ( A lives | guard says C ) = 1/2.

Hence, the generalized Bayes rule results in posterior probability intervals strictly
containing the prior probability in all situations.
Our use of the vocabulary “optimism”, “pessimism” and “conservatism” to
refer to the three updating rules is informed by the interpretation of their respective posterior inference under the effective assumptions they each impose,
and is reminiscent of that of Fygenson (2008) for modeling of extrapolated
probabilities. These ideological differences illuminate the dynamics among the
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

16

GONG AND MENG

Posterior probabilities of A receiving parole
as functions of δB

P(A lives | guard says C)

0.5

0.4

δB
1.00
0.3

0.75
0.50

0.2

0.25
0.00

0.1

0.0
0.0

0.1

0.2

0.3

0.4

0.5

P(A lives | guard says B)

Fig 1. Posterior probabilities of prisoner A receiving parole given the guard’s two possible answers, as a
function of the guard’s reporting bias δB (3.7).

updating rules for imprecise probability, and highlight the pedagogical significance of the three prisoners’ paradox itself. In this example, Dempster’s rule
updates its conditional lower probability to be greater than that of its prior
upper probability thus incurs sure loss of the form (3.5), the Geometric rule
behaves the opposite way and incurs sure loss of the form (3.6), and the generalized Bayesian rule exhibits dilation. As far as unsettling updating goes, there
seems to be no escape regardless of which rule to choose. How on earth then
do we draw a conclusion?
Reading through the literature, the dilated answer supplied by the generalized Bayes rule is the most accepted solution to the paradox. As counterintuitive as it may be, dilation is a professed consequence of an overfitting nature
of the generalized Bayes rule, for the rule is inclusive of all possibilities allowed within the ambiguous model, to the point of simultaneously admitting
assumptions that are incommensurable with one another. As we saw previously,
the upper conditional probability PB ( A lives | guard says Z ) = 1/2 is achieved
under the assumption δZ = 1, where Z = C, B. Similarly, the lower conditional
probability PB ( A lives | guard says Z ) = 0 is achieved when δZ = 0, with
Z = C, B. Since δC + δB = 1, δC and δB cannot simultaneously be 0 or 1: indeed,
when one is 1 the other must be 0. Hence the permissible value of the pair

{ x = P( A lives | guard says B), y = P( A lives | guard says C )}
forms a one-dimensional curve y = (1 − 2x )/(2 − 3x ) inside the square [0, 1/2] ×
[0, 1/2], as depicted in Figure 1. For a given conditioning event Z, the generalized Bayes rule achieves its extremes by seeking a distribution that itself depends on Z, namely, a conditioning-dependent conditional distribution P(Z) (·| Z ),
a clear case of overfitting. Understanding the hidden incommensurability is important for preventing logical fallacies such as reasoning under the (wrong)
assumption that { x, y} can take any value inside the square [0, 1/2] × [0, 1/2].
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

17

We will return to the three prisoners again in Section 6.2 to discuss its inferential implications. In particular, the three prisoners’ paradox is a direct variant
of the Monty Hall problem, which possesses a clean, indisputable decision recommendation.
3.3 What’s unsettled in unsettling updates?
In case some readers are not yet completely put off by the unsettling updates, we would like to offer a few words about when, as well as when not, one
should find dilation or sure loss unsettling. It seems to us that the attitude one
should take towards these phenomena is contingent upon the way the underlying probability model is interpreted.
Specifically, dilation is troubling when the set of probabilities is used as a
description of uncertain inference. For example, if the probability interval is
regarded as an approximation to some underlying true probability state, akin
to a confidence or posterior interval to an estimand, knowing that the interval
will surely grow wider in the posterior is indeed counterproductive since the
goal of inference in most cases is to tighten the interval. But in this sense, the
sure loss phenomenon may just be fine, since it is common to derive disjoint yet
equally valid confidence or posterior intervals from the same sampling posterior distribution, without violating any classic rules of probabilistic calculation.
On the other hand, as explained in Section 3.2, the lower and upper probabilities can be taken as acceptable prices of a gamble. Under this interpretation,
any strategy that induces sure loss is absolutely unacceptable. However in this
case, dilation is much less to be worried about: a strictly wider interval in the
posterior will simply exclude the player from engaging in the conditional bet,
and does not violate coherence in a decision-theoretic sense.
With precise probabilities, by conditioning on an observable event we are
simply imposing a restriction to the sub-space that is defined by that event,
which itself is assumed to be measurable with respect to the original probability
space. With imprecise probabilities, not all events are measurable with respect
to the imprecise probability model specified on the full joint space, and a crucial
way the updating rules differ from one another is how they make use of this
supplied conditioning information. Therefore, for any of the updating rules to
function at all, they must build within themselves a particular “mechanism” of
imposing the mathematical restriction specified by the observable event, when
it is not currently measurable with respect to the set of probabilities the rule
aims to update, much in the same way as a sampling mechanism (Kish, 1965)
or missing-data mechanism (Rubin, 1976). The fact that dilation and sure loss
cannot happen under the precise probability does not necessarily render them
undesirable: the quality of this inference hinges on the quality of the final action
they recommend. Bringing these anomalies to light allows us to study their
implications, especially those unfamiliar or unexpected, on the final action.
4. SIMPSON’S PARADOX: AN IMPRECISE MODEL WITH
AGGREGATION SURE LOSS
One may well think that the examples discussed in Section 1 lie on the
boundary, if not outside, of the realm of mainstream statistical modeling. Imprecise models do not seem to be the kind of thing one just stumbles upon,
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

18

GONG AND MENG

they exist only when one makes them exist. We argue that such is not the case,
that all precise models are really just the tip of an “imprecise model iceberg”.
That is, every precise model is a fully specified margin nested within a larger,
ever-augmentable model, with extended features we have not allowed entrance
to the modeling scene, and possibly lacking the knowledge to do so precisely.
Here is a concrete way to induce an imprecise model
from a precise one.

Take any precise model with dimensions X1 , ..., X p which jointly merit a
known multivariate distribution. If the model is expanded to include a previously unobservable margin X p+1 , all of a sudden the state space becomes
( p + 1)-dimensional. The resulting augmented model becomes imprecise, for
as many as 2 p new marginal relationships are left to
 be specified or learned –
that is, between X p+1 and any subset of X1 , ..., X p . In the regression setting
where a multivariate Normal model is assumed for the previous p variables,

one seemingly straightforward way is to go and model X1 , ..., X p , X p+1 as
jointly Normal, which is already a very strong assumption that takes care of
a vast majority of the 2 p relationships. Even under such drastic simplification,
the mean and p + 1 bivariate covariances are still left to specify, resulting in a
family of ( p + 1)-dimensional Normal models.

In reality, the relationship between the existing X1 , ..., X p and a new X p+1 is
often something the analyst is neither knowledgeable nor comfortable assuming. This is the case when X p+1 is a lurking variable in observational
studies

which may have strong collinearity with subsets of X1 , ..., X p . Using the language of imprecise probability, we now turn to decipher Simpson’s paradox, a
famous and familiar setting with its far-reaching significance. The very occurrence of Simpson’s paradox is proof that we have employed, likely due to lack
of control, an aggregation rule that has incurred sure loss in inference.
Example 4 cont. (Simpson’s paradox). Following the setup in Section 1, Simpson’s paradox refers to an apparent contradiction between an inference on treatment efficacy at an aggregated level, p̄obs < q̄obs , and the inference at the disaggregated level when the covariate type of the patient has been accounted for:
pk > qk for all k = 1, ..., K. Indeed, how can a treatment be superior than its
alternative in every possible way, yet be inferior overall?
4.1 Explicating the aggregation rules underlying the Simpson’s paradox
Denote for k = 1, ..., K,
u k = P (U = k | Z = 1 ) ,

v k = P (U = k | Z = 0 ) .

Here, u and v reflect the demographic distribution of the populations receiving the experimental and control treatments, respectively. By the law of total
probability,
(4.1)

p̄ = p> u

and

q̄ = q> v,

and thus given fixed p and q, p̄ and q̄ are functions of u and v respectively. The
marginal probabilities p̄ and q̄ are meant to describe an event under conditions
of inferential interest, in this case, patient recovery within the two treatment
arms. We refer to u and v as aggregation rules, functions that map conditional
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

19

Fig 2. Ideal aggregating rules guarantee the comparison between treatment arms is made on a fair ground.
Observed Simpson’s paradox is strong evidence that the de-facto aggregating rules are fair for comparison.
Left: if pk > qk for all k, then p> v > q> v for all v; Right: disparate uobs and vobs make possible
pobs < qobs . Note that Π in Theorem 4.1 is the convex hull sandwiched between the blue (p) and red (q)
hyperplanes in the first octant.

probabilities to a marginal probability, which is in reverse direction compared
to updating rules as discussed in the previous sections, which are maps from a
marginal probability to a set of conditional probabilities.
Typically, measurements between different conditions are made for the purpose of a comparison, such as the evaluation of an causal effect of treatment Z
on outcome Y. A comparison between p̄ and q̄ is fair if and only if the aggregation rules they employ are identical, that is, u = v as in (4.1). This is what it
means to say the comparison has been made between apples and apples. Such
is the case if no confounding exists between the covariate U and the propensity
of assignment, i.e., U ⊥ Z.
Clearly, when u = v, p̄ > q̄ if pk > qk for all k. Hence Simpson’s paradox
is mathematically impossible within a fair comparison. However, for a given
observed pair p̄obs and q̄obs , have we been careful enough to enforce the de-facto
aggregation rules to equal the ideal one? That is, do we have that the observed
comparison is fair enough, i.e., there exists a v such that
.
.
(4.2)
uobs = v
and
vobs = v?
For certain values of p and q, it is entirely possible that suitable realizations
of (uobs , vobs ) within the K-simplex could result in p̄obs < q̄obs . To be exact, these
are p and q values satisfying maxk qk > mink pk . At least one, and possibly both
realizations of uobs and vobs play differentially to the relative weaknesses of p,
i.e., coordinates of smaller magnitude, and the strengths of q accordingly. When
this preferential weighting, also known as confounding, is strong enough to reverse the perceived stochastic dominance of the outcome variable under either
treatment, an apparent paradox is induced. Randomization procedures effectively put quality guarantees on the fairness of comparison; as the sample size
n grows larger, (4.2) holds with high probability with deviations quantifiable
with respect to p and q that is immune against all U, observed or unobserved.
4.2 The paradox is sure loss
Simpson’s paradox is reminiscent of the “sure loss” phenomenon we saw
in earlier sections. Indeed, when not conditioned on U, if asked to pick a bet
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

20

GONG AND MENG

between the experimental and control treatments, we would prefer the control
treatment over the experimental one. But once conditioned on U, the experimental treatment suddenly became the superior bet regardless of U’s value.
One is thus set to surely lose money by engaging in a combination of these
two bets. That is, We formalize this idea in the following theorem, where SK
is the standard K-simplex defined by {(v1 , . . . , vK ) : ∑kK=1 vk = 1; vk ≥ 0, k =
1, . . . , K }.
Theorem 4.1 (Equivalence of Simpson’s paradox and aggregation sure loss). Let
Λ be a convex hull in [0, 1]K characterized by the pair of element-wise upper and lower
bounds (p, q). That is,
n
o
Λ = λ ∈ [0, 1]K : qk ≤ λk ≤ pk , k = 1, ..., K .
Let V ⊆ SK be a closed set (with respect to the Euclidean distance on RK ) of desirable
aggregation rules, and let u ∈ SK . Then, u incurs sure loss on Λ relative to V if and
only if (u, v) induces Simpson’s paradox in (p, q) for all v ∈ V .
Proof. Denote the set of marginal nprobability derived ofrom Λ under the
desirable aggregating rule V as PV = λ> v : λ ∈ Λ, v ∈ V . By the closeness
of both Λ and V (with respect to the Euclidean distance on RK ), we have
(4.3)

inf PV = inf q> v
v∈V

and

sup PV = sup p> v,
v∈V

and
(4.4)

p> u = sup λ> u

and

q> u = inf λ> u.
λ∈Λ

λ∈Λ

Employing Definition 3.3, to say that u incurs sure loss on Λ relative to V means
that
(4.5)

sup λ> u < inf PV

or

λ∈Λ

inf λ> u > sup PV .

λ∈Λ

On the other hand, to say that for every v ∈ V , (u, v) induces Simpson’s paradox in (p, q) means that
(4.6)

p> u < inf q> v
v∈V

or

q> u > sup p> v.
v∈V

By (4.3)-(4.4), conditions (4.5) and (4.6) are trivially the same, and hence the
theorem.
We remark that, in Definition 3.3, sure loss is defined with respect to a single
conditioning rule because the prior/marginal lower and upper probabilities P
and P are treated as given. Such is not the case with the sure loss concept in
Theorem 4.1. This is why we must first define a set of desirable rules v ∈ V
which implies a prior/marginal probability interval, before discussing the behavior of the other aggregation rule u relative to it. One can check that the
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

21

relationship between u and v is reciprocal, that is, if u induces sure loss relative to v, then v induces sure loss relative to u. Thus, we can talk about an
aggregation scheme as an ordered pair of rules (u, v) ∈ K-simplex2 , and its characteristics as whether it incurs sure loss with respect to itself, whether it induces
the paradox in (p, q), and so on.
Indeed, this is the case with respect to the atomic lower and upper probability
(ALUP) model of Herron et al. (1997). A set of probabilities Π(p,q) is an ALUP
generated by (p, q), where p, q ∈ [0, 1]K , if
(4.7)

Π(p,q) = {π ∈ SK : sup πk = pk , inf πk = qk }

A connection between ALUP model and Simpson’s paradox is made below.
Lemma 4.2 (ALUP models). If an aggregation scheme (u, v) induces Simpson’s
paradox in (p, q), it incurs sure loss with respect to itself on the ALUP model Π(p,q) ,
as defined in (4.7).
Proof. Without loss of generality, suppose an aggregation scheme (u, v)
induces Simpson’s paradox in (p, q) in the form of p> u = supλ∈Λ λ> u <
infλ∈Λ λ> v = q> v. But since Π(p,q) is a closed and convex subset of Λ, we
have supλ∈Λ λ> u ≥ supπ ∈Π( p,q) π > u and infλ∈Λ λ> v ≤ infπ ∈Π( p,q) π > v, hence
the “only if” part of Theorem 4.1 still holds.
4.3 Implication on inference
In Example 4, the description of the model is precise with the conditional
values p and q, as well as the marginal values p̄obs and q̄obs . The model is
imprecise, and in fact completely vacuous, on the aggregation rules (uobs , vobs )
which gave rise to the observed marginal values.
In order for the observed marginal probabilities p̄obs and q̄obs to yield a meaningful comparison, we must have clear answers to the following two questions
regarding uobs and vobs :
1. Are they equal?
2. What is the mutual value v they both should be equal to?
An affirmative answer to the first question ensures that p̄obs and q̄obs are at least
on a comparable footing. For example, for the evaluation of an causal effect of
Z on Y, regardless of the population of interest, it must be ensured that no confounding between the covariate U and the propensity of assignment took place,
i.e., U ⊥ Z. That is why Simpson’s paradox is a sanity check for any apparent
causal relationship, as the paradox constitutes sufficient (but not necessary) evidence there is non-negligible confounding between U and Z, a telltale sign that
one is comparing apples with oranges. Much classic and contemporary literature on causal inference sensitivity analysis, e.g., Cornfield et al. (1959); Ding
& VanderWeele (2016), hinge on establishing deterministic bounds to exclude
scenarios that are in essence Simpson’s paradoxes, as well as quantifying the
probability of population-level paradox given observed paradox in the sample,
e.g., Pavlides & Perlman (2009). If the assignment Z cannot be controlled in
one or both treatment arms, the aggregation rule is no longer chosen by the
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

22

GONG AND MENG

investigator/statistician but rather left self-selected, in all or in part by the observational mechanism. In particular, if arbitrary confounding can be present
in both treatment arms, u and v can take up any value in the K-simplex. It is
also entirely possible that controlled randomization or weighting is available
in only one of the treatment arms, or on a subset of levels of U, reflecting an
aggregation rule as a mixture of intentional choice and self-selection.
It is also crucial that the ideal aggregation rule v, the mutual value for both
uobs and vobs , is a conscious choice made by the investigator to reflect the scientific question of interest. Two typical situations that give rise to natural choices
of v are:
• to make inference about population average treatment effect, choose v to
be the oracle probability distribution of patients’ covariates in the population;
• to make inference about a particular patient’s treatment effect, choose
v=

0 ···

0 1(Ui =k) 0 · · ·

0

0

the indicator vector matching the patient’s covariate value Ui with its level
k.
One can devise a range of choices of v to reflect any amount of intermediate
pooling within what is deemed as the relevant subpopulation. As discussed in
Liu & Meng (2014, 2016), what defines the game of individualized inference is
picking the v at the appropriate resolution level while subject to the tradeoff
between population relevance and estimation robustness.
Choosing the right v and enforcing uobs = vobs = v is not merely a mathematical decision to be made on paper, but rather entails action in a real-life
observational environment, one that likely involves the physical activities of
stratification and randomization such as controlled experiments and survey
designs. Only through doing so can we make sure the de-facto aggregation
rules are equal to the ideal rule, or equivalently that we know executable ways
to adjust for the differences between these quantities should there be any, e.g.,
through retrospective weighting. Failure to acknowledge the distinction and
potential differences among v, uobs , and vobs paves the way not only for Simpson’s paradox, but also equivalently for endorsing mythical statistical aggregation rules with the potential to exhibit incoherent behavior, and the worst of all,
to mislead ourselves in making the wrong (treatment) decisions, a sure loss in
real sense.
5. BEHAVIOR OF UPDATING RULES: SOME CHARACTERIZATIONS
This section presents some results on the behavior of the three updating
rules discussed in this paper. We begin with the intuitive ones and progress
towards those that are perhaps surprising. Unless otherwise noted, this section
assumes that P is a Choquet capacity of order 2 on B (Ω), and Π = { P :
P ≥ P}, the closed and convex set of probabilities compatible with it. Recall
PB , PD , and PG are the conditional lower probability functions according to
the generalized Bayes (Def. 2.5), Dempster’s (Def. 2.6) and the Geometric rules
(Def. 2.7) respectively.
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

23

5.1 Generalized Bayes rule cannot contract nor induce sure loss
Lemma 5.1. Let B = { B1 , B2 , ...} be a measurable and denumerable partition of
Ω. For any A ∈ B (Ω), we have
inf PB ( A | Bi ) ≤ P ( A) ,

Z ∈B

and

sup PB ( A | Bi ) ≥ P ( A) .
Bi ∈B

Proof. The proof is by contradiction. Assume that infBi ∈B PB ( A | Bi ) > P ( A).
For the given A, because Π is a closed set, there exists a P( A) ∈ Π such that
P( A) ( A) = P ( A). Note that the superscript in P( A) reminds us that this probability measure can vary with the choice of A. But this does not affect the validity
of applying the total probability law under this chosen P( A) , which leads to
P ( A ) = P( A) ( A ) =

∞

∑ P( A) ( A | Bi ) P( A) ( Bi )

i =1
∞

≥

∞

PB ( A | Bi ) P( A) ( Bi )
∑ PB ( A | Bi ) P( A) ( Bi ) ≥ ∑ inf
B

i =1
∞

>

i =1

i

∑ P ( A) P( A) ( Bi ) = P ( A) ,

i =1

resulting in a contradiction. The same argument applies for the upper probability of A, that if supBi ∈B PB ( A | Bi ) < P ( A), then using P ( A) = P̃( A) ( A),
P ( A) ≤

∞

∞

i =1

i =1

∑ PB ( A | Bi ) P̃( A) ( Bi ) < ∑ P ( A) P̃( A) ( Bi ) = P ( A) ,

and hence again a contradiction.
A direct consequence of Lemma 5.1 is the following thorem.
Theorem 5.2. Let B be a denumerable and measurable partition of Ω, and Π be
the set of probability measures compatible with P. For any event A ∈ B (Ω), under the
generalized Bayes rule,
• B cannot induce sure loss in A,
• B cannot contract A.
The first part of Theorem 5.2, that the generalized Bayes rule avoids sure loss,
is well-known in the literature and is the very reason that many authors such
as Walley (1991) and Jaffray (1992) consider it to be the sole choice as coherent
updating rule, or the “conditioning proper”. However, as we will show next,
the generalized Bayes rule is also the most prone to dilation.
5.2 Generalized Bayes rule produces a superset of probability measures
Lemma 5.3 (Generalized Bayes rule produces the widest intervals). For all A, B ∈
B (Ω) such that the following quantities are defined, we have
(5.1)

PB ( A | B) ≤ PD ( A | B) ≤ PD ( A | B) ≤ PB ( A | B)

and
(5.2)

PB ( A | B) ≤ PG ( A | B) ≤ PG ( A | B) ≤ PB ( A | B) .
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

24

GONG AND MENG

That is, the conditional probability intervals resulting from Dempster’s rule
and the Geometric rule are always contained within those of the generalized
Bayes rule. The fact that Dempster’s rule produces shorter posterior intervals
than that of the generalized Bayesian rule was discussed in Dempster (1967)
and Kyburg (1987). Here we supply a straightforward proof that applies to
both sharper rules.
Proof. For Dempster’s rule, the conditional plausibility function satisfies
PD ( A | B) =

supP∈Π P ( A ∩ B)
P ( A ∩ B)
≤ sup
= PB ( A | B )
supP∈Π P ( B)
P ( B)
P∈Π

and by conjugacy, also PD ( A | B) ≥ PB ( A | B). Similarly for the Geometric
rule, the conditional lower probability function satisfies
PG ( A | B) =

infP∈Π P ( A ∩ B)
P ( A ∩ B)
≥ inf
= PB ( A | B )
P∈Π
infP∈Π P ( B)
P ( B)

and by conjugacy, also PG ( A | B) ≤ PB ( A | B).
Theorem 5.4 (Generalized Bayes rule). Let B ∈ B (Ω) be such that P( B) > 0.
Denote sets of posterior probability measures ΠB = { P : P ≥ PB (· | B)}, ΠD =
{ P : P ≥ PD (· | B)} and ΠG = { P : P ≥ PG (· | B)}. Then,
(5.3)

ΠG ⊆ ΠB

and

ΠD ⊆ ΠB .

Theorem 5.4 is a direct consequence of Lemma 5.3, noting that ΠG , ΠB and
ΠD are all convex and closed. Two more consequences of Lemma 5.3 are stated
below; in particular, Examples 3 and 5 are respective embodiments of the two
corollaries below.
Corollary 5.5. If B incurs sure loss in A under Dempster’s rule and sure gain
under the Geometric rule, or vice versa, then B strictly dilates A under generalized
Bayesian rule.
Corollary 5.6. If B (strictly) dilates A under either Dempster’s rule or the Geometric rule, then B (strictly) dilates A under generalized Bayesian rule.
Theorem 2.1 of Seidenfeld & Wasserman (1993) stated that, if dilation occurs
with the generalized Bayesian rule, the associated set of probabilities Π has a
non-empty intersection with that of the independence plane between A and B.
Thus following Corollary 5.6 we have
Corollary 5.7. If B = { B, Bc } dilates A under either Dempster’s rule or the
Geometric rule, then there exists P∗ ≥ P such that
(5.4)

P∗ ( A ∩ B) = P∗ ( A) P∗ ( B) .

That is, dilation of an event by a binary partition under either Dempster’s or
the Geometric rules is a necessary condition for the posited set of probabilities
to contemplate event independence, since posterior intervals under both rules
are contained within the generalized Bayes posterior interval.
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

25

5.3 Generalized Bayes rule and Geometric rule cannot sharpen vacuous prior
intervals
Theorem 5.8 (Sharpening of vacuous intervals). Let P be such that P ( A) = 0,
P ( A) = 1. For any B ∈ B (Ω) such that P ( B) > 0, we have
(5.5)

PG ( A | B) = 0 ,

PG ( A | B) = 1

PB ( A | B ) = 0 ,

PB ( A | B) = 1.

and
(5.6)

Proof. Notice that if P ( A) = 0 and P ( A) = 1, then P ( A ∩ B) = P ( Ac ∩ B) =
0 for any B. Therefore, by (2.9) we have
PG ( A | B) = P ( A ∩ B) /P ( B) = 0
and PG ( A | B) = 1 − PG ( Ac | B) = 1 provided that the denominator is greater
than zero. Furthermore, by (5.1) we have PB ( A | B) ≤ PG ( A | B) = 0 and
PB ( A | B) ≥ PG ( A | B) = 1.
The liberty to express partially lacking, and vacuous, prior knowledge is a
prized advantage of imprecise probability over their precise, or full Bayesian,
counterparts. Theorem 5.8 shows that both the generalized Bayes rule and Geometric rule are incapable of revising a vacuous prior interval to something
informative for any possible outcome in the event space, whereas Dempster’s
rule is capable of such revision, Example 1 being an instance. This again highlights the non-negligible influence imposed by the rule itself, as well as the
difficulty to deliver all desirable properties in one single rule. Avoiding sure
loss and being able to update from complete ignorance both seem to be rather
basic requirements, but together they are sufficient to eliminate all three rules
studied here. The following result perhaps is even more disturbing, because it
says that in the world of imprecise probabilities, not only must we live with
imperfections, but also accept intrinsic contradictions.
5.4 The counteractions of Dempster’s rule and Geometric rule
Theorem 5.9. If B = { B, Bc } dilates A under the Geometric rule, then it must
contract A under Dempster’s rule. Similarly, if B dilates A under Dempster’s rule,
then it must contract A under the Geometric rule. In both cases, the contraction is
strict if the corresponding dilation is strict.
Proof. If B strictly dilates A under the Geometric rule, then for either Z ∈ B
(5.7)

PG ( A | Z ) =

P ( A ∩ Z)
< P ( A) ,
P (Z)

(5.8)

PG ( A | Z ) =

P ( A ∪ Zc ) − P (Zc )
> P ( A) .
P (Z)

imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

26

GONG AND MENG

It follows then
PD ( A | B)
P ( A)

=

P ( A ∩ B)
P ( A ∩ B)
=
P ( A) · P ( B)
P ( A) · (1 − P ( Bc ))

<

P ( A ∩ B)


P ( A) · 1 − P ( A ∪ B) − P ( B) /P ( A)

=

P ( A ∩ B)
≤ 1,
P ( A) + P ( B) − P ( A ∪ B)



where the first inequality follows from (5.8) with Z = Bc , and the second inequality is based on the 2-alternating nature of P. (The 2-alternating nature was
also implicitly used in the first inequality to ensure P ( A ∪ B) − P ( B) < P ( A),
hence the positivity of the denominator after replacing P( Bc ) with an upper
bound.) In a similar vein,
PD ( A | B)
P ( A)

=
≥
=

P ( A ∪ Bc ) − P ( Bc )
P ( B) − P ( Ac ∩ B)
=
P ( B) · P ( A)
P ( B) · P ( A)
c
P ( A) − P ( A ∩ B )
(1 − P ( Bc )) · P ( A)
P ( A) − P ( A ∩ Bc )
> 1,
P ( A) − P ( Bc ) · P ( A)

where the first inequality uses the 2-monotone nature of P and the second
inequality is based on (5.7) with Z = B. Thus we have PD ( A | B) < P ( A) and
PD ( A | B) > P ( A), and clearly both inequalities still hold when we replace
B by Bc because (5.7)-(5.8) hold for both Z = B and Z = Bc , hence B strictly
contracts A under Dempster’s rule. If B dilates A under the Geometric rule
but not strictly, the inequality in either (5.7) or (5.8), but not both, may hold
with equality, hence B contracts A under Dempster’s rule but not strictly. This
completes the proof for the first half of the statement.
For the second half, when B strictly dilates A under Dempster’s rule, we
have for any Z ∈ B ,
PD ( A | Z ) =
PD ( A | Z ) =

P ( A ∩ Z)
> P ( A) ,
P (Z)
P ( A ∪ Zc ) − P (Zc )
< P ( A) .
P (Z)

Noting both inequalities hold for Z and Z c , we have
1>

P ( A ∪ Z) − P (Z)
P ( A) − P ( A ∩ Z )
≥
.
c
P
( A) − P ( A) · P ( Z )
P ( A) · P ( Z )

Hence P ( A) < P ( A ∩ Z ) /P ( Z ) = PG ( A | Z ). On the other hand,

P ( A) − P ( A ∪ Zc ) − P ( Zc )
P ( A ∩ Zc )
1<
≤
.
P ( A) · P ( Zc )
P ( A) − P ( A) · P ( Z )

Hence P ( A) > P ( A ∪ Z c ) − P ( Z c ) /P ( Z ) = PG ( A | Z ). The same argument
applies that if B dilates A under Dempster’s rule but not strictly, it contracts
A under the Geometric rule but not strictly. This completes the proof for the
second half of the statement.
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

27

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

5.5 Visualizing Relationships and Complications
Example 5 (Pre-election poll). Suppose that we intend to study the voter
intention prior to the 2016 US election. For simplicity, assume there are only
two parties, represented respectively by Clinton and Trump, with one to be
elected. The pre-election poll consists of two questions:
1. Do you intend to vote for Trump or Clinton?
2. Do you identify more as a Republican or a Democrat?
Among all surveyed individuals, some answered both questions, some only
one, and the rest did not respond. Let Q1 = {Trump, Clinton} denote votes
for Trump and Clinton, respectively, and Q2 = {Republican, Democrat} denote identification with the Republican and Democratic parties, respectively.
If all the percentages of response patterns are fully known, this model can be
represented as a belief function. Assume the mass function m (·) reflecting the
coarsened sampling distribution for these set-valued observations appears as
Table 3 (of course, the numbers are for illustrations only):
Table 3
Hypothetical data from a voter poll consisting of two questions

Q1

C

T

C

T

C

T

(n/a)

(n/a)

(n/a)

Q2

Dem

Dem

Rep

Rep

(n/a)

(n/a)

Dem

Rep

(n/a)

m(·)

0.1 − e

0.2 + 8e

A “tuning parameter” e ∈ [−0.025, 0.1] is installed to create a family of mass
function specifications in order to investigate the differential behavior among
updating rules as a function of the coarseness of the data. The smaller the e, the
more the mass function concentrates on the precise observations (more survey
questions answered). The larger the e, the closer the random set approaches the
vacuous belief function. As a function of e, the prior lower and upper probabilities for Clinton are
P (C) = 0.3 − 3e ,

P (C) = 0.7 + 3e.

The prior lower and upper probabilities for Trump, as well as for identification
of either parties are numerically identical to the above, since the setup is fully
symmetric with respect to both voting intention and partisanship. For example,
when e = 0, the table above shows that altogether 40% of the respondents
diligently answered both questions, 20% only identified prior partisanship, 20%
only expressed current voting intentions, and another 20% did not respond at
all. Thus, m (·) determines a pair of belief and plausibility functions which
bounds the vote share for both Clinton and Trump to be within 30% and 70%.
How will information on partisanship affect the knowledge on voting intention? According to the three updating rules, the lower and upper probabilities

imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

28

GONG AND MENG

Prior and posterior intervals for Clinton, given partisanship
1.00

probability

0.75

Prior
Dempster's

0.50

gen. Bayes
Geometric
0.25

0.00
−0.025

0.000

0.025

ε

0.050

0.075

0.100

Fig 3. Prior probability interval for Clinton’s voter support (black) and posterior probability intervals
given reported partisanship according to the three updating rules (blue: generalized Bayes, red: Dempster’s, green: Geometric). Due to full symmetry of the setup, contraction happens under an updating rule
whenever the corresponding posterior interval depicted is contained within the prior interval; vice versa
for dilation.

for Clinton conditional on either values of partisanship Q2 , are as follows:
0.1 − e
,
0.6 + 4e
0.2 − 2e
PD (C | Q2 ) =
,
0.7 + 3e
1
PG (C | Q2 ) = ,
3

PB (C | Q2 ) =

0.5 + 5e
,
0.6 + 4e
0.5 + 5e
PD (C | Q2 ) =
,
0.7 + 3e
2
PG (C | Q2 ) = .
3
PB (C | Q2 ) =

See Figure 3 for the above quantities as functions of e. We observe that
• Under the generalized Bayes rule, knowledge about partisanship strictly dilates voting intention for either candidate for all e < 0.1. That is to say,
learning the prior partisanship of an individual dilates our inference of
her current voting intention, and vice versa, and this is true no matter
which party or candidate is said to be favored;
• Under Dempster’s rule, partisanship strictly dilates voting intention for
either candidate for −0.011 < e < 0.1, and strictly contracts both for
−0.025 < e < −0.011;
• Under the Geometric rule, partisanship strictly dilates voting intention for
either candidate for −0.025 < e < −0.011, and strictly contracts both for
−0.011 < e < 0.1. Moreover, the absolute value of the lower and upper
posterior probability remained constant regardless of the value of e.
Furthermore, we observe some of the phenomena discussed previously in
this section. For example, the extent of dilation exhibited by the generalized
Bayes rule is to a strictly larger extent than that of both Dempster’s rule and
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

29

the Geometric rule, if either of them does dilate. The dilation-contraction status
of Dempster’s rule and the Geometric rule are in full opposition to each other,
switching precisely at e = −0.011.
6. FOOD FOR THOUGHTS
6.1 Assumption incommensurability and conditioning protocol
As revealed in Section 3.3, each imprecise probability updating rule is constantly faced with the problem that the conditioning information may not be
measurable with respect to the very imprecise probability it is trying to update.
As a consequence, they each effectively build within themselves a mechanism
for imposing mathematical restrictions generated by a given event B. This is
why, as far as we can see, the situation in the world of imprecise probability
is more confusing and clearer at the same time. It is more confusing because
the notations P• and P• carry meanings contingent upon the •-rule we choose.
Yet, different rules are built upon different mechanisms for imposing the mathematical restriction specified by an event partition B , in a much similar vein to
the sampling and missing data mechanisms mentioned previously, potentially
supplying a variety of options suitable for different situations that users may
choose from, as long as they are well-informed of the implied assumptions of
each rule. In this sense the situation is clearer, because the imprecise nature
should compel the users to be explicit about the imposed mechanisms in order
to proceed. Below we illustrate this point.
Example 2 cont. (The boxer, the wrestler, and the God’s coin). Recall the boxer
and wrestler example in which there exists a priori, a fair coin and vacuous
knowledge of the two fighters. Our analysis in Section 3 showed that upon
knowing X = Y, Dempster’s rule will judge the posterior probability of boxer’s
win to be precisely half, whereas generalized Bayes rule will remain that the
chance is anywhere within [0, 1]. We realize that the witness who relayed the
message X = Y could have meant it in (at least) two different ways:
1. that he happened to see both the coin flip and the match between the two
fighters, and the results of the two events were identical;
2. that he, somehow miraculously, knew that the coin toss decides the outcome of the match, as if the coin is God’s pseudorandom number generator.
If the first meaning is taken, as most of us naturally do, it seems that the
generalized Bayes answer makes sense. After all, since we do not know the relationship between two co-observed phenomenon, the worst case scenario would
be to admit all possibilities, including the most extreme forms of dependence,
when deriving the probability interval.
However, if the head of the coin dictates the triumph of the boxer, and the
former event is known precisely as a toss-up, it makes sense to think of the
match as a true toss-up as well. In this case, it is rightful to call for a transferral
of the a priori precise probability of X onto the a priori vacuous Y. The same
logic would apply had we been told X 6= Y, in the sense that the head of the
coin dictates the triumph of the wrestler. In both cases, the update is akin to
adding another piece of structural knowledge to the model itself.
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

30

GONG AND MENG

This example reflects well a point made by Shafer (1985). In order for probabilistic conditioning to be properly interpreted, it is crucial to have a “protocol”
specifying what information can be learned, in addition to learning the actual
information itself. Updating in absence of a protocol, or worse, under an unacknowledged, implicit protocol can produce dangerous complications to the
interpretation of the output inference. Dilation and sure loss, phenomena exclusive to imprecise probability, are striking instances that demonstrate such
danger. The difference among the three imprecise model updating rules discussed in this paper is precisely the way the same incoming message might
be interpreted. Each conditioning rule effectively creates a world of alternative
possible observations, hence a protocol is de-facto in place, only hidden behind
these explicit-looking rules.
When performing updates in the boxer and wrestler’s case, the distinction
between conditioning protocols underlying the solutions we have offered so far
is one between factual versus incidental knowledge spaces. Knowing X = Y is
a possible outcome and by chance observing it constitutes incidental knowledge. Knowing that X = Y is the factual state of the nature is knowledge of a
fundamentally different type, one that is much more restrictive and powerful
at the same time: in other words, X 6= Y cannot, could not, and will not happen. Unlike their incidental counterparts, claiming either X = Y or X 6= Y as
factual necessarily makes them incommensurable with one another, even over
sampling repetitions. That is to say, if either X = Y or X 6= Y are to be hardcoded into the model, they will each result in a model distinct from the other
in a way that their respective posterior judgments about the same event, say
Y = 1, are not meant to enter the same law of total probability calculation. If
we are willing to admit either X = Y or X 6= Y as factual evidence to condition
on, they can no longer be regarded as a partition of the full space like they did
back in Section 3.1; the model must also anticipate to deal with a whole range
of other possible relationships between X and Y that are non-deterministic, as
part of the conditioning protocol in Shafer’s sense.
The distinction drawn here between the judgments of factual knowledge versus incidental knowledge are referred to as revision versus focusing in the imprecise probability literature; see Smets (1991) for a discussion on the matter and
its reflected ideologies of update rules. Whether a rule is applicable to a particular imprecise model would consequently depend on the type of knowledge
we have and what questions we want to answer.
Within a precise modeling framework, the knowledge type for conditioning can typically be coded into the conditioning event itself, which might be
on an enhanced probabilistic space but without increasing the resolution of
the original (marginal) model because it is already at the highest possible resolution. Hence one universal updating rule is sufficient. Under an imprecise
model, such a resolution-preserving encoding may not be possible because of
the low resolution nature of the original model. Various rules then were/are
invented to carry out the update as a qualitative rescue for the model’s inability to quantify the knowledge types within its original resolution. This makes
the judgment of knowledge types particularly pronounced, which serves well
as a loud reminder of the central and precise nature of conditioning in statistical learning. But this also increases vulnerability and confusion when we do
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

31

not explicate the applicability and subtitles of each updating rule, leading to
seemingly paradoxical phenomena studied in this paper.
6.2 Imprecise probability, precise decisions
Seeing a myriad of sensible and non-sensible answers produced by the updating rules of imprecise models, one may wonder if anything certain, or close
to certain, can be inferred from these models at all without stirring up a controversy. To this end, we discuss a final twist to the three prisoners’ story.
Example 3 cont. (Three prisoners’ Monty Hall). Having heard from the guard
that B will not receive parole, prisoner A is presented with an option to switch
his identity with prisoner C: that is, the next morning A will be met with the
fate of C (and C that of A), both having been decided unbeknownst to them. Is
this a good idea for A?
The answer is unequivocally yes. The above is a recast of the Monty Hall
problem in which you, the contestant standing in front of a randomly chosen
door (prisoner A), have just been shown a door with a goat behind it (“B will
be executed”), and are contemplating a switch to the other unopened door (the
identity of prisoner C) for a better chance of winning the new car (parole). By
the calculations in (3.9), we know that under the generalized Bayes rule
PB ( A lives | guard says B) = PB (C lives | guard says B)
suggesting that a switch will under no circumstance hurt the chance of A’s survival, because without switching, A’s best chance of surviving does not exceed
C’s worst chance of living. Moreover, as the most conservative rule of all, the
(almost) separation of the two posterior probability intervals from the generalized Bayes rule guarantees the same for the other updating rules as well. As far
as A is concerned, the action of identity switching can be recommended without reservation, regardless of the choice of rule among the three discussed. This
possibility is due to the (very) low resolution nature of the action space, often
binary (e.g., switching or not), allowing different high-resolution probabilistic
statements to admit the same low resolution classification in the action space.
6.3 Beyond associations: extended types of information contribution in
imprecise models
As discussed at the beginning of Section 1, in precise probability models, association is the fundamental means through which observed information contribute to the model, as it characterizes how probabilistic information should
change with one after the other has been learned, and vice versa. The sign of
the association gives the sense of direction, such as seen from the coefficients
in regression models. The magnitude of the association implies an order of
priority, such as in large scale genome-wide association studies and elsewhere
where correlation coefficients are used as test statistics. Plentiful association
is the indication of signal strength, potential discovery, and the prospect of a
causal relationship. The absence of association, on the other hand, is just as
desirable when used to justify independence assumptions, creating a blanket
of simplicity on which small-world models can be built and trusted. The three
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

32

GONG AND MENG

types of associations (positive, negative, and independence) are in one-to-one
correspondence with the three possible directions of change as the probability
of an event updates from the prior to the posterior according to the Bayes rule.
In precise probabilities, these three types of associations exhaust all possibilities
of information contribution from one event to another.
Imprecise probabilities complicate the landscape of information contribution
fundamentally because the probabilistic description assigned to each event is
no longer singular. For coherent upper and lower probabilities, this description
is now a closed interval [ P( A), P( A)] of possibly non-negligible width. As a
consequence, characterizations of the direction of change from prior to posterior becomes a complicated topic. Generalized versions of the classical notion
of association and independence are yet to be defined for sets of probabilities. Furthermore, novel phenomena like dilation, contraction and sure loss as
explored in this paper are hinting at extended types of information contribution waiting to be harnessed. As imprecise probability updating is contingent
upon the choice of rules, can we use some of these rules, capable of an array
of novel behavior such as Dempster’s rule, to characterize a generalized set of
definitions of information contribution in imprecise probabilities?
ACKNOWLEDGEMENTS
We thank Arthur Dempster, Keli Liu, Glenn Shafer and Teddy Seidenfeld for
very helpful discussions and comments, and Steve Finch for proofreading.
REFERENCES
Blyth, C. R. (1972). On Simpson’s paradox and the sure-thing principle. Journal of the American
Statistical Association, 67(338), 364–366.
Cornfield, J., Haenszel, W., Hammond, E. C., Lilienfeld, A. M., Shimkin, M. B., & Wynder, E. L.
(1959). Smoking and lung cancer: recent evidence and a discussion of some questions. Journal
of the National Cancer Institute, 22, 173–203.
Dempster, A. P. (1967). Upper and lower probabilities induced by a multivalued mapping. The
Annals of Mathematical Statistics, 38(2), 325–339.
Diaconis, P. (1978). Review of “A mathematical theory of evidence” (G. Shafer). Journal of the
American Statistical Association, 73(363), 677–678.
Diaconis, P., & Zabell, S. (1983). Some alternatives to Bayes’ rule (Tech. Rep.). Stanford University,
CA, Department of Statistics.
Ding, P., & VanderWeele, T. J. (2016). Sensitivity analysis without assumptions. Epidemiology,
27(3), 368.
Fagin, R., & Halpern, J. Y. (1991). A new approach to updating beliefs. In Proceedings of the Sixth
Conference on Uncertainty in Artificial Intelligence (pp. 317–325).
Fygenson, M. (2008). Modeling and predicting extrapolated probabilities with outlooks. Statistica
Sinica, 18(1), 9–40.
Gelman, A. (2006). The boxer, the wrestler, and the coin flip: a paradox of robust Bayesian
inference and belief functions. The American Statistician, 60(2), 146–150.
Gong, R., & Meng, X.-L. (2018). Probabilistic revelations of the versatile belief function: the good,
the bad, and the unsure.
(in preparation)
Good, I. (1974). A little learning can be dangerous. The British Journal for the Philosophy of Science,
25(4), 340–342.
Hannig, J. (2009). On generalized fiducial inference. Statistica Sinica, 19(2), 491–544.
Hannig, J., & Xie, M. G. (2012). A note on Dempster-Shafer recombination of confidence distributions. Electronic Journal of Statistics, 6, 1943–1966.
Herron, T., Seidenfeld, T., & Wasserman, L. (1994). The extent of dilation of sets of probabilities
and the asymptotics of robust Bayesian inference. In PSA: Proceedings of the Biennial Meeting of
the Philosophy of Science Association (Vol. 1994, pp. 250–259).
imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

DILATION, SURE LOSS, AND SIMPSON’S PARADOX

33

Herron, T., Seidenfeld, T., & Wasserman, L. (1997). Divisive conditioning: further results on
dilation. Philosophy of Science, 64(3), 411–444.
Huber, P. J., & Strassen, V. (1973). Minimax tests and the Neyman-Pearson lemma for capacities.
The Annals of Statistics, 1(2), 251–263.
Jaffray, J.-Y. (1992). Bayesian updating and belief functions. IEEE transactions on Systems, Man,
and Cybernetics, 22(5), 1144–1152.
Kish, L. (1965). Survey sampling. John Wiley and Sons, New York, NY.
Kohlas, J. (1991). The reliability of reasoning with unreliable arguments. Annals of Operations
Research, 32(1), 67–113.
Kruse, R., & Schwecke, E. (1990). Specialization–a new concept for uncertainty handling with
belief functions. International Journal Of General System, 18(1), 49–60.
Kyburg, H. E. (1987). Bayesian and non-Bayesian evidential updating. Artificial Intelligence, 31(3),
271–293.
Liu, K., & Meng, X.-L. (2014). Comment: A fruitful resolution to Simpsons Paradox via multiresolution inference. The American Statistician, 68(1), 17–29.
Liu, K., & Meng, X.-L. (2016). There is individualized treatment. Why not individualized inference? The Annual Review of Statistics and Its Applications, 3, 79-111.
Martin, R., & Liu, C. (2015). Inferential Models: reasoning with uncertainty. CRC Press, Boca Raton,
FL.
Morgan, J. P., Chaganty, N. R., Dahiya, R. C., & Doviak, M. J. (1991). Let’s make a deal: The
player’s dilemma. The American Statistician, 45(4), 284–287.
Mosteller, F. (1965). Fifty challenging problems in probability with solutions. Courier Corporation,
North Chelmsford, MA.
Pavlides, M. G., & Perlman, M. D. (2009). How likely is Simpson’s Paradox? The American
Statistician, 63(3), 226–233.
Pearl, J. (1990). Reasoning with belief functions: An analysis of compatibility. International Journal
of Approximate Reasoning, 4(5-6), 363–389.
Rubin, D. B. (1976). Inference and missing data. Biometrika, 63(3), 581–592.
Schweder, T., & Hjort, N. L. (2016). Confidence, likelihood, probability. Cambridge University Press,
Cambridge, UK.
Seidenfeld, T., & Wasserman, L. (1993). Dilation for sets of probabilities. The Annals of Statistics,
21(3), 1139–1154.
Shafer, G. (1976). A mathematical theory of evidence. Princeton University Press, Princeton, NJ.
Shafer, G. (1979). Allocations of probability. The Annals of Probability, 7(5), 827–839.
Shafer, G. (1985). Conditional probability. International Statistical Review/Revue Internationale de
Statistique, 261–275.
Simpson, E. H. (1951). The interpretation of interaction in contingency tables. Journal of the Royal
Statistical Society. Series B (Methodological), 13(2), 238–241.
Smets, P. (1991). About updating. In Proceedings of the Seventh Conference on Uncertainty in
Artificial Intelligence (pp. 378–385).
Smets, P. (1993). Belief functions: The disjunctive rule of combination and the generalized
Bayesian theorem. International Journal of Approximate Reasoning, 9(1), 1–35.
Suppes, P., & Zanotti, M. (1977). On using random relations to generate upper and lower
probabilities. Synthese, 36(4), 427–440.
Walley, P. (1991). Statistical reasoning with imprecise probabilities. Taylor & Francis, Oxford, UK.
Walley, P. (2000). Towards a unified theory of imprecise probability. International Journal of
Approximate Reasoning, 24(2-3), 125–148.
Wasserman, L. (1990). Prior envelopes based on belief functions. The Annals of Statistics, 18(1),
454–464.
Wasserman, L., & Kadane, J. B. (1990). Bayes theorem for Choquet capacities. The Annals of
Statistics, 18(3), 1328–1339.
Yager, R. R. (1987). On the Dempster-Shafer framework and new combination rules. Information
Sciences, 41(2), 93–137.
Yager, R. R., & Liu, L. (2008). Classic works of the Dempster-Shafer theory of belief functions (Vol. 219).
Springer, New York, NY.

imsart-sts ver. 2014/10/16 file: DS_and_dilation.tex date: December 27, 2017

