Natural extension as
a limit of regular extensions
Enrique Miranda (U. of Oviedo, Spain) and Marco Zaffalon (IDSIA, Switzerland)
Abstract Basic notions Weak natural extension
What is coherence of lower previsions? And Variables: X1 , . . . , Xn , taking values in respec- Weak natural extension: The smallest CLP
their natural extension? These questions have a tive finite sets X1 , . . . , Xn . P m+1 (XOm+1 |XIm+1 ) with domain
very clear answer when we work with desir- The vector of variables (Xj )j∈J is de- Km+1 which is weakly coherent with
able gambles but not that much from the dual noted by XJ . P 1 (XO1 |XI1 ), . . . , P m (XOm |XIm ) is
viewpoint of probability. The space of possibilities is X n := given, for every f ∈ Km+1 , zm+1 ∈ XIm+1 ,
We show that both coherence and the natural ×j∈{1,...,n} Xj . by P m+1 (f |zm+1 ) :=
extension can be regarded as related to the exis-
tence of a sequence of unconditional credal sets Conditional lower prevision (CLP):
P (XO |XI ), with domain H ⊆ K, where minx∈π−1 (zm+1 ) f (x) if P (zm+1 ) = 0
from which, by Bayes’ rule, the original assess- Im+1
ments can recovered as well as all their natural K is the set of all gambles that depend min{P (f |zm+1 ) : P ≥ P } otherwise,
extensions. on XO , XI , represents a subject’s beliefs
Furthermore, we discuss the difference be- about the gambles that depend on the where P is the smallest one in (WC3) or
tween the natural extension, and what we call outcome of the variables {Xj , j ∈ O}, (WC4).
the weak natural extension (i.e., that based after coming to know the outcome of the The weak natural extension can be too lit-
on weak coherence). We argue that most ap- variables {Xj , j ∈ I}. tle informative.
proaches in the literature compute weak natu- We always take P (XO |XI ) to be separately
coherent. Example: Consider X1 , X2 taking values in
ral extensions, which we show are not enough X := {1, 2}. Define coherent linear previ-
informative compared to natural extensions. Basic model: A collection of CLPs, i.e., sions P (X1 ), P (X2 |X1 ) using the assess-
Our results are valid for finite spaces and con- P 1 (XO1 |XI1 ), . . . , P m (XOm |XIm ). ments P (X1 = 1) := 1, P (X2 = 1|X1 =
ditional lower previsions with non-linear do- 1) := 0.5, P (X2 = 1|X1 = 2) := 1. We ob-
mains. tain a unique coherent linear joint by total
Initial results + weak coherence probability. It assigns probability zero to
Introduction Avoiding uniform sure loss: X1 = 2. As a consequence, the weak nat-
P 1 (XO1 |XI1 ), . . . , P m (XOm |XIm ) avoid ural extension of P (X1 ), P (X2 |X1 ) is vac-
Tools: Variables X1 , . . . , Xn taking finitely
uniform sure loss iff there are dom- uous for X2 conditional on X1 = 2. On
many values, and coherent lower previ-
inating weakly coherent conditional the other hand, it follows from the coher-
sions P 1 (XO1 |XI1 ), . . . , P m (XOm |XIm ).
linear previsions with (full) domains ence of P (X1 ), P (X2 |X1 ) that the natural
Preminary results: We give new characterisa- K1 , . . . , Km . extension yields back the original linear
tions of avoiding uniform sure and partial prevision P (X2 |X1 = 2), which tells us
loss based on the existence of dominating Avoiding partial loss: that X2 = 1 with certainty given X1 = 2.
conditional linear previsions. P 1 (XO1 |XI1 ), . . . , P m (XOm |XIm ) avoid
partial loss iff there are dominating
Weak natural extension: How can we ex- coherent conditional linear previsions Main results, strong coherence
tend weakly coherent lower previsions with domains K1 , . . . , Km . M( ): For every > 0, let M( ) be the set of
P 1 (XO1 |XI1 ), . . . , P m (XOm |XIm ) to new unconditional linear previsions satisfying
ones? Result: the (so-called weak nat- Weak natural extension: Let
ural) extension can be made through P 1 (XO1 |XI1 ), . . . , P m (XOm |XIm ) be P (fj |zj ) ≥ P j (fj |zj ) − R(fj )
conditioning the smallest unconditional separately coherent conditional lower
prevision P (X1 , . . . , Xn ) that is weakly previsions with domains H1 , . . . , Hm . whenever P (zj ) > 0, and for every
coherent with them. The following are equivalent: fj ∈ Hj , zj ∈ XIj , j = 1, . . . , m, where
R(fj ) := max fj − min fj .
Main result, (strong) coherence: (WC1) They are weakly coherent.
P 1 (XO1 |XI1 ), . . . , P m (XOm |XIm ) are (WC2) They are the lower envelopes Approximating conditionals: For every >
jointly coherent if and only if there is a se- of a class of weakly coherent 0, define the approximating condition-
quence of unconditional lower previsions conditional linear previsions, als Rm+1 (f |zm+1 ) := inf{P (f |zm+1 ) :
P (X1 , . . . , Xn ), ∈ R+ , s.t. by applying λ λ
{P1 (XO1 |XI1 ), . . . , Pm (XOm |XIm ) : P ∈ M( ), P (zm+1 ) > 0} and
Bayes’ rule whenever possible to the λ ∈ Λ}. their limits F m+1 (XOm+1 |XIm+1 ) :=
mass functions in the set equivalent to lim →0 Rm+1 (XOm+1 |XIm+1 ).
(WC3) There is a coherent lower prevision
P (X1 , . . . , Xn ), we recover the original
P on L(X n ) which is weakly coher- Main result: P 1 (XO1 |XI1 ), . . . , P m (XOm |XIm )
conditional lower previsions in the limit.
ent with them. weakly coherent and avoiding par-
Main result, natural extension: the nat- (WC4) There is a coherent lower prevision tial loss. Then the natural extension
ural extension of the original as- P on L(X n ) which is pairwise coher- E m+1 (XOm+1 |XIm+1 ) coincides with
sessments to a new lower prevision ent with them. F m+1 (XOm+1 |XIm+1 ).
P m+1 (XOm+1 |XIm+1 ) is nothing else
but the application of Bayes’ rule to Moreover, we give an explicit formula for Characterising coherence: Let
P (X1 , . . . , Xn ) with → 0. the smallest coherent lower prevision P P 1 (XO1 |XI1 ), . . . , P m (XOm |XIm ) be
in (WC3) and (WC4). separately coherent CLPs. They are
Zero probabilities: We relate the need of the coherent if and only if they are the point-
sequence P (X1 , . . . , Xn ), ∈ R+ , to the We summarise the relationships between the wise limits of a sequence of coherent
existence of events with zero lower prob- different consistency conditions when all the CLPs defined by regular extension.
ability and show that in this case the weak referential spaces are finite in the following fig-
natural extension can be much less infor- ure. SC Env. of SC Precise
mative than the natural extension.
-
Essential references
/ ^ - E. Miranda and G. de Cooman, Coherence and
Credits: Walley, Pelessoni and Vicig have in- WC Env. of WC Precise APL Dom. by SC Precise
- -
independence in non-linear spaces, T.R., 2005.
troduced these ideas while restricting the
- P. Walley, Statistical Reasoning with Imprecise
attention to events (rather than gambles) ^
AUL Dom. by WC Precise
/
- Probabilities, Chapman and Hall, 1991.
and therefore to finitely many probabilis-
Keys: SC = strongly coherent; WC = weakly co- - P. Walley, R. Pelessoni, and P. Vicig, Direct
tic assessments. Our work builds upon
herent; AUL = avoiding uniform sure loss; APL algorithms for checking consistecy and mak-
those ideas, while generalising them so
= avoiding partial loss; Env. = envelope; Dom. ing inferences for conditional probability as-
that the only actual restriction now is the
= dominated. sessments, JSPI, 126:119–151, 2004.
finiteness of the spaces.