Evidential Probability
and
Objective Bayesian Epistemology
Gregory Wheeler, CENTRIA, New University of Lisbon
Jon Williamson, Philosophy, University of Kent
To appear in
Prasanta S. Bandyopadhyay and Malcolm Forster (eds):
Handbook of the Philosophy of Statistics, Elsevier
Draft of July 16, 2010
Abstract
In this chapter we draw connections between two seemingly opposing
approaches to probability and statistics: evidential probability on the one
hand and objective Bayesian epistemology on the other.
Contents
1
Introduction
2
2
Evidential Probability
2.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Calculating Evidential Probability . . . . . . . . . . . . . . . . .
2
2
6
3
Second-order Evidential Probability
3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Calculating Second-order EP . . . . . . . . . . . . . . . . . . . .
9
9
11
4
Objective Bayesian Epistemology
15
4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Calculating Objective Bayesian Degrees of Belief . . . . . . . . . 18
5
EP-Calibrated Objective Bayesianism
19
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Calculating EP-Calibrated Objective Bayesian Probabilities . . . 21
6
Conclusion
22
1
1
Introduction
Evidential probability (EP), developed by Henry Kyburg, offers an account of
the impact of statistical evidence on single-case probability. According to this
theory, observed frequencies of repeatable outcomes determine a probability
interval that can be associated with a proposition. After giving a comprehensive
introduction to EP in §2, in §3 we describe a recent variant of this approach,
second-order evidential probability (2oEP). This variant, introduced in Haenni
et al. (2008), interprets a probability interval of EP as bounds on the sharp
probability of the corresponding proposition. In turn, this sharp probability can
itself be interpreted as the degree to which one ought to believe the proposition
in question.
At this stage we introduce objective Bayesian epistemology (OBE), a theory
of how evidence helps determine appropriate degrees of belief (§4). OBE might
be thought of as a rival to the evidential probability approaches. However, we
show in §5 that they can be viewed as complimentary: one can use the rules
of EP to narrow down the degree to which one should believe a proposition to
an interval, and then use the rules of OBE to help determine an appropriate
degree of belief from within this interval. Hence bridges can be built between
evidential probability and objective Bayesian epistemology.
2
2.1
Evidential Probability
Motivation
Rudolf Carnap (Carnap, 1962) drew a distinction between probability1 , which
concerned rational degrees of belief, and probability2 , which concerned statistical regularities. Although he claimed that both notions of probability were
crucial to scientific inference, Carnap practically ignored probability2 in the
development of his systems of inductive logic. Evidential probability (EP) (Kyburg, 1961; Kyburg and Teng, 2001), by contrast, is a theory that gives primacy
to probability2 , and Kyburg’s philosophical program was an uncompromising
approach to see how far he could go with relative frequencies. Whereas Bayesianism springs from the view that probability1 is all the probability needed for
scientific inference, EP arose from the view that probability2 is all that we really
have.
The theory of evidential probability is motivated by two basic ideas: probability assessments should be based upon relative frequencies, to the extent that
we know them, and the assignment of probability to specific individuals should
be determined by everything that is known about that individual. Evidential
probability is conditional probability in the sense that the probability of a sentence χ is evaluated given a set of sentences Γδ . But the evidential probability
of χ given Γδ , written Prob(χ, Γδ ), is a meta-linguistic operation similar in kind
to the relation of provability within deductive systems.
The semantics governing the operator Prob(·, ·) is markedly dissimilar to
axiomatic theories of probability that take conditional probability as primitive,
such as the system developed by Lester Dubbins (Dubbins, 1975; Arló-Costa
and Parikh, 2005), and it also resists reduction to linear (de Finetti, 1974) as
well as lower previsions (Walley, 1991). One difference between EP and the
2
first two theories is that EP is interval-valued rather than point-valued, because
the relative frequencies that underpin assignment of evidential probability are
typically incomplete and approximate. But more generally, EP assignments
may violate coherence. For example, suppose that χ and ϕ are sentences in the
object language of evidential probability. The evidential probability of χ ∧ ϕ
given Γδ might fail to be less than or equal to the evidential probability that
χ given Γδ .1 A point to stress from the start is that evidential probability is a
logic of statistical probability statements, and there is nothing in the activity
of observing and recording statistical regularities that guarantees that a set of
statistical probability statements will comport to the axioms of probability. So,
EP is neither a species of Carnapian logical probability nor a kind of Bayesian
probabilistic logic.23 EP is instead a logic for approximate reasoning, thus it is
more similar in kind to the theory of rough sets (Pawlak, 1991) and to systems
of fuzzy logic (Dubois and Prade, 1980) than to probabilistic logic.
The operator Prob(·, ·) takes as arguments a sentence χ in the first coordinate and a set of statements Γδ in the second. The statements in Γδ represent
a knowledge base, which includes categorical statements as well as statistical
generalities. Theorems of logic and mathematics are examples of categorical
statements, but so too are contingent generalities. One example of a contingent
categorical statement is the ideal gas law. EP views the propositions “2+2 = 4”
and “P V = nRT ” within a chemistry knowledge base as indistinguishable analytic truths that are built into a particular language adopted for handling
statistical statements to do with gasses. In light of EP’s expansive view of analyticity, the theory represents all categorical statements as universally quantified
sentences within a guarded fragment of first-order logic (Andréka et al., 1998).4
Statistical generalities within Γδ , by contrast, are viewed as direct inference
statements and are represented by syntax that is unique to evidential probability. Direct inference, recall, is the probability assigned a target subclass
given known frequency information about a reference population, and is often
contrasted to indirect inference, which is the assignment of probability to a population given observed frequencies in a sample. Kyburg’s ingenious idea was to
solve the problem of indirect inference by viewing it as a form of direct inference. Since the philosophical problems concerning direct inference are much less
contentious than those raised by indirect inference, the unusual properties and
behavior of evidential probability should be weighed against this achievement
(Levi, 2007).
Direct inference statements are statements that record the observed frequency of items satisfying a specified reference class that also satisfy a particular
target class, and take the form of
1 Specifically, the lower bound of Prob(χ ∧ ϕ, Γ ) may be strictly greater than the lower
δ
bound of Prob(χ, Γδ ).
2 See the essays by Levi and by Seidenfeld in (Harper and Wheeler, 2007) for a discussion
of the sharp differences between EP and Bayesian approaches, particularly on the issue of
conditionalization. A point sometimes overlooked by critics is that there are different systems
of evidential probability corresponding to different conditions we assume to hold. Results
pertaining to a qualitative representation of EP inference, for instance, assume that Γδ is
consistent. A version of conditionalization holds in EP given that there is specific statistical
statement pertaining to the relevant joint distribution. See (Kyburg, 2007) and (Teng, 2007).
3 EP does inherit some notions from Keynes’s (Keynes, 1921), however, including that
probabilities are interval-valued and not necessarily comparable.
4 A guarded fragment of first-order logic is a decidable fragment of first-order logic.
3
%�x(τ (�x), ρ(�x), [l, u]).
This schematic statement says that given a sequence of propositional variables
�x that satisfies the reference class predicate ρ, the proportion of ρ that also
satisfies the target class predicate τ is between l and u.
Syntactically, ‘τ (�x), ρ(�x), [l, u]’ is an open formula schema, where ‘τ (·)’ and
‘ρ(·)’ are replaced by open first-order formulas, ‘�x’ is replaced by a sequence
of propositional variables, and ‘[l, u]’ is replaced by a specific sub-interval of
[0, 1]. The binding operator ‘%’ is similar to the ordinary binding operators
(∀, ∃) of first-order logic, except that ‘%’ is a 3-place binding operator over
the propositional variables appearing the target formula τ (�x) and the reference
formula ρ(�x), and binding those formulas to an interval.5 The language Lep
of evidential probability then is a guarded first-order language augmented to
include direct inference statements. There are additional formation rules for
direct inference statements that are designed to block spurious inference, but
we shall pass over these details of the theory.6 An example of a direct inference
statement that might appear in Γδ is
%x(B(x), A(x), [.71, .83]),
which expresses that the proportion of A’s that are also B’s lies between 0.71
and 0.83.
As for semantics, a model M of Lep is a pair, �D, I�, where D is a two-sorted
domain consisting of mathematical objects, Dm , and a finite set of empirical
objects, De . EP assumes that there is a first giraffe and a last carbon molecule.
I is an interpretation function that is the union of two partial functions, one
defined on Dm and the other on De . Otherwise M behaves like a first-order
model: the interpretation function I maps (empirical/mathematical) terms into
the (empirical/mathematical) elements of D, monadic predicates into subsets of
D, n-arity relation symbols into Dn , and so forth. Variable assignments also
behave as one would expect, with the only difference being the procedure for
assigning truth to direct inference statements.
The basic idea behind the semantics for direct inference statements is that
the statistical quantifier ‘%’ ranges over the finite empirical domain De , not
the field terms l, u that denote real numbers in Dm . This means that the only
free variables in a direct inference statement range over a finite domain, which
will allow us to look at proportions of models in which a sentence is true. A
satisfaction set of an open formula ϕ whose only free n variables are empirical
in the subset of Dn that satisfies ϕ.
A direct inference statement %x(τ (x), ρ(x), [l, u]) is true in M under variable
assignment v iff the cardinality of the satisfaction sets for the open formula ρ
under v is greater than 0 and the ratio of the cardinality of satisfaction sets for
τ (x∗ ) ∧ ρ(x∗ ) over the cardinality of the satisfaction sets for ρ(x) (under v) is
in the closed interval [l, u], where all variables of x occur in ρ, all variables of τ
occur in ρ, and x∗ is the sequence of variables free in ρ but not bound by %x
(Kyburg and Teng (2001)).
The operator Prob(·, ·) then provides a semantics for a nonmonotonic consequence operator (Wheeler, 2004; Kyburg et al., 2007). The structural properties
5
6
Hereafter we relax notation and simply use an arbitrary variable ‘x’ for ‘�
x’.
See (Kyburg and Teng, 2001).
4
enjoyed by this consequence operator are as follows:7
Properties of EP Entailment: Let |= denote classical consequence and
let ≡ denote classical logical equivalence. Whenever µ ∧ ξ, ν ∧ ξ are sentences
of Lep ,
Right Weakening: if µ |≈ ν and ν |= ξ then µ |≈ ξ.
Left Classical Equivalence: if µ |≈ ν and µ ≡ ξ then ξ |≈ ν.
(KTW) Cautious Monotony: if µ |= ν and µ |≈ ξ then µ ∧ ξ |≈ ν.
(KTW) Premise Disjunction: if µ |= ν and ξ |≈ ν then µ ∨ ξ |≈ ν.
(KTW) Conclusion Conjunction: if µ |= ν and µ |≈ ξ then µ |≈ ν ∧ ξ.
As an aside, this qualitative EP-entailment relation presents challenges in handling disjunction in the premises since the KTW disjunction property admits
a novel reversal effect similar to, but distinct from, Simpson’s paradox (Kyburg et al. (2007); Wheeler (2007)). This raises a question over how best to
axiomatize EP. One approach, which is followed by (Hawthorne and Makinson
(2007)) and considered in (Kyburg et al. (2007)), is to replace Boolean disjunction by ‘exclusive-or’. While this route ensures nice properties for |≈, it does
so at the expense of introducing a dubious connective into the object language
that is neither associative nor compositional.8 Another approach explored in
(Kyburg et al. (2007)) is a weakened disjunction axiom (KTW Or) that yields a
sub-System P nonmonotonic logic and preserves the standard properties of the
positive Boolean connectives.
Now that we have a picture of what EP is, we turn to consider the inferential behavior of the theory. We propose to do this with a simple ball-draw
experiment before considering the specifics of the theory in more detail in the
next section.
Example 1. Suppose the proportion of white balls (W ) in an urn (U ) is
known to be within [.33, 4], and that ball t is drawn from U . These facts are
represented in Γδ by the sentences, %x(W (x), U (x), [.33, .4]) and U (t).
(i) If these two statements are all that we know about t, i.e., they are the
only statements in Γδ pertaining to t, then Prob(W (t), Γδ ) = [.33, .4].
(ii) Suppose additionally that the proportion of plastic balls (P ) that are white
is observed to be between [.31, .36], t is plastic, and that every plastic
ball is a white ball. That means that %x(P (x), U (x), [.31, .36]), P (t),
and ∀x.P (x) → W (x) are added to Γδ as well. Then there is conflicting
statistical knowledge about t, since either:
1. the probability that ball t is white is between [.33, .4], by reason of
%x(W (x), U (x), [.33, .4]), or
7 Note that these properties are similar to, but strictly weaker than, the properties of the
class of cumulative consequence relations specified by System P (Kraus et al. (1990)). To
yield the axioms of System P, replace the nonmonotonic consequence operator |∼ for |= in
the premise position of [And*], [Or*], and [Cautious Monotonicity*].
8 Example: ‘A xor B xor C’ is true if A, B, C are; and ‘(A xor B) xor C’ is not equivalent
to ‘A xor (B xor C)’ when A is false but B and C both true.
5
2. the probability that ball t is white is between [.31, .36], by reason of
%x(W (x), P (x), [.31, .36]),
may apply. There are several ways that statistical statements may conflict
and there are rules for handling each type, which we will discuss in the
next section. But in this particular case, because it is known that the
class of plastic balls is more specific than the class of balls in U and we
have statistics for the proportion of plastic balls that are also white balls,
the statistical statement in (2) dominates the statement in (1). So, the
probability that t is white is in [.31, .36].
(iii) Adapting an example from (Kyburg and Teng, 2001, 216), suppose U is
partitioned into three cells, u1 , u2 , and u3 , and that the following compound experiment is performed. First, a cell of U is selected at random.
Then a ball is drawn at random from that cell. To simplify matters, suppose that there are 25 balls in U and 9 are white such that 3 of 5 balls
from u1 are white, but only 3 of 10 balls in u2 and 3 of 10 in u3 are white.
The following table summarizes this information.
Table 1: Compound Experiment
W
W
u1
3
2
5
u2
3
7
10
u3
3
7
10
9
16
25
We are interested in the probability that t is white, but we have a conflict.
9
Given these over all precise values, we would have Prob(W (t), Γδ ) = 25
.
However, since we know that t was selected by performing this compound
experiment, then we also have the conflicting direct inference statement
%x, y(W ∗ (x, y), U ∗ (x, y), [.4, .4]), where U ∗ is the set of compound two
stage experiments, and W ∗ is the set of outcomes in which the ball selected
is white.9 We should prefer the statistics from the compound experiment
because they are richer in information. So, the probability that t is white
is .4.
(iv) Finally, if there happens to be no statistical knowledge in Γδ pertaining to
t, then we would be completely ignorant of the probability that t is white.
So in the case of total ignorance, Prob(W (t), Γδ ) = [0, 1].
We now turn to a more detailed account of how EP calculates probabilities.
2.2
Calculating Evidential Probability
In practice an individual may belong to several reference classes with known
statistics. Selecting the appropriate statistical distribution among the class of
9 Γ should also include the categorical statements ∀x, y(U ∗ �x, y� → W (y)), which says that
δ
the second stage of U concerns the proportion of balls that are white, and three statements of
the form � W ∗ (µ, t) ↔ W (t)� , where µ is replaced by u1 , u2 , u3 , respectively. This statement
tells us that everything that’s true of W ∗ is true of W , which is what ensures that this conflict
is detected.
6
potential probability statements is the problem of the reference class. The task
of assigning evidential probability to a statement χ relative to a set of evidential
certainties relies upon a procedure for eliminating excess candidates from the
set of potential candidates. This procedure is described in terms of the following
definitions.
Potential Probability Statement: A potential probability statement for χ
with respect to Γδ is a tuple �t, τ (t), ρ(t), [l, u]�, such that instances of χ ↔ τ (t),
ρ(t), and %x(τ (x), ρ(x), [l, u]) are each in Γδ .
Given χ, there are possibly many target statements of form τ (t) in Γδ that
have the same truth value as χ. If it is known that individual t satisfies ρ,
and known that between .7 and .8 of ρ’s are also τ ’s, then �t, τ (t), ρ(t), [.7, .8]�
represents a potential probability statement for χ based on the knowledge base
Γδ . Our focus will be on the statistical statements %x(τ (x), ρ(x), [l, u]) in Γδ
that are the basis for each potential probability statement.
Selecting the appropriate probability interval for χ from the set of potential
probability statements reduces to identifying and resolving conflicts among the
statistical statements that are the basis for each potential probability statement.
Conflict: Two intervals [l, u] and [l� , u� ] conflict iff neither [l, u] ⊂ [l� , u� ] nor
[l, u] ⊃ [l� , u� ]. Two statistical statements conflict iff their intervals conflict.
Note that conflicting intervals may be disjoint or intersect. For technical reasons
an interval is said to conflict with itself.
Cover: Let X be a set of intervals. An interval [l, u] covers X iff for every
[l� , u� ] ∈ X, l ≤ l� and u� ≤ u. A cover [l, u] of X is the smallest cover, Cov(X),
iff for all covers [l∗ , u∗ ] of X, l∗ ≤ l and u ≤ u∗ .
Difference Set: (i) Let X be a non-empty set of intervals and P(X) be the
powerset of X. A non-empty Y ∈ P(X) is a difference set of X iff Y includes
every x ∈ X that conflicts with some y ∈ Y . (ii) Let X be the set of intervals
associated with a set Γ of statistical statements, and Y be the set of intervals
associated with a set Λ of statistical statements. Λ is a difference set to Γ iff Y
is closed under difference with respect to X.
Example 2. An example might help. Let X be the set of intervals [.30, .40],
[.35, .45], [.325, .475], [.50, .55], [.30, .70], [.20, .60], [.10, .90]. There are three sets
closed under difference with respect to X:
(i) {[.30, .40], [.35, .45], [.325, .475], [.50, .55]},
(ii) {[.30, .70], [.20, .60]},
(iii) {[.10, .90]}.
The intuitive idea behind a difference set is to eliminate intervals from a set that
are broad enough to include all other intervals in that set. The interval [.10, .90]
is the broadest interval in X. So, it only appears as a singleton difference set
and is not included in any other difference set of X. It is not necessary that all
intervals in a difference set X be pairwise conflicting intervals. Difference sets
identify the set of all possible conflicts for each potential probability statement
in order to find that conflicting set with the shortest cover.
Minimal Cover Under Difference: (i) Let X be a non-empty set of
intervals and Y = {Y1 , . . . , Yn } the set of all difference sets of X. The minimal
cover under difference of X is the smallest cover of the elements of Y, i.e., the
shortest cover in {Cov(Y1 ), . . . , Cov(Yn )}.
7
(ii) Let X be the set of intervals associated with a set Γ of statistical statements, and Y be the set of all difference sets of X associated with a set Λ
of statistical statements. Then the minimal cover under difference of Γ is the
minimal cover under difference of X.
EP resolves conflicting statistical data concerning χ by applying two principles to the set of potential probability assignments, Richness and Specificity,
to yield a class of relevant statements. The (controversial) principle of Strength
is then applied to this set of relevant statistical statements, yielding a unique
probability interval for χ. For discussion of these principles, see (Teng (2007)).
We illustrate these principles in terms of a pair (ϕ, ϑ) of conflicting statistical
statements for χ, and represent their respective reference formulas by ρϕ and
ρϑ . The probability interval assigned to χ is the shortest cover of the relevant
statistics remaining after applying these principles.
1. [Richness] If ϕ and ϑ conflict and ϑ is based on a marginal distribution
while ϕ is based on the full joint distribution, eliminate ϑ.
2. [Specificity] If ϕ and ϑ both survive the principle of richness, and if
ρϕ ⊂ ρϑ , then eliminate �τ, ρϑ , [l, u]� from all difference sets.
The principle of specificity says that if it is known that the reference class
ρϕ is included in the reference class ρϑ , then eliminate the statement ϑ. The
statistical statements that survive the sequential application of the principle of
richness followed by the principle of specificity are called relevant statistics.
3. [Strength] Let ΓRS be the set of relevant statistical statements for χ
with respect to Γδ , and let the set {Λ1 , . . . , Λn } be the set of difference
sets of ΓRS . The principle of strength is the choosing of the minimal
cover under difference of ΓRS , i.e., the selection of the shortest cover in
{Cov(Λ1 ), . . . , Cov(Λn )}.
The evidential probability of χ is the minimal cover under difference of ΓRS .
We may define Γ� , the set of practical certainties, in terms of a body of
evidence Γδ :
Γ� = {χ : ∃ l, u (Prob(¬χ, Γδ ) = [l, u] ∧ u ≤ �)},
or alternatively,
Γ� = {χ : ∃ l, u (Prob(χ, Γδ ) = [l, u] ∧ l ≥ 1 − �)}.
The set Γ� is the set of statements that the evidence Γδ warrants accepting; we
say a sentence χ is �-accepted if χ ∈ Γ� . Thus we may add to our knowledge
base statements that are nonmonotonic consequences of Γδ with respect to a
threshold point of acceptance.
Finally, we may view the evidence Γδ to provide real-valued bounds on ‘degrees of belief’ owing to the logical structure of sentences accepted into Γδ .
However, the probability interval [l, u] associated with χ does not specify a
range of equally rational degrees of belief between l and u: the interval [l, u]
itself is not a quantity, only l and u are quantities, which are used to specify
bounds. On this view, no degree of belief within [l, u] is defensible, which is in
marked contrast to the view offered by Objective Bayesianism.
8
3
3.1
Second-order Evidential Probability
Motivation
Second-order evidential probability—developed in Haenni et al. (2008)—goes
beyond Kyburg’s evidential probability in two ways. First, it treats an EP
interval as bounds on sharp probability. Second, it disentangles reasoning under
uncertainty from questions of acceptance and rejection. Here we explain both
moves in more detail.
Bounds on Degrees of Belief. Kyburg maintained that one can interpret
an evidential probability interval for proposition χ as providing bounds on the
degree to which an agent should believe χ, but he had reservations about this
move:
Should we speak of partial beliefs as ‘degrees of belief’ ? Although
probabilities are intervals, we could still do so. Or we could say
that any ‘degree of belief’ satisfying the probability bounds was ‘rational’. But what would be the point of doing so? We agree with
Ramsey that logic cannot determine a real-valued a priori degree of
belief in pulling a black ball from an urn. This seems a case where
degrees of belief are not appropriate. No particular degree of belief is
defensible. We deny that there are any appropriate a priori degrees
of belief, though there is a fine a priori probability: [0, 1]. There are
real valued bounds on degrees of belief, determined by the logical
structure of our evidence. (Kyburg Jr, 2003, p. 147)
Kyburg is making the following points here. Evidence rarely determines a unique
value for an agent’s degree of belief—rather, it narrows down rational belief to
an interval. One can view this interval as providing bounds on rational degree of
belief, but since evidence can not be used to justify the choice of one point over
another in this interval, there seems to be little reason to talk of the individual
points and one can instead simply treat the interval itself as a partial belief.
This view fits very well with the interpretation of evidential probability
as some kind of measure of weight of evidence. (And evidential probability
provides a natural measure of weight of evidence: the narrower the interval, the
weightier the evidence.) Hence if evidence only narrows down probability to
an interval, then there does indeed seem to be little need to talk of anything
but the interval when measuring features of the evidence. But the view does
not address how to fix a sharp degree of belief—intentionally so, since Kyburg’s
program was designed in part to show us how far one can go with relative
frequency information alone. Even so, we may ask whether there is a way to
use the resources of evidential probability to fix sharp degrees of belief. In other
words, we might return to Carnap’s original distinction between probability1
and probability2 and ask how a theory of the latter can be used to constrain
the former. If we want to talk not only of the quality of our evidence but also
of our disposition to act on that evidence, then it would appear that we need
a richer language than that provided by EP alone: while evidence—and hence
EP—cannot provide grounds to prefer one point in the interval over another as
one’s degree of belief, there may be other, non-evidential grounds for some such
preference, and formalising this move would require going beyond EP.
9
Step
Step
Step
Step
1
2
3
4
Evidence
Acceptance
Uncertain reasoning
Acceptance
{P (ϕ) ∈ [lϕ , uϕ ]}
Γδ = {ϕ : lϕ ≥ 1 − δ}
{P (χ) ∈ [lχ , uχ ]}
Γε = {χ : lχ ≥ 1 − ε}
Figure 1: The structure of (first-order) EP inferences.
Reconciling EP with a Bayesian approach has been considered to be highly
problematic (Levi, 1977, 1980; Seidenfeld, 2007), and was vigorously resisted by
Kyburg throughout his life. On the other hand, Kyburg’s own search for an EPcompatible decision theory was rather limited (Kyburg, 1990). It is natural then
to explore how to modify evidential probability in order that it might handle
point-valued degrees of belief and thereby fit with Bayesian decision theory.
Accordingly second-order EP departs from Kyburg’s EP by viewing evidential
probability intervals as bounds on rational degree of belief, P (χ) ∈ Prob(χ, Γδ ).
In §5 we will go further still by viewing the results of EP as feeding into objective
Bayesian epistemology.
Acceptance and Rejection. If we allow ourselves the language of pointvalued degrees of belief, (first-order) EP can be seen to work like this. An agent
has evidence which consists of some propositions ϕ1 , . . . , ϕn and information
about their risk levels. He then accepts those propositions whose risk levels are
below the agent’s threshold δ. This leaves him with the evidential certainties,
Γδ = {ϕi : P (ϕi ) ≥ 1 − δ}. From Γδ the agent infers propositions ψ of the form
P (χ) ∈ [l, u]. In turn, from these propositions the agent infers the practical
certainties Γε = {χ : l ≥ 1 − ε}. This sequence of steps is depicted in Figure 1.
There are two modes of reasoning that are intermeshed here: on the one hand
the agent is using evidence to reason under uncertainty about the conclusion
proposition ψ, and on the other he is deciding which propositions to accept and
reject. The acceptance mode appears in two places: deciding which evidential
propositions to accept and deciding whether to accept the proposition χ to
which the conclusion ψ refers.
With second-order EP, on the other hand, acceptance is delayed until all reasoning under uncertainty is completed. Then we treat acceptance as a decision
problem requiring a decision-theoretic solution—e.g., accept those propositions
whose acceptance maximises expected utility.10 Coupling this solution with the
use of point-valued probabilities we have second-order evidential probability
(2oEP), whose inferential steps are represented in Figure 2.
There are two considerations that motivate this more strict separation of
uncertain reasoning and acceptance.
First, such a separation allows one to chain inferences—something which is
not possible in 1oEP. By ‘chaining inferences’ we mean that the results of step 2
of 2oEP can be treated as an input for a new round of uncertain reasoning, to be
10 Note that maximising expected utility is not straightforward in this case since bounds on
probabilities, rather than the probabilities themselves, are input into the decision problem.
EP-calibrated objective Bayesianism (§5) goes a step further by determining point-valued
probabilities from these bounds, thereby making maximisation of expected utility straightforward. See Williamson (2008a) for more on the combining objective Bayesianism with a
decision-theoretic account of acceptance.
10
Step 1
Step 2
Step 3
Evidence
Uncertain reasoning
Acceptance
Φ = {P (ϕ) ∈ [lϕ , uϕ ]}
Ψ = {P (χ) ∈ [lχ , uχ ]}
{χ : decision-theoretically optimal}
Figure 2: The structure of 2oEP inferences.
recombined with evidence and to yield further inferences. Only once the chain
of uncertain reasoning is complete will the acceptance phase kick in. Chaining
of inferences is explained in further detail in §3.2.
Second, such a separation allows one to keep track of the uncertainties that
attach to the evidence. To each item of evidence ϕ attaches an interval [lϕ , uϕ ]
representing the risk or reliability of that evidence. In 1oEP, step 2 ensures that
one works just with those propositions ϕ whose risk levels meet the threshold
of acceptance. But in 2oEP there is no acceptance phase before uncertain
reasoning is initiated, so one works with the entirety of the evidence, including
the risk intervals themselves. While the use of this extra information makes
inference rather more complicated, it also makes inference more accurate since
the extra information can matter—the results of 2oEP can differ from the results
of 1oEP.
We adopt a decision-theoretic account of acceptance for the following reason.
In 1oEP, each act of acceptance uniformly accepts those propositions whose
associated risk is less than some fixed threshold: δ in step 2 and ε in step 4.
(This allows statements to detach from their risk levels and play a role as logical
constraints in inference.) But in practice thresholds of acceptance depend not so
much on the step in the chain of reasoning as on the proposition concerned, and,
indeed, the whole inferential set-up. To take a favourite example of Kyburg’s,
consider a lottery. The threshold of acceptance of the proposition the lottery
ticket that the seller is offering me will lose may be higher than that of the
coin with a bias in favour of heads that I am about to toss will land heads and
lower than that of the moon is made of blue cheese. This is because nothing
may hang on the coin toss (in which case a 60% bias in favour of heads may
be quite adequate for acceptance), while rather a lot hangs on accepting that
the moon is made of blue cheese—many other propositions that I have hitherto
granted will have to be revisited if I were to accept this proposition. Moreover,
if I am going to use the judgement to decide whether to buy a ticket then the
threshold of acceptance of the lottery proposition should plausibly depend on
the extent to which I value the prize. Given these considerations, acceptance of
a proposition can fruitfully be viewed as a decision problem, depending on the
decision set-up including associated utilities (Williamson, 2008a). Again, while
this is more complicated than the 1oEP solution of modelling acceptance using
a fixed threshold, the subtleties of a full-blown decision-theoretic account can
matter to the resulting inferences.
3.2
Calculating Second-order EP
In this section we will be concerned with developing some machinery to perform
uncertain reasoning in second-order evidential probability (step 2 in Figure 2).
See Haenni et al. (2008) for further details of this approach.
11
Entailment. Let L� be a propositional language whose propositional variables
are of the form ϕ[a,b] for atomic propositions ϕ ∈ L.11 Here L is the language
of (first-order) EP extended to include statements of the form P (χ) ∈ [l, u],
and, for proposition ϕ of L, ϕ[a,b] is short for P (ϕ) ∈ [a, b]. Hence in L� we
can express propositions about higher-order probabilities, e.g., P (χ) ∈ [l, u][a,b]
which is short for P (P (χ) ∈ [l, u]) ∈ [a, b]. We write ϕa as an abbreviation of
ϕ[a,a] .
For µ, ν ∈ L� write µ|≈2o ν if ν deductively follows from µ by appealing to
the axioms of probability and the following EP-motivated axioms:
A1: Given ϕ11 , . . . , ϕ1n , if Prob(χ, {ϕ1 , . . . , ϕn }) = [l, u] is derivable by (firstorder) EP then infer ψ 1 , where ψ ∈ L is the statement P (χ) ∈ [l, u].
A2: Given ψ 1 then infer χ[l,u] , where ψ ∈ L is the statement P (χ) ∈ [l, u].
Axiom A1 ensures that EP inferences carry over to 2oEP, while axiom A2 ensures that probabilities at the first-order level can constrain those at the secondorder level.
The entailment relation |≈2o will be taken to constitute core second-order
EP . The idea is that when input evidence Φ consisting of a set of sentences of
L� , one infers a set Ψ of further such sentences using the above consequence
relation. Note that although |≈2o is essentially classical consequence with extra
axioms, it is a nonmonotonic consequence relation since 1oEP is nonmonotonic.
But 2oEP yields a strong logic inasmuch as it combines the axioms of probability with the rules of EP, and so questions of consistency arise. Will there always
be some probability function that satisfies the constraints imposed by 1oEP
consequences of evidence? Not always: see Seidenfeld (2007) for some counterexamples. Consequently, some consistency-maintenance procedure needs to
be invoked to cope with such cases. (Of course, some consistency maintenance
procedure will in any case be required to handle certain inconsistent sets of
evidential propositions, so there may be no extra burden here.) One option
is to consider probability functions satisfying (EP consequences of) maximal
satisfiable subsets of evidential statements, for example. In this paper we will
not commit to a particular consistency-maintenance procedure; we leave this
interesting question as a topic for further research.
Credal Networks. This entailment relation can be implemented using probabilistic networks, as we shall now explain. For efficiency reasons, we make the
following further assumptions. First we assume that P is distributed uniformly
over the EP interval unless there is evidence otherwise:
�
�
A3: If Φ|≈2o χ[l,u] then P (χ[l ,u ] |Φ) =
with other consequences of Φ.
|[l,u]∩[l� ,u� ]|
,
|[l,u]|
as long as this is consistent
Second, we assume that items of evidence are independent unless there is evidence of dependence:
[a ,b ]
[a ,b ]
[a ,b ]
[a ,b ]
[a ,b1 ]
A4: If ϕ1 1 1 , . . . , ϕk k k ∈ Φ then P (ϕ1 1 1 , . . . , ϕk k k ) = P (ϕ1 1
as long as this is consistent with other consequences of Φ.
11 As it stands L� contains uncountably many propositional variables, but restrictions can
be placed on a, b to circumscribe the language if need be.
12
[a ,bk ]
) · · · P (ϕk k
),
These assumptions are not essential to second-order EP, but they make the
probabilistic network implementation particularly straightforward.12 Note that
these assumptions are default rules; when determined by A1-4, the consequence
relation |≈2o is nonmonotonic.
A credal network can be used to represent and reason with a set of probability
functions (Cozman, 2000). A credal network consists of (i) a directed acyclic
graph whose nodes are variables A1 , . . . , An and (ii) constraints on conditional
probabilities of the form P (ai | par i ) ∈ [l, u] where ai is an assignment of a value
to a variable and par i is an assignment of values to its parents in the graph.
It is assumed that each variable is probabilistically independent of its nondescendants conditional on its parents in the graph, written Ai ⊥
⊥ ND i | Par i ;
this assumption is known as the Markov Condition.
Credal networks are of fundamental importance for inference in probabilistic
logic (Haenni et al., 2008). A logic is a probabilistic logic if its semantic interpretations are probability functions; the entailment relation of first-order EP
does not constitute a probabilistic logic in this sense, but the entailment relation |≈2o of second-order EP does. In a probabilistic logic we are typically faced
with the following sort of question: given premiss propositions ϕ1 , . . . , ϕn and
their respective probabilities X1 , . . . , Xn , what probability should we attach to
a conclusion proposition ψ? This question can be written in the form
Xn
?
1
ϕX
1 , . . . , ϕn |≈ ψ
where |≈ is the entailment relation of the probabilistic logic. For example,
in second-order evidential probability we might be faced with the following
question
%x(F x, Rx, [.2, .4])[.9,1] , Rt|≈2o P (F t) ∈ [.2, .4]?
This asks, given evidence that (i) the proposition that the frequency of attribute
F in reference class R is between .2 and .4 has probability at least .9, and
(ii) t falls in reference class R, what probability interval should attach to the
proposition that the probability that t has attribute F is between .2 and .4? In
first-order EP, if 1 − δ ≥ .9 then Prob(F t, Γδ ) = [.2, .4] would be conclusively
inferred (and hence treated as if it had probability 1). Clearly this disregards
the uncertainty that attaches to the statistical evidence; the question is, what
uncertainty should attach to the conclusion as a consequence? (This is a secondorder uncertainty; hence the name second-order evidential probability.) One can
construct a credal network to answer this question as follows. Let ϕ1 be the
proposition %x(F x, Rx, [.2, .4]), ϕ2 be Rt and ψ be P (F t) ∈ [.2, .4]. These can
all be thought of as variables that take possible values True and False. The
structure of 1oEP calculations determines the structure of the directed acyclic
graph in the credal net:
12 If items of evidence are known to be dependent then the corresponding nodes will be connected by arrows in the credal network representation outlined below. Any information that
helps to quantify the dependence will help determine the conditional probability distributions
associated with these arrows. If P is known to be distributed non-uniformly over the EP
intervals then information about its distribution will need to be used to determine conditional
probability distributions in the credal net.
13
✓✏
ϕ1
❍❍
✒✑
❍
❍
❥✓✏
❍
ψ
✯✒✑
✟
✟
✓✏
✟✟
ϕ2 ✟
✒✑
The conditional probability constraints involving the premiss propositions are
simply their given risk levels:
P (ϕ1 ) ∈ [.9, 1],
P (ϕ2 ) = 1.
Turning to the conditional probability constraints involving the conclusion proposition, these are determined by 1oEP inferences via axioms A1-3:
P (ψ|ϕ1 ∧ ϕ2 ) = 1,
P (ψ|¬ϕ1 ∧ ϕ2 ) = P (ψ|ϕ1 ∧ ¬ϕ2 ) = P (ψ|¬ϕ1 ∧ ¬ϕ2 ) = .2.
Finally, the Markov condition holds in virtue of A4, which implies that ϕ1 ⊥
⊥
ϕ2 . Inference algorithms for credal networks can then be used to infer the
uncertainty that should attach to the conclusion, P (ψ) ∈ [.92, 1]. Hence we
have:
%x(F x, Rx, [.2, .4])[.9,1] , Rt|≈2o P (F t) ∈ [.2, .4][.92,1]
Chaining Inferences. While it is not possible to chain inferences in 1oEP,
this is possible in 2oEP, and the credal network representation can just as readily
be applied to this more complex case. Consider the following question:
%x(F x, Rx, [.2, .4])[.9,1] , Rt, %x(Gx, F x, [.2, .4])[.6,.7] |≈2o P (Gt) ∈ [0, .25]?
As we have just seen, the first two premisses can be used to infer something
about F t, namely P (F t) ∈ [.2, .4][.92,1] . But now this inference can then be used
in conjunction with the third premiss to infer something about Gt. To work out
the probability bounds that should attach to an inference to P (Gt) ∈ [0, .25],
we can apply the credal network procedure. Again, the structure of the graph
in the network is given by the structure of EP inferences:
✓✏
ϕ1
❍
✒✑
❍❍
❍
❥✓✏
❍
ψ
❍❍
✯✒✑
✟
✟
✟
❍ ✓✏
✓✏
✟
❍
❥
❍
ϕ2 ✟
ψ�
✒✑
✯✒✑
✟
✟
✓✏
✟✟
✟
ϕ3
✒✑
�
Here ϕ3 is %x(Gx, F x, [.2, .4]) and ψ is P (Gt) ∈ [0, .25]; other variables are
as before. The conditional probability bounds of the previous example simply
carry over
P (ϕ1 ) ∈ [.9, 1], P (ϕ2 ) = 1,
14
P (ψ|ϕ1 ∧ ϕ2 ) = 1, P (ψ|¬ϕ1 ∧ ϕ2 ) = .2 = P (ψ|ϕ1 ∧ ¬ϕ2 ) = P (ψ|¬ϕ1 ∧ ¬ϕ2 ).
But we need to provide further bounds. As before, the risk level associated with
the third premiss ϕ3 provides one of these:
P (ϕ3 ) ∈ [.6, .7],
and the constraints involving the new conclusion ψ � are generated by A3:
P (ψ � |ψ ∧ ϕ3 ) =
|[.2 × .6 + .8 × .1, .4 × .7 + .6 × .1] ∩ [0, .25]|
= .31,
|[.2 × .6 + .8 × .1, .4 × .7 + .6 × .1]|
P (ψ � |¬ψ ∧ ϕ3 ) = .27, P (ψ � |ψ ∧ ¬ϕ3 ) = P (ψ � |¬ψ ∧ ¬ϕ3 ) = .25.
The Markov Condition holds in virtue of A4 and the structure of EP inferences.
Performing inference in the credal network yields P (ψ � ) ∈ [.28, .29]. Hence
%x(F x, Rx, [.2, .4])[.9,1] , Rt, %x(Gx, F x, [.2, .4])[.6,.7] |≈2o P (Gt) ∈ [0, .25][.28,.29] .
This example shows how general inference in 2oEP can be: we are not asking
which probability bounds attach to a 1oEP inference in this example, but rather
which probability bounds attach to an inference that cannot be drawn by 1oEP.
The example also shows that the probability interval attaching to the conclusion
can be narrower than intervals attaching to the premisses.
4
4.1
Objective Bayesian Epistemology
Motivation
We saw above that evidential probability concerns the impact of evidence upon
a conclusion. It does not on its own say how strongly one should believe the
conclusion. Kyburg was explicit about this, arguing that evidential probabilities
can at most be thought of as ‘real-valued bounds on degrees of belief, determined
by the logical structure of our evidence’ (Kyburg Jr, 2003, p. 147). To determine
rational degrees of belief themselves, one needs to go beyond EP, to a normative
theory of partial belief.
Objective Bayesian epistemology is just such a normative theory (Rosenkrantz,
1977; Jaynes, 2003; Williamson, 2005). According to the version of objective
Bayesianism presented in Williamson (2005), one’s beliefs should adhere to three
norms:
Probability: The strengths of one’s beliefs should be representable by probabilities. Thus they should be measurable on a scale between 0 and 1, and
should be additive.
Calibration: These degrees of belief should fit one’s evidence. For example,
degrees of belief should be calibrated with frequency: if all one knows
about the truth of a proposition is an appropriate frequency, one should
believe the proposition to the extent of that frequency.
Equivocation: One should not believe a proposition more strongly than the
evidence demands. One should equivocate between the basic possibilities
as far as the evidence permits.
These norms are imprecisely stated: some formalism is needed to flesh them
out.
15
Probability. In the case of the Probability norm, the mathematical calculus
of probability provides the required formalism. Of course mathematical probabilities attach to abstract events while degrees of belief attach to propositions,
so the mathematical calculus needs to be tailored to apply to propositions. It is
usual to proceed as follows (see, e.g., Paris, 1994). Given a predicate language
L with constants ti that pick out all the members of the domain, and sentences
θ, ϕ of L, a function P is a probability function if it satisfies the following axioms:
P1: If |= θ then P (θ) = 1;
P2: If |= ¬(θ ∧ ϕ) then P (θ ∨ ϕ) = P (θ) + P (ϕ);
�n
P3: P (∃xθ(x)) = limn→∞ P ( i=1 θ(ti )).
P1 sets the scale, P2 ensures that probability is additive, and P3, called Gaifman’s condition, sets the probability of ‘θ holds of something’ to be the limit
of the probability of ‘θ holds of one or more of t1 , ..., tn ’, as n tends to infinity.
The Probability norm then requires that the strengths of one’s beliefs be representable by a probability function P over (a suitable formalisation of) one’s
language. Writing P for the set of probability functions over L, the Probability
norm requires that one’s beliefs be representable by some P ∈ P.
Calibration. The Calibration norm says that the strengths of one’s beliefs
should be appropriately constrained by one’s evidence E. (By evidence we
just mean everything taken for granted in the current operating context—
observations, theory, background knowledge etc.) This norm can be explicated
by supposing that there is some set E ⊆ P of probability functions that satisfy constraints imposed by evidence and that one’s degrees of belief should be
representable by some PE ∈ E. Now typically one has two kinds of evidence:
quantitative evidence that tells one something about physical probability (frequency, chance etc.), and qualitative evidence that tells one something about
how one’s beliefs should be structured. In Williamson (2005) it is argued that
these kinds of evidence should be taken into account in the following way. First,
quantitative evidence (e.g., evidence of frequencies) tells us that the physical
probability function P ∗ must lie in some set P∗ of probability functions. One’s
degrees of belief ought to be similarly constrained by evidence of physical probabilities, subject to a few provisos:
C1: E �= ∅.
If evidence is inconsistent this tells us something about our evidence rather than
about physical probability, so one cannot conclude that P∗ = ∅ and one can
hardly insist that PE ∈ ∅. Instead P∗ must be determined by some consistency
maintenance procedure—one might, for example, take P∗ to be determined by
maximal consistent subsets of one’s evidence—and neither P∗ nor E can ever be
empty.
C2: If E is consistent and implies proposition θ that does not mention physical
probability P ∗ , then P (θ) = 1 for all P ∈ E.
This condition merely asserts that categorical evidence be respected—it prevents
E from being too inclusive. The qualification that θ must not mention physical
probability is required because in some cases evidence of physical probability
should be treated more pliably:
16
C3: If P, Q ∈ P∗ and R = λP + (1 − λ)Q for λ ∈ [0, 1] then, other things being
equal, one should be permitted to take R as one’s belief function PE .
Note in particular that C3 implies that, other things being equal, if P ∈ P∗
then P ∈ E; it also implies C1 (under the understanding that P∗ �= ∅). C3 is
required to handle the following kind of scenario. Suppose for example that you
have evidence just that an experiment with two possible outcomes, a and ¬a,
has taken place. As far as you are aware, the physical probability of a is now
1 or 0 and no value in between. But this does not imply that your degree of
belief in a should be 1 or 0 and no value in between—a value of 21 , for instance,
is quite reasonable in this case. C3 says that, in the absence of other overriding
evidence, �P∗ � ⊆ E where �P∗ � is the convex hull of P∗ . The following condition
imposes the converse relation:
C4: E ⊆ �P∗ �.
Suppose for example that evidence implies that either P ∗ (a) = 0.91 or P ∗ (a) =
0.92. While C3 permits any element of the interval [0.91, 0.92] as a value for
one’s degree of belief PE (a), C4 confines PE (a) to this interval—indeed a value
outside this interval is unwarranted by this particular evidence. Note that C4
implies C2: θ being true implies that its physical probability is 1, so P (θ) = 1
for all P ∈ P∗ , hence for all P ∈ �P∗ �, hence for all P ∈ E.
In the absence of overriding evidence the conditions C1–4 set E = �P∗ �. This
sheds light on how quantitative evidence constrains degrees of belief, but one
may also have qualitative evidence which constrains degrees of belief in ways
that are not mediated by physical probability. For example, one may know
about causal influence relationships involving variables in one’s language: this
may tell one something about physical probability, but it also tells one other
things—e.g., that if one extends one’s language to include a new variable that
is not a cause of the current variables, then that does not on its own provide
any reason to change one’s beliefs about the current variables. These constraints
imposed by evidence of influence relationships, discussed in detail in Williamson
(2005), motivate a further principle:
C5: E ⊆ S where S is the set of probability functions satisfying structural
constraints.
We will not dwell on C5 here since structural constraints are peripheral to
the theme of this paper, namely to connections between objective Bayesian
epistemology and evidential probability. It turns out that the set S is always
non-empty, hence C1–5 yield:
Calibration: One’s degrees of belief should be representable by PE ∈ E =
�P∗ � ∩ S.
Equivocation. The third norm, Equivocation, can be fleshed out by requiring that PE be a probability function, from all those that are calibrated with
evidence, that is as close as possible to a totally equivocal probability function
P= called the equivocator on L. But we need to specify the equivocator and also
what we mean by ‘as close as possible’. To specify the equivocator, first create
an ordering a1 , a2 , . . . of the atomic sentences of L—sentences of the form U t
17
where U is a predicate or relation and t is a tuple of constants of corresponding
arity—such that those atomic sentences involving constants t1 , . . . tn−1 occur
earlier in the ordering than those involving tn . Then we can define the equivoej−1
e
) = 1/2 for all j and all e1 , . . . ej ∈ {0, 1},
cator P= by P= (aj j | ae11 ∧ · · · ∧ aj−1
0
1
where ai is just ai and ai is ¬ai . Clearly P= equivocates between each atomic
sentence of L and its negation. In order to explicate ‘as close as possible’ to P=
we shall appeal to the standard notion of distance between probability functions,
the n-divergence of P from Q:
df
dn (P, Q) =
1
�
e
e
P (ae11 ∧ · · · ∧ arnrn ) log
e1 ,...,ern =0
P (ae11 ∧ · · · ∧ arnrn )
.
e
Q(ae11 ∧ · · · ∧ arnrn )
Here a1 , ..., arn are the atomic sentences involving constants t1 , ..., tn ; we follow
the usual convention of taking 0 log 0 to be 0, and note that the n-divergence
is not a distance function in the usual mathematical sense because it is not
symmetric and does not satisfy the triangle inequality—rather, it is a measure
of the amount of information that is encapsulated in P but not in Q. We then
say that P is closer to the equivocator than Q if there is some N such that for
n ≥ N , dn (P, P= ) < dn (Q, P= ). Now we can state the Equivocation norm as
follows. For a set Q of probability functions, denote by ↓Q the members of Q
that are closest to the equivocator P= .13 Then,
E1: PE ∈ ↓E.
This principle is discussed at more length in Williamson (2008b). It can be
construed as a version of the maximum entropy principle championed by Edwin
Jaynes. Note that while some versions of objective Bayesianism assume that an
agent’s degrees of belief are uniquely determined by her evidence and language,
we make no such assumption here: ↓E may not be a singleton.
4.2
Calculating Objective Bayesian Degrees of Belief
Just as credal nets can be used for inference in 2oEP, so too can they be used
for inference in OBE. The basic idea is to use a credal net to represent ↓E,
the set of rational belief functions, and then to perform inference to calculate
the range of probability values these functions ascribe to some proposition of
interest. These methods are explained in detail in Williamson (2008b); here we
shall just give the gist.
For simplicity we shall describe the approach in the base case in which the
evidence consists of interval bounds on the probabilities of sentences of the
agent’s language L, E = {P ∗ (ϕi ) ∈ [li , ui ] : i = 1, . . . , k}, E is consistent
and does not admit infinite descending chains; but these assumptions can all
be relaxed. In this case E = �P∗ � ∩ S = P∗ . Moreover, the evidence can be
[l ,u ]
[l ,u ]
written in the language L� introduced earlier: E = {ϕ1 1 1 , . . . , ϕk k k }, and
the question facing objective Bayesian epistemology takes the form
[l ,u1 ]
ϕ1 1
[l ,uk ]
, . . . , ϕk k
|≈OBE ψ ?
13 If there are no closest members (i.e., if chains are all infinitely descending: for any member
P of Q there is some P � in Q that is closer to the equivocator than P ) the context may yet
determine an appropriate subset ↓Q ⊆ Q of probability functions that are sufficiently close to
the equivocator; for simplicity of exposition we shall ignore this case in what follows.
18
✓✏
A1
✒✑
✓✏
A2
✒✑
✓✏
A3
✒✑
Figure 3: Constraint graph.
✓✏
A1
✒✑
✓✏
✲ A2
✒✑
✓✏
✲ A3
✒✑
Figure 4: Graph satisfying the Markov Condition.
where |≈OBE is the entailment relation defined by objective Bayesian epistemology as outlined above. As explained in Williamson (2008b), this entailment
relation is nonmonotonic but it is well-behaved in the sense that it satisfies all
the System-P properties of nonmonotonic logic.
The method is essentially this. First construct an undirected graph, the
constraint graph, by linking with an edge those atomic sentences that appear in
the same item of evidence. One can read off this graph a list of probabilistic
independencies that any function in ↓E must satisfy: if node A separates nodes
B and C in this graph then B ⊥
⊥ C | A for each probability function in ↓E. This
constraint graph can then be transformed into a directed acyclic graph for which
the Markov Condition captures many or all of these independencies. Finally one
can calculate bounds on the probability of each node conditional on its parents
in the graph by using entropy maximisation methods: each probability function
in ↓E maximises entropy subject to the constraints imposed by E, and one can
identify the probability it gives to one variable conditional on its parents using
numerical optimisation methods (Williamson, 2008b).
To take a simple example, suppose we have the following question:
∀x(U x → V x)
3/5
, ∀x(V x → W x)
3/4
, U t1 [0.8,1] |≈OBE W t?1
A credal net can be constructed to answer this question. There is only one
constant symbol t1 , and so the atomic sentences of interest are U t1 , V t1 , W t1 .
Let A1 be U t1 , A2 be V t1 and A3 be W t1 . Then the constraint graph G is
depicted in Fig. 3 and the corresponding directed acyclic graph H is depicted in
Fig. 4. It is not hard to see that P (A1 ) = 4/5, P (A2 |A1 ) = 3/4, P (A2 |¬A1 ) =
1/2, P (A3 |A2 ) = 5/6, P (A3 |¬A2 ) = 1/2; together with H, these probabilities
yield a credal network. (In fact, since the conditional probabilities are precisely
determined rather than bounded, we have a special case of a credal net called a
Bayesian net.) The Markov Condition holds since separation in the constraint
graph implies probabilistic independence. Standard inference methods then give
us P (A3 ) = 11/15 as an answer to our question.
5
EP-Calibrated Objective Bayesianism
5.1
Motivation
At face value, evidential probability and objective Bayesian epistemology are
very different theories. The former concerns the impact of evidence of physical
19
probability, Carnap’s probability2 , and concerns acceptance and rejection; it appeals to interval-valued probabilities. The latter theory concerns rational degree
of belief, probability1 , and invokes the usual point-valued mathematical notion
of probability. Nevertheless the core of these two theories can be reconciled, by
appealing to second-order EP as developed above.
2oEP concerns the impact of evidence on rational degree of belief. Given
statistical evidence, 2oEP will infer statements about rational degrees of belief.
These statements can be viewed as constraints that should be satisfied by the
degrees of belief of a rational agent with just that evidence. So 2oEP can be
thought of as mapping statistical evidence E to a set E of rational belief functions that are compatible with that evidence. (This is a non-trivial mapping
because frequencies attach to a sequence of outcomes or experimental conditions
that admit repeated instantiations, while degrees of belief attach to propositions.
Hence the epistemological reference-class problem arises: how can one determine
appropriate single-case probabilities from information about generic probabilities? Evidential probability is a theory that tackles this reference-class problem
head on: it determines a probability interval that attaches to a sentence from
statistical evidence about repetitions.)
But this mapping from E to E is just what is required by the Calibration
norm of OBE. We saw in §4 that OBE maps evidence E to E = �P∗ � ∩ S, a set
of probability functions calibrated with that evidence. But no precise details
were given as to how �P∗ �, nor indeed P∗ , is to be determined. In special cases
this is straightforward. For example, if one’s evidence is just that the chance
of a is 12 , P ∗ (a) = 1/2, then �P∗ � = P∗ = {P ∈ P : P (a) = 1/2}. But in
general, determining �P∗ � is not a trivial enterprise. In particular, statistical
evidence takes the form of information about generic frequencies rather than
single-case chances, and so the reference-class problem arises. It is here that
2oEP can be plugged in: if E consists of propositions of L� —i.e., propositions,
including statistical propositions, to which probabilities or closed intervals of
probabilities attach—then �P∗ � is the set of probability functions that satisfy
the |≈2o consequences of E.
C6: If E is a consistent set of propositions of L� then �P∗ � = {P : P (χ) ∈ [l, u]
for all χ, l, u such that E|≈2o χ[l,u] }.
We shall call OBE that appeals to calibration principles C1–6 epistemicprobability-calibrated objective Bayesian epistemology, or EP-OBE for short. We
shall denote the corresponding entailment relation by |≈EP
OBE .
We see then that there is a sense in which EP and OBE can be viewed as
complementary rather than in opposition. Of course, this isn’t the end of the
matter. Questions still arise as to whether EP-OBE is the right way to flesh out
OBE. One can, for instance, debate the particular rules that EP uses to handle
reference classes (§2.2). One can also ask whether EP tells us everything we need
to know about calibration. As mentioned in §4.1, further rules are needed in
order to handle structural evidence, fleshing out C5. Moreover, both 1oEP and
2oEP take statistical statements as input; these statements themselves need to
be inferred from particular facts—indeed EP, OBE and EP-OBE each presume
a certain amount of statistical inference. Consequently we take it as understood
that Calibration requires more than just C1–6.
And questions arise as to whether the alterations to EP that are necessary
to render it compatible with OBE are computationally practical. Second-order
20
EP replaces the original theory of acceptance with a decision theoretic account
which will incur a computational burden. Moreover, some thought must be given
as to which consistency maintenance procedure should be employed in practice.
Having said this, we conjecture that there will be real inference problems for
which the benefits will be worth the necessary extra work.
5.2
Calculating EP-Calibrated Objective Bayesian Probabilities
Calculating EP-OBE probabilities can be achieved by combining methods for
calculating 2oEP probabilities with methods for calculating OBE probabilities.
Since credal nets can be applied to both formalisms independently, they can also
be applied to their unification. In fact in order to apply the credal net method to
OBE, some means is needed of converting statistical statements, which can be
construed as constraints involving generic, repeatably-instantiatable variables,
to constraints involving the single-case variables which constitute the nodes of
the objective Bayesian credal net; only then can the constraint graph of §4.2 be
constructed. The 2oEP credal nets of §3.2 allow one to do this, since this kind
of net incorporates both statistical variables and single-case variables as nodes.
Thus 2oEP credal nets are employed first to generate single-case constraints, at
which stage the OBE credal net can be constructed to perform inference. This
fits with the general view of 2oEP as a theory of how evidence constrains rational
degrees of belief and OBE as a theory of how further considerations—especially
equivocation—further constrain rational degrees of belief.
Consider the following very simple example:
%x(F x, Rx, [.2, .5]), Rt, ∀x(F x → Gx)
3/4
?
|≈EP
OBE Gt
Now the first two premisses yield F t[.2,.5] by EP. This constraint combines with
the third premiss to yield an answer to the above question by appealing to OBE.
This answer can be calculated by constructing the following credal net:
✓✏
ϕ
❍❍
✒✑
❍❍✓✏
✓✏
❥
❍
✲ Gt
Ft
✒✑
✯✒✑
✟
✟
✓✏
✟✟
✟
Rt
✒✑
Here ϕ is the first premiss. The left-hand side of this net is the 2oEP net, with
associated probability constraints
P (ϕ) = 1, P (Rt) = 1,
P (F t|ϕ ∧ Rt) ∈ [.2, .5], P (F t|¬ϕ ∧ Rt) = 0 = P (F t|ϕ ∧ ¬Rt) = P (F t|¬ϕ ∧ ¬Rt).
The right-hand side of this net is the OBE net with associated probabilities
P (Gt|F t) = 7/10, P (Gt|¬F t) = 1/2.
21
Standard inference algorithms then yield an answer of 7/12 to our question:
%x(F x, Rx, [.2, .5]), Rt, ∀x(F x → Gx)
6
3/4
7/12
|≈EP
OBE Gt
Conclusion
While evidential probability and objective Bayesian epistemology might at first
sight appear to be chalk and cheese, on closer inspection we have seen that their
relationship is more like horse and carriage—together they do a lot of work,
covering the interface between statistical inference and normative epistemology.
Along the way we have taken in an interesting array of theories—first-order
evidential probability, second-order evidential probability, objective Bayesian
epistemology and EP-calibrated OBE—that can be thought of as nonmonotonic
logics.
2oEP and OBE are probabilistic logics in the sense that they appeal to
the usual mathematical notion of probability. More precisely, their entailment
relations are probabilistic: premisses entail the conclusion if every model of
the premisses satisfies the conclusion, where models are probability functions.
This connection with probability means that credal networks can be applied
as inference machinery. Credal nets yield a perspicuous representation and the
prospect of more efficient inference (Haenni et al., 2008).
Acknowledgements
We are grateful to the Leverhulme Trust for supporting this research, and to
Prasanta S. Bandyopadhyay, Teddy Seidenfeld and an anonymous referee for
helpful comments.
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