Constraints on credences in two
not mutually exclusive propositions:
the search for the best belief update function. . .
Leszek Wroński
Institute of Philosophy
Jagiellonian University
Kraków, Poland
’Full and Partial Belief’ Workshop
The Tilburg Center for Logic, General Ethics, and Philosophy of Science
20 X 2014
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
1 / 23
. . . or . . .
Leszek Wroński
Institute of Philosophy
Jagiellonian University
Kraków, Poland
’Full and Partial Belief’ Workshop
The Tilburg Center for Logic, General Ethics, and Philosophy of Science
20 X 2014
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
1 / 23
Some problems with
minimizing expected epistemic inaccuracy
Leszek Wroński
Institute of Philosophy
Jagiellonian University
Kraków, Poland
’Full and Partial Belief’ Workshop
The Tilburg Center for Logic, General Ethics, and Philosophy of Science
20 X 2014
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
1 / 23
Some problems with
minimizing expected epistemic inaccuracy:
logarithmic measures to the rescue
Leszek Wroński
Institute of Philosophy
Jagiellonian University
Kraków, Poland
’Full and Partial Belief’ Workshop
The Tilburg Center for Logic, General Ethics, and Philosophy of Science
20 X 2014
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
1 / 23
Outline:
the accuracy-centered perspective in formal epistemology: the basic
notions;
a problem from Leitgeb & Pettigrew (2010b) regarding updating one’s
belief function in response to a certain type of evidence;
the solution in the simplest non-trivial case and its intransferability to
different cases;
a better (?) solution obtained using a different measure of inaccuracy
from the one suggested by Leitgeb & Pettigrew.
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
2 / 23
Alethic inaccuracy
An important notion in formal epistemology: Inaccuracy as distance
from truth.
Properly fleshed out, this notion has been used to justify numerous
epistemic norms.
In this talk assume that belief functions are probability functions on the
power sets of finite sets of epistemically possible worlds. (It follows that
we assume probabilism; with an agent at a given time we associate a
finite probability space hW , P(W ), bi.)
An inaccuracy measure should be a function giving us a real number for a
possible world and a belief function:
I (w , b) = r
(“If w is the actual world, then the inaccuracy of the belief function b is r ”).
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
3 / 23
I (w , b) = r
(“If w is the actual world, then the inaccuracy of the belief function b is r ”).
The actual shape of an inaccuracy measure follows from which scoring
rule is used. Two widely used types of scoring rules lead to the so called
quadratic and logarithmic inaccuracy measures.
We can also consider the expected inaccuracy of a belief function b0
given a belief function b:
X
b({w }) · I (w , b0 )
w ∈W
and require from an ideally rational agent to, in response to the incoming
evidence, aim to minimise expected inaccuracy from the perspective of
their current belief function.
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
4 / 23
Updating your credences
Suppose, starting with a belief function b, you learn (only) that a
proposition E is true. How should your new belief function b0 look like?
The traditional Bayesian answer: conditionalise. That is, for any A,
b0 (A) = b(A|E ).
This way of updating is sometimes called full conditionalisation. It only
makes sense if b(E ) 6= 0. And it will never raise your credence from 0 or
lower it from 1.
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
5 / 23
But what if our evidence is not certain? E.g. we learn our new credences in
disjoint propositions E1 , · · · , Em , which form a partition of W ?
The traditional answer: use Jeffrey Conditionalisation. That is, for any
A,
m
X
0
b (A) =
b0 (Ei ) · b(A|Ei ).
i=1
Full conditionalisation is a special case of Jeffrey conditionalisation (JC).
JC will also never make your extreme credences non-extreme. It is also
order-dependent: two sets of constraints applied in reverse order may
result in a different belief function.
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
6 / 23
The updating problem
What if the constraints implied by your evidence are (only) your new
credences in some two propositions A and B such that A ∩ B 6= ∅?
This is an open problem in Leitgeb & Pettigrew (2010b).
I will show a partial answer in the L & P approach and a general answer in
a rival framework.
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
7 / 23
A sketch of (a part) of Leitgeb & Pettigrew (2010):
a series of arguments for using quadratic inaccuracy measures;
a proof that full conditionalisation minimizes expected inaccuracy;
an example showing that Jeffrey conditionalisation does not!
a proof that an Alternative Jeffrey Conditionalisation (AJC) rule does.
AJC sometimes dictates that you raise your credence in a proposition from
0. The way of updating depends not only on your new credences in the
elements of a partition, but also on the cardinality of the elements.
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
8 / 23
Levinstein (2012):
the AJC rule leads to unintuitive updating behaviour;
if the logarithmic inaccuracy measure is used, Jeffrey
Conditionalisation follows as the way to minimise one’s expected
inaccuracy in cases of the type discussed above.
So: what happens in our case? To repeat: you learn just the new credences
in some two propositions A and B such that A ∩ B 6= ∅.
Let us tackle the simplest non-trivial—four-world—setup first. Assume
then that W = {w1 , w2 , w3 , w4 }, A = {w1 , w2 } and B = {w2 , w3 } and you
start with a belief function b; you learn your new credences b0 (A) = p and
b0 (B) = q. Notice that the task reduces to figuring out b0 (A ∩ B).
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
9 / 23
Levinstein (2012):
the AJC rule leads to unintuitive updating behaviour;
if the logarithmic inaccuracy measure is used, Jeffrey
Conditionalisation follows as the way to minimise one’s expected
inaccuracy in cases of the type discussed above.
So: what happens in our case? To repeat: you learn just the new credences
in some two propositions A and B such that A ∩ B 6= ∅.
Let us tackle the simplest non-trivial—four-world—setup first. Assume
then that W = {w1 , w2 , w3 , w4 }, A = {w1 , w2 } and B = {w2 , w3 } and you
start with a belief function b; you learn your new credences b0 (A) = p and
b0 (B) = q. Notice that the task reduces to figuring out b0 (A ∩ B).
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
9 / 23
Fact 1
Let A be an agent with the belief function b at time t. Let
W = {w1 , w2 , w3 , w4 } be the set of epistemically possible worlds for A.
Let A be the proposition {w1 , w2 }; let B be {w2 , w3 }. Suppose that
between times t and t 0 the agent learns (only) the following constraints on
her belief function b0 at t 0 : b0 (A) = p and b0 (B) = q. Let us label the
as K . The belief update function which
number b(A ∩ B) + p−b(A)+q−b(B)
2
minimizes global expected inaccuracy in the sense of Leitgeb & Pettigrew
(2010) is fully determined by the two constraints and the following
condition:

if K < 0

0
0
b (A ∩ B) = min {p, q} if min {p, q} < K


K
otherwise.
So: the change in your credence in A ∩ B is the average of the changes in
your credences in A and in B (whenever it makes mathematical sense).
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
10 / 23
At this point a prima facie reasonable way to proceed would be the
following:
given any similar situation with an arbitrary finite W , calculate
b0 (A ∩ B) using the above formula;
this, together with the two constraints, gives us the credences at time
t 0 in all propositions in the set {A ∩ B, A ∩ ¬B, ¬A ∩ B, ¬A ∩ ¬B},
which is a partition of W ;
We can (it would seem) now use AJC to derive the full shape of b0 .
But no. It turns out the answer depends on the cardinality of the
propositions. (Not just the AJC does; but also the method of calculating
the input for AJC.)
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
11 / 23
At this point a prima facie reasonable way to proceed would be the
following:
given any similar situation with an arbitrary finite W , calculate
b0 (A ∩ B) using the above formula;
this, together with the two constraints, gives us the credences at time
t 0 in all propositions in the set {A ∩ B, A ∩ ¬B, ¬A ∩ B, ¬A ∩ ¬B},
which is a partition of W ;
We can (it would seem) now use AJC to derive the full shape of b0 .
But no. It turns out the answer depends on the cardinality of the
propositions. (Not just the AJC does; but also the method of calculating
the input for AJC.)
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
11 / 23
Consider the following example (adapted from Osherson 2002): “Listening
to the radio I hear a forecast for rain but I’m not sure whether it comes
from the chief meteorologist or from his unreliable deputy.” At that
moment I have the following belief function b1 , where R is “It rains today”
and C is “The chief was speaking”;
b1
¬RC
.2
RC
.4
R¬C
.1
¬R¬C
.3
Now, a glance at the sky raises my credence in R to .7. Assume that the
forecast is rebroadcasted and even though I strain my ears, I conclude I
should not change my credence in C . The following is my new credence
dictated by minimizing my expected epistemic inaccuracy in the sense of
Leitgeb & Pettigrew (i.e. by Fact 1):
0
b1
Leszek Wroński (JU, Poland)
¬RC
.1
RC
.5
R¬C
.2
¬R¬C
.2
Problems with minimizing GExpInacc
20 X 2014
12 / 23
I’d like to raise two problems in this situation.
a problem in the 4-world case, with reacting to a certain type of
evidence;
a problem with extending it to a 5-world case and reacting to the
same evidence as on the previous slide.
Let us start with the second problem.
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
13 / 23
Suppose I owe the chief meteorologist money and I don’t want to meet
him. I know that whenever it doesn’t rain, he always walks home through a
park he never visits otherwise and in which I walk my dog (which I do
regardless of the weather); not wanting to disturb my dog’s routine, I will
also go to the park today. (If it rains, the chief meteorologist takes a bus
home.) He can traverse the park via one of two paths, call them “1” and
“2”. Let B be the proposition “the chief meteorologist will walk via path 1
today”. My four epistemic possible worlds
¬RC
RC
R¬C
¬R¬C
become
¬RCB
¬RC ¬B
RC ¬B
R¬C ¬B
¬R¬C ¬B
and my initial credence becomes, say,
b2
¬RCB
.1
¬RC ¬B
.1
RC ¬B
.4
R¬C ¬B
.1
¬R¬C ¬B
.3
The ¬RC world has effectively split in two. (Think of this as of two agents
with a similar space of epistemically possible worlds.)
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
14 / 23
Consider the same evidential situation: a glance at the sky raises my
credence in R to .7; the forecast is rebroadcasted and even though I strain
my ears I conclude I should not change my credence in C . The following is
my new credence dictated by minimizing the expected inaccuracy in the
L & P sense:
0
b2
¬RCB
.043
¬RC ¬B
.043
RC ¬B
.514
R¬C ¬B
.186
¬R¬C ¬B
.214
Let us bring together the initial and updated credences for the two agents:
b1
0
b1
b2
0
b2
¬RCB
.1
.043
Leszek Wroński (JU, Poland)
¬RC
.2
.1
¬RC ¬B
.1
.043
RC
.4
.5
R¬C
.1
.2
RC ¬B
.4
.514
¬R¬C
.3
.2
R¬C ¬B
.1
.186
Problems with minimizing GExpInacc
¬R¬C ¬B
.3
.214
20 X 2014
15 / 23
Notice that the two agents
started with the same credences in R, C , and RC ;
faced the same evidence implying (only) that they should increase their
credence in R in the same way and not to change their credence in C ;
ended with a different credence in RC ;
and it seems the only difference was whether the park was considered
or not.
Note: there are cases in which not only the credence in a suitably chosen
proposition ends up being different, but the two agents end up with
different probability rankings regarding two propositions which they
initially believe to the same degree: that is, cases in which
b1 (φ) = b2 (φ) and b1 (ψ) = b2 (ψ)
but
0
0
0
0
b2 (φ) < b2 (ψ) while b1 (φ) > b1 (ψ).
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
16 / 23
Now, for a problematic example in the 4-world case. Start again with the
same credence function:
b1
¬RC
.2
RC
.4
R¬C
.1
¬R¬C
.3
Consider now two pieces of evidence:
a glance at the sky lowers my credence in R to .4 (by .1);
a rebroadcasting of the forecast increases my credence in C to .7 (by
.1).
The following is my new credence dictated by Fact 1:
00
b1
¬RC
.3
RC
.4
R¬C
.0
¬R¬C
.3
Notice that somehow I ended up with credence 0 in R¬C .
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
17 / 23
What happened is I updated my credences in R and C as if they were
perfectly anticorrelated. I interpreted an increase in the credence in C as
an increase in the credence in ¬RC ; I interpreted a decrease in the
credence in R as a decrease in the credence in R¬C .
In general:
increase in R=increase in C : update as if they are perfectly correlated
(increase RC by the same amount);
increase in C =−increase in R: update as if they are perfectly
anticorrelated (leave RC as it was, increase ¬RC and decrease R¬C
by the same amount).
Even if some would like to protest against using the word “correlation”
here, I think we should find e.g. the example involving the decrease of
credence in R¬C to 0 troubling.
This behaviour extends beyond the 4-world case.
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
18 / 23
Logarithmic measures to the rescue
Levinstein (2012): JC does minimise expected inaccuracy if a logarithmic
inaccuracy measure is used.
The measure used by Levinstein is Ilog (w , b) = − ln(b({w }).
What happens if we use this measure in our case?
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
19 / 23
Fact 2
Let A be an agent with the belief function b at time t. Let A and B be
two propositions such that A ∩ B 6= ∅. Suppose that the evidence received
by A between times t and t 0 entails (only) her credences b0 (A) and b0 (B).
To minimise expected inaccuracy calculated by the logarithmic measure A
should calculate her new credences in the logical combinations of A and B
so that the following is true:
b(¬A ∩ B)
b(A ∩ B)
b(¬A ∩ ¬B)
b(A ∩ ¬B)
+ 0
= 0
+ 0
0
b (A ∩ ¬B) b (¬A ∩ B)
b (A ∩ B) b (¬A ∩ ¬B)
and then use Jeffrey Conditionalisation.
This does not depend on the cardinality of the propositions.
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
20 / 23
...or, in other words:
Fact 2
Let A be an agent with the belief function b at time t. Let A and B be
two propositions such that A ∩ B 6= ∅. Suppose that the evidence received
by A between times t and t 0 entails (only) the following constraints on her
belief function b0 at t 0 : b0 (A) = p and b0 (B) = q. To minimise expected
inaccuracy calculated by the logarithmic measure A should calculate her
new credences in the logical combinations of A and B so that the following
is true:
b(¬A ∩ B)
b(A ∩ B)
b(¬A ∩ ¬B)
b(A ∩ ¬B)
+
= 0
+ 0
0
0
p − b (A ∩ B) q − b (A ∩ B)
b (A ∩ B) b (A ∩ B) + 1 − p − q
and then use Jeffrey Conditionalisation.
This does not depend on the cardinality of the propositions.
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
20 / 23
Conclusions
I have shown:
the solution to Leitgeb & Pettigrew’s problem in the simplest
non-trivial case;
that the solution in their framework is not transferable to different
cases;
a general solution using the logarithmic inaccuracy measure.
This might be an argument against using quadratic inaccuracy measures.
But what to do with L & P’s powerful arguments in their 2010 paper?
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
21 / 23
Thank you!
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
22 / 23
References:
Leitgeb, H., and Pettigrew, R. (2010), “An Objective Justification of
Bayesianism I: Measuring Inaccuracy”, Philosophy of Science 77(2);
Leitgeb, H., and Pettigrew, R. (2010), “An Objective Justification of
Bayesianism II: The Consequences of Minimizing Inaccuracy”,
Philosophy of Science 77(2);
Levinstein, B. (2012), “Leitgeb and Pettigrew on Accuracy and
Updating”, Philosophy of Science 79(3).
Leszek Wroński (JU, Poland)
Problems with minimizing GExpInacc
20 X 2014
23 / 23