Updating Beliefs via Maximization of Expected Epistemic Utility Ted Shear Branden Fitelson
Updating Beliefs via Maximization of Expected
Epistemic Utility
Ted Shear 1 Branden Fitelson 2
1
University of California, Davis ebshear@ucdavis.edu
2
Rutgers/MCMP branden@fitelson.org
Workshop in Full and Partial Belief – TiLPS 2014 www.tedshear.com/tilps_slides.pdf
www.tedshear.com/tilps_handout.pdf
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 1 / 24
Question:
When are the beliefs of an epistemically rational agent diachronically coherent?
Answer:
When they maximize expected epistemic utility relative to her credence function conditional on new evidence!
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 2 / 24
We will presuppose that:
1. the only thing that agents care about is epistemic value in the
Jamesian sense [8, 5];
2. each agent has a probabilistic credence function b ( · ) ;
3. each agent takes precisely one of three qualitative doxastic attitudes – belief ( B ϕ
) , disbelief ( D ϕ
) , or suspension ( S ϕ
) – towards every proposition in some finite agenda, A ; and
4. the truth values of B ( ϕ
) and ϕ will not alter b [3, 2, 7, 9].
We do not assume that: there is any reduction between (qualitative) beliefs and credences.
Our proposal should be seen as a joint constraint on the update of full beliefs and credences.
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 3 / 24
Let an agent’s qualitative belief state be represented by the set B contains every statement of the agent’s beliefs, disbeliefs, or suspensions.
B
D
S
{ ϕ
: B ϕ
∈ B }
{ ϕ
: D ϕ ∈ D }
{ ϕ
: S ϕ ∈ S }
We will only consider reasonable agents with symmetric beliefs and disbeliefs in the sense that:
B ( ϕ
) iff B ϕ
∈ B
D ( ϕ
) iff B ¬ ϕ
∈ B
S ( ϕ
) iff B ϕ
<
B and B ¬ ϕ
<
B
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 4 / 24
Let r and − w be the epistemic utilities an agent receives for qualitative accuracy and inaccuracy respectively following the scoring rule developed by Easwaran [4].
Presently, we place no restriction on these values aside from assuming that 0
< r ≤ w . In the present framework, if we these values were otherwise, then (assuming act-state independence) agents would be required to be opinionated.
That is not to say that there is nothing further to say about these values. For example, Pruss [11] argues that the value of be at least 2
.
588 times the value of r .
w needs to
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 5 / 24
The following is a piecewise definition of the agent’s epistemic utility function, u ( ·
, w ) : u ( B ( ϕ
)
, w ) : u ( D ( ϕ
)
, w ) : u ( S ( ϕ
)
, w ) :
r if ϕ is true at w
− w if ϕ is false at w
− w if ϕ is true at w
r if ϕ is false at w
0
0 if if ϕ is true at w ϕ is false at w
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 6 / 24
Using the agents credal function, define expected epistemic utility
(EEU) of a belief set:
EEU ( B ( ϕ
)
, b ) :
X X b ( w ) · u ( B ( ϕ
)
, w ) ϕ
∈A w ∈ W
Theorem [4]: An agent with credal function b and beliefs B , disbeliefs D , and suspentions S will maximize EEU relative to her credence function iff for every ϕ
∈ A
B ( ϕ
) iff b ( ϕ
)
>
D ( ϕ
) iff b ( ϕ
)
<
S ( ϕ
) iff b ( ϕ
) ∈ w r + w r r + w f r r + w
, w r + w g
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 7 / 24
Define the EEU maximizing qualitative update on belief set, B , by E as:
B C E : { B ϕ,
D
ψ,
S
θ
: b b (
θ
| E ) ∈
( ϕ | E )
> f r r + w
, w r + w g
} w r + w
, b (
ψ | E )
< r r + w
,
An agent with a belief set, B , before learning E and a belief set, B
0
, after learning E is diachronically coherent iff
B
0
B C E
How does this kind of qualitative updating compare with AGM revision?
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 8 / 24
Basic Gärdenfors [1, 6] postulates for AGM revision ( ∗ ):
1.
Closure: B
2.
Success:
∗ ϕ
Cn ( B ∗ ϕ
) ϕ
∈ B ∗ ϕ
3.
Inclusion: B ∗ ϕ
⊆ Cn ( B ∪ { ϕ
} )
4.
Vacuity: If ϕ is consistent with B , then B ∗ ϕ ⊇ Cn ( B ∪ { ϕ } )
5.
Consistency: If ϕ is not self-contradictory, then B ∗ ϕ is consistent
6.
Extensionality: If ϕ ≡ ψ ∈ Cn (
∅
) , then B ∗ ϕ
B ∗ ψ
Supplementary postulates:
7.
Superexpansion: B ∗ ( ϕ
∧
ψ
) ⊆ Cn (( B ∗ ϕ
) ∪ {
ψ
} )
8.
Subexpansion: If
Cn
ψ is consistent with B ∗ ϕ
, then
(( B ∗ ϕ
) ∪ {
ψ
} ) ⊆ B ∗ ( ϕ
∧
ψ
)
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 9 / 24
One principle of particular interest is the following one that is implied by the AGM postulates:
Accretiveness: If ϕ is consistent with B , then B ⊂ B ∗ ϕ
We still see that agents who maximize EEU will not necessarily satisfy Accretiveness .
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 10 / 24
Assume the following readings for the atomics:
P ‘Nixon is a pacifist’
S ‘Nixon will win the next election’
The agent, Pat, has the credal function b p which assigns the following values to the atomics: b p
( P ) 0
.
5 b p
( S ) 0
.
75 generated as a vector on the smallest incompossible worlds: b p
( S ∧ P ) 0
.
375 b p
( ¬ S ∧ P ) 0
.
125 b p b p
( S ∧ ¬ P ) 0
.
375
( ¬ S ∧ ¬ P ) 0
.
125
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 11 / 24
P
0
.
5
S ∧ P
0
.
375
S ≡ ¬ P
0
.
5
S ≡ P
0
.
5
S ∧ ¬ P
0
.
375
¬ S ∧ P
0
.
125
⊥
0
B { S
,
S ∨ P
,
S ∨ ¬ P }
¬ P
0
.
5
¬ S ∧ ¬ P
0
.
125
Where r 1 and w 2 and the agenda, A p
, is the algebra on atomics P and S , the lattice below represents the belief set that maximizes EEU relative to b p
:
>
1
S ∨ P
0
.
875
S ∨ ¬ P
0
.
875
¬ S ∨ P
0
.
625
¬ S ∨ ¬ P
0
.
625
0
S
.
75
¬ S
0
.
25
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 12 / 24
Suppose that Pat learns (with certainty) that if Nixon will win the next election, then he is not a pacifist. The effect of this update under our rule is represented below:
>
1
S ∨ P
0
.
8
S ∨ ¬ P
0
.
8
¬ S ∨ P
0
.
4
¬ S ∨ ¬ P
1
¬ S
0
.
4
S
0
.
6
P
0
.
2
S ≡ ¬ P
0
.
8
S ≡ P
0
.
2
¬ P
0
.
8
S ∧ P
0
S ∧ ¬ P
0
.
6
¬ S ∧ P
0
.
2
¬ S ∧ ¬ P
0
.
2
⊥
0
B C ( ¬ S ∨ ¬ P ) { P
,
S ∨ P
,
S ∨ ¬ P
,
¬ S ∨ ¬ P
,
P ≡ ¬ S }
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 13 / 24
Compare her original beliefs:
B { S
,
S ∨ P
,
S ∨ ¬ P } with her new beliefs:
B C ( ¬ S ∨ ¬ P ) { P
,
S ∨ P
,
S ∨ ¬ P
,
¬ S ∨ ¬ P
,
P ≡ ¬ S }
.
She gives up belief in S because:
Pat’s old evidence that she will win the election is now also evidence that he is not a pacifist; and
Pat’s old evidence that he is a pacifist is now also evidence that he will not win the election.
Thus, EEU maximizing updates may not satisfy Accretiveness .
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 14 / 24
We have sketched the basic proposal for the obvious naïve diachronic coherence requirement of EEU maximization conditional on new evidence.
We have gestured at some differences between EEU maximizing updates and AGM updates.
In particular, we have demonstrated that EEU maximizing agents will not necessarily be accretive.
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 15 / 24
Future work:
Establish purely qualitative postulates for EEU maximizing revision (See [13] for r w )
Jeffrey conditionalization
Defining contraction operators ([12])
Deal with act-state dependence and higher-order beliefs
Incorporate comparative confidence
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 16 / 24
Thank you!
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 17 / 24
[1] C. E. Alchourron, P. Gärdenfors, and D. Makinson. On the logic of theory change: Partial meet contraction and revision functions.
The Journal of Symbolic Logic , 50(2):510–530, June 1985.
[2] M. Caie. Rational probabilistic incoherence.
Philosophical Review , 122(4):527–575, 2013.
[3] J. Carr. Epistemic utility and the aim of belief. 2014.
[4] K. Easwaran. Dr. Truthlove, or how I learned to stop worrying and love Bayesian probabilities.
http://dl.dropbox.com/u/10561191/Unpublished/Truthlove.pdf
, 2014.
[5] K. Easwaran and B. Fitelson. Accuracy, coherence and evidence. In T. S. Gendler and J. Hawthorne, editors, Oxford
Studies in Epistemology , volume 5. Oxford University Press, Oxford, 2014.
[6] P. Gärdenfors. Rules for rational changes of belief. In T. Pauli, editor, Philosophical essays dedicated to Lennart
Åqvist on His Fiftieth Birthday , pages 88–101. Departmento de Filosofía, Universidad de Uppsala, Uppsala, 1982.
[7] H. Greaves. Epistemic decision theory.
Mind , forthcoming.
[8] W. James. The will to believe.
The New World , 5:327–347, 1896.
[9] J. Konek and B. Levinstein. The foundations of epistemic decision theory. 2014.
[10] H. Leitgeb. Reducing Belief Simpliciter to Degrees of Belief.
Annals of Pure and Applied Logic , 164:1338–1389, 2013.
[11] A. R. Pruss. Independent tests and the log-likelihood-ratio measure of confirmation.
Thought , 3:124–135, 2014.
[12] M. Titelbaum.
Quitting Certainties: A Bayesian Framework Modeling Degrees of Belief . Oxford University Press,
Oxford, 2014.
[13] J. van Eijck and B. Renne. Belief as willingness to bet. 2014.
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 18 / 24
Consider the two core philosophical principles that underly AGM revision:
Conservativity: When performing belief-contravening revision, agents should only make the minimal change that is consistent with the new information.
Entrenchment: Agents ought to give up least entrenched beliefs before giving up more entrenched beliefs.
We will show that EEU maximization can violate Conservativity and
Entrenchment (given some minimal assumptions).
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 19 / 24
Assume the following readings for the atomics:
T ‘Taylor will attend the party’
R ‘Ryan will attend the next party’
The agent, Morgan, has the credal function b m which assigns the following values to the atomics: b m
( T ) 0
.
86 b m
( R ) 0
.
84 generated as a vector on the smallest incompossible worlds: b m
( T ∧ R ) 0
.
7 b m
( ¬ T ∧ R ) 0
.
14 b m b m
( T ∧ ¬ R ) 0
.
16
( ¬ T ∧ ¬ R ) 0
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 20 / 24
Where r 1 and w 2 and the agenda, A p
, is the algebra on atomics T and R , the lattice below represents the belief set that maximizes EEU relative to b p
:
>
1
T ∨ R
1
T ∨ ¬ R
0
.
86
¬ T ∨ R
0
.
84
¬ T ∨ ¬ R
0
.
3
0
T
.
86
¬ T
0
.
14
R
0
.
84
T ≡ ¬ R
0
.
3
T ≡ R
0
.
7
¬ R
0
.
16
T ∧ R
0
.
7
T ∧ ¬ R
0
.
16
¬ T ∧ R
0
.
14
¬ T
⊥
0
B { T
,
R
,
T ∨ R
,
T ∨ ¬ T
,
T ∨ ¬ R
,
¬ T ∨ R }
∧ ¬
0
R
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 21 / 24
Suppose that Morgan learns (with certainty) that either Taylor or
Ryan will show up but not both.
T
0
.
533
>
1
T ∨ R
1
R
0
.
466
T ∨ ¬ R
0
.
533
T ≡ ¬ R
1
¬ T ∨ R
0
.
466
T ≡ R
0
¬ T ∨ ¬ R
1
¬ R
0
.
533
T ∧ R
0
T ∧ ¬ R
0
.
533
¬ T ∧ R
0
.
466
¬ T ∧ ¬ R
0
⊥
0
B C ( T ≡ ¬ R ) { T ∨ R
,
T ≡ ¬ R
,
¬ T ∨ ¬ R }
¬ T
0466
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 22 / 24
Compare her original beliefs:
B { T
,
R
,
T ∨ R
,
T ∨ ¬ T
,
T ∨ ¬ R
,
¬ T ∨ R } with her new beliefs:
B C ( T ≡ ¬ R ) { T ∨ R
,
T ≡ ¬ R
,
¬ T ∨ ¬ R }
Although she was initially highly confident that they would both be at the party, when she learns that only one of them will show up, she no longer accepts that either of them will be there.
While she is slightly more confident that Morgan will show up than that Ryan will, she suspends belief about both even though the minimal change consistent with the new information would have been to only give up one belief.
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 23 / 24
A majority of the remainder of the talk will be devoted to contrasting
EEU maximizing qualitative updates with AGM revision.
It is worth noting though that this sort of updating will violate the following principle from Leitgeb [10]:
( P 2 ) For any ϕ
∈ B
,
B ∗ ϕ
B
The reason that this will fail can be seen intuitively since there may be some proposition threshold, but
ψ
∈ S such that b (
ψ
) ϕ is a probability raiser for
ψ
.
is just shy of the
So, updating on ϕ with certainty may push b ( ψ ) above the threshold.
Shear, Fitelson (UCD, Rutgers/MCMP)
Updating Beliefs via Maximization of EEU
October 20, 2014 24 / 24
