Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras 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
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras 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
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras 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
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras 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 { ϕ : B ϕ ∈ B } � D { ϕ : D ϕ ∈ D } � S { 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 ) B ¬ ϕ ∈ B iff 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
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras 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 w needs to be at least 2 . 588 times the value of r . Shear, Fitelson (UCD, Rutgers/MCMP) Updating Beliefs via Maximization of EEU October 20, 2014 5 / 24
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras The following is a piecewise definition of the agent’s epistemic utility function, u ( · , w ) : if ϕ is true at w r u ( B ( ϕ ) , w ) : � if ϕ is false at w − w if ϕ is true at w − w u ( D ( ϕ ) , w ) : � if ϕ is false at w r if ϕ is true at w 0 u ( S ( ϕ ) , w ) : � if ϕ is false at w 0 Shear, Fitelson (UCD, Rutgers/MCMP) Updating Beliefs via Maximization of EEU October 20, 2014 6 / 24
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras Using the agents credal function, define expected epistemic utility (EEU) of a belief set: � � EEU ( B ( ϕ ) , b ) : � b ( w ) · u ( B ( ϕ ) , w ) w ∈ W ϕ ∈A 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 ) > B ( ϕ ) w iff r + w b ( ϕ ) < D ( ϕ ) r iff r + w � � b ( ϕ ) ∈ S ( ϕ ) r w iff r + w , r + w Shear, Fitelson (UCD, Rutgers/MCMP) Updating Beliefs via Maximization of EEU October 20, 2014 7 / 24
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras Define the EEU maximizing qualitative update on belief set, B , by E as: B � E : � { B ϕ, D ψ, S θ : b ( ϕ | E ) > r + w , b ( ψ | E ) < w r r + w , � � b ( θ | E ) ∈ r w } r + w , r + w An agent with a belief set, B , before learning E and a belief set, B ′ , after learning E is diachronically coherent iff B ′ � B � 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
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras Basic Gärdenfors [1, 6] postulates for AGM revision ( ∗ ): 1. Closure: B ∗ ϕ � Cn ( B ∗ ϕ ) 2. Success: 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 ψ is consistent with B ∗ ϕ , then Cn (( B ∗ ϕ ) ∪ { ψ } ) ⊆ B ∗ ( ϕ ∧ ψ ) Shear, Fitelson (UCD, Rutgers/MCMP) Updating Beliefs via Maximization of EEU October 20, 2014 9 / 24
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras 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
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras 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 . 375 b p ( ¬ S ∧ P ) � 0 . 125 b p ( ¬ S ∧ ¬ P ) � 0 . 125 Shear, Fitelson (UCD, Rutgers/MCMP) Updating Beliefs via Maximization of EEU October 20, 2014 11 / 24
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras 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 S ∨ ¬ P ¬ S ∨ P ¬ S ∨ ¬ P 0 . 875 0 . 875 0 . 625 0 . 625 S P S ≡ ¬ P S ≡ P ¬ P ¬ S 0 . 75 0 . 5 0 . 5 0 . 5 0 . 5 0 . 25 S ∧ P S ∧ ¬ P ¬ S ∧ P ¬ S ∧ ¬ P 0 . 375 0 . 375 0 . 125 0 . 125 ⊥ 0 B � { S , S ∨ P , S ∨ ¬ P } Shear, Fitelson (UCD, Rutgers/MCMP) Updating Beliefs via Maximization of EEU October 20, 2014 12 / 24
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras 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 S ∨ ¬ P ¬ S ∨ P ¬ S ∨ ¬ P 0 . 8 0 . 8 0 . 4 1 S P S ≡ ¬ P S ≡ P ¬ P ¬ S 0 . 6 0 . 2 0 . 8 0 . 2 0 . 8 0 . 4 S ∧ P S ∧ ¬ P ¬ S ∧ P ¬ S ∧ ¬ P 0 0 . 6 0 . 2 0 . 2 ⊥ 0 B � ( ¬ 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
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras Compare her original beliefs: B � { S , S ∨ P , S ∨ ¬ P } with her new beliefs: B � ( ¬ 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
Background EEU Maximization Contrastive cases with AGM Conclusions and future work References Extras 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
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