Credal States • At any time t , your opinions are representable with a credal possible worlds; opinionated; 8 state < W , A , c t > • W = { w 1 , w 2 , . . . , w N } is a fjnite set of doxastically • A ⊆ W is a proposition ; • A ⊆ ℘ ( W ) is the set of propositions about which you are • c t : A → [ 0 , 1 ] is your time t credence function.
Credal States • At any time t , your opinions are representable with a credal possible worlds; opinionated; 8 state < W , A , c t > • W = { w 1 , w 2 , . . . , w N } is a fjnite set of doxastically • A ⊆ W is a proposition ; • A ⊆ ℘ ( W ) is the set of propositions about which you are • c t : A → [ 0 , 1 ] is your time t credence function.
Credal States • At any time t , your opinions are representable with a credal possible worlds; opinionated; 8 state < W , A , c t > • W = { w 1 , w 2 , . . . , w N } is a fjnite set of doxastically • A ⊆ W is a proposition ; • A ⊆ ℘ ( W ) is the set of propositions about which you are • c t : A → [ 0 , 1 ] is your time t credence function.
Bayesianism Probabilism
Probabilism Probabilism 9 At all times t , c t should be a probability function.
Bayesianism Conditionalization
Conditionalization adopt, at t , upon receiving the total evidence E . Conditionalization Tiere should be some credence function c such that, for all evidence, 10 • Use ‘ c t , E ’ for the credence function you are disposed to times t and all A , E ∈ A such that E could be your total time t c t , E ( A ) = c ( A | E )
Conditionalization • Use ‘ c E ’ for the credence function you are disposed to adopt, at t , upon receiving the total evidence E . Conditionalization Tiere should be some credence function c such that, for all evidence, 10 times t and all A , E ∈ A such that E could be your total time t c t , E ( A ) = c ( A | E )
Conditionalization adopt, at t , upon receiving the total evidence E . Conditionalization Tiere should be some credence function c such that, for all evidence, 10 • Use ‘ c t , E ’ for the credence function you are disposed to times t and all A , E ∈ A such that E could be your total time t c t , E ( A ) = c ( A | E )
Bayesianism • Tie Bayesian account of rational learning: you should be a probabilistic conditionalizer. 11
Daniel is not a probabilistic conditionalizer If Daniel were a conditionalizer, then So Daniel isn’t probabilistic 12 1. c a Dem. φ -ed ( φ -ing is wrong ) is low. 2. c a Rep. φ -ed ( φ -ing is wrong ) is high. c ( φ -ing is wrong | a Dem. φ -ed ) is low and c ( φ -ing is wrong | a Rep. φ -ed ) is high But Daniel thinks whether φ -ing is wrong is independent of whether a Dem. or a Rep. φ -ed. So c ( φ -ing is wrong ) is low and c ( φ -ing is wrong ) is high
Daniel is not a probabilistic conditionalizer If Daniel were a conditionalizer, then So Daniel isn’t probabilistic 12 1. c a Dem. φ -ed ( φ -ing is wrong ) is low. 2. c a Rep. φ -ed ( φ -ing is wrong ) is high. c ( φ -ing is wrong | a Dem. φ -ed ) is low and c ( φ -ing is wrong | a Rep. φ -ed ) is high But Daniel thinks whether φ -ing is wrong is independent of whether a Dem. or a Rep. φ -ed. So c ( φ -ing is wrong ) is low and c ( φ -ing is wrong ) is high
Daniel is not a probabilistic conditionalizer If Daniel were a conditionalizer, then So Daniel isn’t probabilistic 12 1. c a Dem. φ -ed ( φ -ing is wrong ) is low. 2. c a Rep. φ -ed ( φ -ing is wrong ) is high. c ( φ -ing is wrong | a Dem. φ -ed ) is low and c ( φ -ing is wrong | a Rep. φ -ed ) is high But Daniel thinks whether φ -ing is wrong is independent of whether a Dem. or a Rep. φ -ed. So c ( φ -ing is wrong ) is low and c ( φ -ing is wrong ) is high
Daniel is not a probabilistic conditionalizer If Daniel were a conditionalizer, then So Daniel isn’t probabilistic 12 1. c a Dem. φ -ed ( φ -ing is wrong ) is low. 2. c a Rep. φ -ed ( φ -ing is wrong ) is high. c ( φ -ing is wrong | a Dem. φ -ed ) is low and c ( φ -ing is wrong | a Rep. φ -ed ) is high But Daniel thinks whether φ -ing is wrong is independent of whether a Dem. or a Rep. φ -ed. So c ( φ -ing is wrong ) is low and c ( φ -ing is wrong ) is high
Daniel is not a probabilistic conditionalizer If Daniel were a conditionalizer, then So Daniel isn’t probabilistic 12 1. c a Dem. φ -ed ( φ -ing is wrong ) is low. 2. c a Rep. φ -ed ( φ -ing is wrong ) is high. c ( φ -ing is wrong | a Dem. φ -ed ) is low and c ( φ -ing is wrong | a Rep. φ -ed ) is high But Daniel thinks whether φ -ing is wrong is independent of whether a Dem. or a Rep. φ -ed. So c ( φ -ing is wrong ) is low and c ( φ -ing is wrong ) is high
Daniel is not a probabilistic conditionalizer • Tie accuracy-fjrster likes this diagnosis of Daniel’s irrationality. • Tiey wish to show that Probabilism and Conditionalization follow from: • the axiological claim that accuracy is the sole epistemic good; • a claim about how to properly value accuracy; and • the consequentialist deontic norm that it is rational to maximize expected epistemic value. 13
Daniel is not a probabilistic conditionalizer • Tie accuracy-fjrster likes this diagnosis of Daniel’s irrationality. Conditionalization follow from: • the axiological claim that accuracy is the sole epistemic good; • a claim about how to properly value accuracy; and • the consequentialist deontic norm that it is rational to maximize expected epistemic value. 13 • Tiey wish to show that Probabilism and
Daniel is not a probabilistic conditionalizer • Tie accuracy-fjrster likes this diagnosis of Daniel’s irrationality. Conditionalization follow from: • the axiological claim that accuracy is the sole epistemic good; • a claim about how to properly value accuracy; and • the consequentialist deontic norm that it is rational to maximize expected epistemic value. 13 • Tiey wish to show that Probabilism and
Daniel is not a probabilistic conditionalizer • Tie accuracy-fjrster likes this diagnosis of Daniel’s irrationality. Conditionalization follow from: • the axiological claim that accuracy is the sole epistemic good; • a claim about how to properly value accuracy; and • the consequentialist deontic norm that it is rational to maximize expected epistemic value. 13 • Tiey wish to show that Probabilism and
Daniel is not a probabilistic conditionalizer • Tie accuracy-fjrster likes this diagnosis of Daniel’s irrationality. Conditionalization follow from: • the axiological claim that accuracy is the sole epistemic good; • a claim about how to properly value accuracy; and • the consequentialist deontic norm that it is rational to maximize expected epistemic value. 13 • Tiey wish to show that Probabilism and
Epistemic Value
Epistemic Value • Write the epistemic value of a credence function c , under the supposition that w is actual, as: accuracy of c in w . • E.g., one accuracy measure is the quadratic or ‘Brier’ def 14 V ( c , w ) • For the accuracy-fjrster, V ( c , w ) is entirely a function of the measure, Q = − ∑ Q ( c , w ) ( ν A ( w ) − c ( A )) 2 A ∈ A
Epistemic Value • Write the epistemic value of a credence function c , under the supposition that w is actual, as: accuracy of c in w . • E.g., one accuracy measure is the quadratic or ‘Brier’ def 14 V ( c , w ) • For the accuracy-fjrster, V ( c , w ) is entirely a function of the measure, Q = − ∑ Q ( c , w ) ( ν A ( w ) − c ( A )) 2 A ∈ A
Epistemic Value • Write the epistemic value of a credence function c , under the supposition that w is actual, as: accuracy of c in w . • E.g., one accuracy measure is the quadratic or ‘Brier’ def 14 V ( c , w ) • For the accuracy-fjrster, V ( c , w ) is entirely a function of the measure, Q = − ∑ Q ( c , w ) ( ν A ( w ) − c ( A )) 2 A ∈ A
Epistemic Value • Write the epistemic value of a credence function c , under the supposition that w is actual, as: accuracy of c in w . • E.g., one accuracy measure is the quadratic or ‘Brier’ def 14 V ( c , w ) • For the accuracy-fjrster, V ( c , w ) is entirely a function of the measure, Q = − ∑ Q ( c , w ) ( ν A ( w ) − c ( A )) 2 A ∈ A
Epistemic Value • Write the epistemic value of a credence function c , under the supposition that w is actual, as: accuracy of c in w . • E.g., one accuracy measure is the quadratic or ‘Brier’ def 14 V ( c , w ) • For the accuracy-fjrster, V ( c , w ) is entirely a function of the measure, Q = − ∑ Q ( c , w ) ( ν A ( w ) − c ( A )) 2 A ∈ A
Epistemic Value • Write the epistemic value of a credence function c , under the supposition that w is actual, as: accuracy of c in w . • E.g., one accuracy measure is the quadratic or ‘Brier’ def 14 V ( c , w ) • For the accuracy-fjrster, V ( c , w ) is entirely a function of the measure, Q = − ∑ Q ( c , w ) ( ν A ( w ) − c ( A )) 2 A ∈ A
Epistemic Value • Write the epistemic value of a credence function c , under the supposition that w is actual, as: accuracy of c in w . • E.g., one accuracy measure is the quadratic or ‘Brier’ def 14 V ( c , w ) • For the accuracy-fjrster, V ( c , w ) is entirely a function of the measure, Q = − ∑ Q ( c , w ) ( ν A ( w ) − c ( A )) 2 A ∈ A
Other Accuracy Measures • Tie Absolute Value measure def • Tie Logarithmic measure • Tie Euclidean distance measure def def 15 √ ∑ E ( c , w ) = − ( ν A ( w ) − c ( A )) 2 A ∈ A = ∑ A ( c , w ) | ν A ( w ) − c ( A ) | A ∈ A = ∑ L ( c , w ) ln [ | ( 1 − ν A ( w )) − c ( A ) | ] A ∈ A
Other Accuracy Measures • Tie Absolute Value measure def • Tie Logarithmic measure • Tie Euclidean distance measure def def 15 √ ∑ E ( c , w ) = − ( ν A ( w ) − c ( A )) 2 A ∈ A = ∑ A ( c , w ) | ν A ( w ) − c ( A ) | A ∈ A = ∑ L ( c , w ) ln [ | ( 1 − ν A ( w )) − c ( A ) | ] A ∈ A
Other Accuracy Measures • Tie Absolute Value measure def • Tie Logarithmic measure • Tie Euclidean distance measure def def 15 √ ∑ E ( c , w ) = − ( ν A ( w ) − c ( A )) 2 A ∈ A = ∑ A ( c , w ) | ν A ( w ) − c ( A ) | A ∈ A = ∑ L ( c , w ) ln [ | ( 1 − ν A ( w )) − c ( A ) | ] A ∈ A
Evaluating credence functions • Leitgeb & Pettigrew: if your credence function is a of c ’s epistemic value. • Tiis is a general decision-theoretic norm: epistemic acts are choiceworthy to the degree that they maximize expected value. 16 • Use ‘ V c ( c ∗ ) ’ to represent how valuable the credence function c ∗ is, according to the credence function c . probability, p , then, for all c , V p ( c ) should be p ’s expectation = ∑ ! V p ( c ) V ( c , w ) · p ( w ) w ∈ W
Evaluating credence functions • Leitgeb & Pettigrew: if your credence function is a of c ’s epistemic value. • Tiis is a general decision-theoretic norm: epistemic acts are choiceworthy to the degree that they maximize expected value. 16 • Use ‘ V c ( c ∗ ) ’ to represent how valuable the credence function c ∗ is, according to the credence function c . probability, p , then, for all c , V p ( c ) should be p ’s expectation = ∑ ! V p ( c ) V ( c , w ) · p ( w ) w ∈ W
Evaluating credence functions • Leitgeb & Pettigrew: if your credence function is a of c ’s epistemic value. • Tiis is a general decision-theoretic norm: epistemic acts are choiceworthy to the degree that they maximize expected value. 16 • Use ‘ V c ( c ∗ ) ’ to represent how valuable the credence function c ∗ is, according to the credence function c . probability, p , then, for all c , V p ( c ) should be p ’s expectation = ∑ ! V p ( c ) V ( c , w ) · p ( w ) w ∈ W
Evaluating credence functions • Leitgeb & Pettigrew: if your credence function is a of c ’s epistemic value. • Tiis is a general decision-theoretic norm: epistemic acts are choiceworthy to the degree that they maximize expected value. 16 • Use ‘ V c ( c ∗ ) ’ to represent how valuable the credence function c ∗ is, according to the credence function c . probability, p , then, for all c , V p ( c ) should be p ’s expectation = ∑ ! V p ( c ) V ( c , w ) · p ( w ) w ∈ W
Evaluating credence functions • Leitgeb & Pettigrew: if your credence function is a of c ’s epistemic value. • Tiis is a general decision-theoretic norm: epistemic acts expected value. 16 • Use ‘ V c ( c ∗ ) ’ to represent how valuable the credence function c ∗ is, according to the credence function c . probability, p , then, for all c , V p ( c ) should be p ’s expectation = ∑ ! V p ( c ) V ( c , w ) · p ( w ) w ∈ W are choiceworthy to the degree that they maximize causal?
Evaluating credence functions • Leitgeb & Pettigrew: if your credence function is a of c ’s epistemic value. • Tiis is a general decision-theoretic norm: epistemic acts are choiceworthy to the degree that they maximize expected value. 16 • Use ‘ V c ( c ∗ ) ’ to represent how valuable the credence function c ∗ is, according to the credence function c . probability, p , then, for all c , V p ( c ) should be p ’s expectation = ∑ ! V p ( c ) V ( c , w ) · p ( w ) w ∈ W
Valuing Accuracy Properly Propriety 17 Tie epistemic value function V is proper ifg, for every probability p and every credence function c � p , V p ( c ) < V p ( p ) • Q is proper , • L is proper , • A is not proper / • E is not proper /
Valuing Accuracy Properly Propriety 17 Tie epistemic value function V is proper ifg, for every probability p and every credence function c � p , V p ( c ) < V p ( p ) • Q is proper , • L is proper , • A is not proper / • E is not proper /
Valuing Accuracy Properly Propriety 17 Tie epistemic value function V is proper ifg, for every probability p and every credence function c � p , V p ( c ) < V p ( p ) • Q is proper , • L is proper , • A is not proper / • E is not proper /
Valuing Accuracy Properly Propriety 17 Tie epistemic value function V is proper ifg, for every probability p and every credence function c � p , V p ( c ) < V p ( p ) • Q is proper , • L is proper , • A is not proper / • E is not proper /
Valuing Accuracy Properly Propriety 17 Tie epistemic value function V is proper ifg, for every probability p and every credence function c � p , V p ( c ) < V p ( p ) • Q is proper , • L is proper , • A is not proper / • E is not proper /
Why Propriety? Epistemic Conservativism P1. For any probability p , there is some evidence you could have that would make it permissible to have p as your credence function. your own, then it is permissible to adopt c as your credence function, even without receiving any evidence. P3. It is impermissible to change your credences without receiving evidence. C1. So, epistemic value must be proper. 18 P2. If another credence function c is at least as valuable as
Why Propriety? Epistemic Conservativism P1. For any probability p , there is some evidence you could have that would make it permissible to have p as your credence function. your own, then it is permissible to adopt c as your credence function, even without receiving any evidence. P3. It is impermissible to change your credences without receiving evidence. C1. So, epistemic value must be proper. 18 P2. If another credence function c is at least as valuable as
Why Propriety? Epistemic Conservativism P1. For any probability p , there is some evidence you could have that would make it permissible to have p as your credence function. your own, then it is permissible to adopt c as your credence function, even without receiving any evidence. P3. It is impermissible to change your credences without receiving evidence. C1. So, epistemic value must be proper. 18 P2. If another credence function c is at least as valuable as
Why Propriety? Epistemic Conservativism P1. For any probability p , there is some evidence you could have that would make it permissible to have p as your credence function. your own, then it is permissible to adopt c as your credence function, even without receiving any evidence. P3. It is impermissible to change your credences without receiving evidence. C1. So, epistemic value must be proper. 18 P2. If another credence function c is at least as valuable as
Why Propriety? Epistemic Conservativism P1. For any probability p , there is some evidence you could have that would make it permissible to have p as your credence function. your own, then it is permissible to adopt c as your credence function, even without receiving any evidence. P3. It is impermissible to change your credences without receiving evidence. C1. So, epistemic value must be proper. 18 P2. If another credence function c is at least as valuable as
Why Propriety? Immodesty P1. For any probability p , there is some evidence you could have that would make it permissible to have p as your credence function. P4. Rationality requires you to think that your own credences are epistemically better than any other credences you could have held instead. C1. So, epistemic value must be proper. 19
Why Propriety? Immodesty P1. For any probability p , there is some evidence you could have that would make it permissible to have p as your credence function. P4. Rationality requires you to think that your own credences are epistemically better than any other credences you could have held instead. C1. So, epistemic value must be proper. 19
Why Propriety? Immodesty P1. For any probability p , there is some evidence you could have that would make it permissible to have p as your credence function. P4. Rationality requires you to think that your own credences are epistemically better than any other credences you could have held instead. C1. So, epistemic value must be proper. 19
Why Propriety? Immodesty P1. For any probability p , there is some evidence you could have that would make it permissible to have p as your credence function. P4. Rationality requires you to think that your own credences are epistemically better than any other credences you could have held instead. C1. So, epistemic value must be proper. 19
Propriety & Probabilism non-probabilistic credence function is accuracy dominated by a probabilistic credence function, and no probabilistic credence function is so dominated. (Predd et al, 2009) • Assuming that accuracy domination is irrational and that 20 • If V is a proper measure of accuracy, then every V is a proper measure of accuracy, Probabilism follows.
Propriety & Probabilism non-probabilistic credence function is accuracy dominated by a probabilistic credence function, and no probabilistic credence function is so dominated. (Predd et al, 2009) • Assuming that accuracy domination is irrational and that 20 • If V is a proper measure of accuracy, then every V is a proper measure of accuracy, Probabilism follows.
Conditionalization & Accuracy
Conditionalization & Accuracy Take 1
Propriety & Conditionalization • Leitgeb & Pettigrew (2010): Upon learning that E , you should be disposed to adopt a new credence function which maximizes your expected epistemic value in all possibilities consistent with E . p E c 21 { } ! ∑ = arg max p ( w ) · V ( c , w ) w ∈ E
Propriety & Conditionalization • Leitgeb & Pettigrew (2010): Upon learning that E , you should be disposed to adopt a new credence function which maximizes your expected epistemic value in all possibilities consistent with E . p E c 21 { } ! ∑ = arg max p ( w ) · V ( c , w ) w ∈ E
Propriety & Conditionalization Tieorem 1 (Generalized from Leitgeb & Pettigrew, 2010) c arg max p E and any proposition E , 22 c { } ! ∑ = arg max p ( w ) · V ( c , w ) w ∈ E If V is a proper accuracy measure, then, for any probability p { } ∑ p ( w ) · V ( c , w ) = p ( − | E ) w ∈ E ! If V is a proper accuracy measure, then p E = p ( − | E ) .
Propriety & Conditionalization Tieorem 1 (Generalized from Leitgeb & Pettigrew, 2010) c arg max p E and any proposition E , 22 c { } ! ∑ = arg max p ( w ) · V ( c , w ) w ∈ E If V is a proper accuracy measure, then, for any probability p { } ∑ p ( w ) · V ( c , w ) = p ( − | E ) w ∈ E ! If V is a proper accuracy measure, then p E = p ( − | E ) .
Propriety & Conditionalization Tieorem 1 (Generalized from Leitgeb & Pettigrew, 2010) c arg max p E and any proposition E , 22 c { } ! ∑ = arg max p ( w ) · V ( c , w ) w ∈ E If V is a proper accuracy measure, then, for any probability p { } ∑ p ( w ) · V ( c , w ) = p ( − | E ) w ∈ E ! If V is a proper accuracy measure, then p E = p ( − | E ) .
Why only E -possibilities? p E c • We should attempt to maximize expected epistemic value, but this is not an expectation; why should it be maximized? 23 { } ! ∑ = arg max p ( w ) · V ( c , w ) w ∈ E
Why only E -possibilities? p E c • We should attempt to maximize expected epistemic value, but this is not an expectation; why should it be maximized? 23 { } ! ∑ = arg max p ( w ) · V ( c , w ) w ∈ E
Why only E -possibilities? p E value in the remaining worlds. to pick a posterior which maximizes expected epistemic • Stage 2: use your prior (no longer probabilistic) credences • Stage 1: upon learning E , you eliminate worlds • A 2-stage theory of rational learning: 24 c { } ! ∑ = arg max p ( w ) · V ( c , w ) w ∈ E incompatible with E from W ;
Why only E -possibilities? p E value in the remaining worlds. to pick a posterior which maximizes expected epistemic • Stage 2: use your prior (no longer probabilistic) credences • Stage 1: upon learning E , you eliminate worlds • A 2-stage theory of rational learning: 24 c { } ! ∑ = arg max p ( w ) · V ( c , w ) w ∈ E incompatible with E from W ;
Why only E -possibilities? p E value in the remaining worlds. to pick a posterior which maximizes expected epistemic • Stage 2: use your prior (no longer probabilistic) credences • Stage 1: upon learning E , you eliminate worlds • A 2-stage theory of rational learning: 24 c { } ! ∑ = arg max p ( w ) · V ( c , w ) w ∈ E incompatible with E from W ;
Why only E -possibilities? p E value in the remaining worlds. to pick a posterior which maximizes expected epistemic • Stage 2: use your prior (no longer probabilistic) credences • Stage 1: upon learning E , you eliminate worlds • A 2-stage theory of rational learning: 24 c { } ! ∑ = arg max p ( w ) · V ( c , w ) w ∈ E incompatible with E from W ;
Why only E -possibilities? p E value in the remaining worlds. to pick a posterior which maximizes expected epistemic • Stage 2: use your prior (no longer probabilistic) credences • A 2-stage theory of rational learning: 24 c { } ! ∑ = arg max p ( w ) · V ( c , w ) w ∈ E • Stage 1: upon learning E , you eliminate worlds incompatible with E from W ;
Accuracy-fjrst? • Why eliminate worlds at stage 1? • Because they are incompatible with your evidence. • Tiis answer relies upon a norm like “do not treat a world as epistemically possible if it is incompatible with your evidence” • Tiis is a distinctively evidential norm • It has not been justifjed in terms of the rational pursuit of accuracy alone. • Moreover, no such justifjcation is possible, if we assume that accuracy is properly measured. 25
Accuracy-fjrst? • Why eliminate worlds at stage 1? • Because they are incompatible with your evidence. • Tiis answer relies upon a norm like “do not treat a world as epistemically possible if it is incompatible with your evidence” • Tiis is a distinctively evidential norm • It has not been justifjed in terms of the rational pursuit of accuracy alone. • Moreover, no such justifjcation is possible, if we assume that accuracy is properly measured. 25
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