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 � W , P ( W ) , b � .) 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 b ′ given a belief function b : � b ( { w } ) · I ( w , b ′ ) 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 b ′ look like? The traditional Bayesian answer: conditionalise. That is, for any A , b ′ ( A ) = b ( A | E ) . This way of updating is sometimes called full conditionalisation . It only makes sense if b ( E ) � = 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 E 1 , · · · , E m , which form a partition of W ? The traditional answer: use Jeffrey Conditionalisation . That is, for any A , m � b ′ ( A ) = b ′ ( E i ) · b ( A | E i ) . 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 � = ∅ ? 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 � = ∅ . Let us tackle the simplest non-trivial—four-world—setup first. Assume then that W = { w 1 , w 2 , w 3 , w 4 } , A = { w 1 , w 2 } and B = { w 2 , w 3 } and you start with a belief function b ; you learn your new credences b ′ ( A ) = p and b ′ ( B ) = q . Notice that the task reduces to figuring out b ′ ( 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 � = ∅ . Let us tackle the simplest non-trivial—four-world—setup first. Assume then that W = { w 1 , w 2 , w 3 , w 4 } , A = { w 1 , w 2 } and B = { w 2 , w 3 } and you start with a belief function b ; you learn your new credences b ′ ( A ) = p and b ′ ( B ) = q . Notice that the task reduces to figuring out b ′ ( 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 = { w 1 , w 2 , w 3 , w 4 } be the set of epistemically possible worlds for A . Let A be the proposition { w 1 , w 2 } ; let B be { w 2 , w 3 } . Suppose that between times t and t ′ the agent learns (only) the following constraints on her belief function b ′ at t ′ : b ′ ( A ) = p and b ′ ( B ) = q . Let us label the number b ( A ∩ B ) + p − b ( A )+ q − b ( B ) as K . The belief update function which 2 minimizes global expected inaccuracy in the sense of Leitgeb & Pettigrew (2010) is fully determined by the two constraints and the following condition: 0 if K < 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 b ′ ( A ∩ B ) using the above formula; this, together with the two constraints, gives us the credences at time t ′ 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 b ′ . 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 b ′ ( A ∩ B ) using the above formula; this, together with the two constraints, gives us the credences at time t ′ 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 b ′ . 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
Recommend
More recommend
