Constraints on credences in two not mutually exclusive propositions: - - PowerPoint PPT Presentation

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:

• w∈W

b({w}) · I(w, b′) 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.

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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, b′(A) =

m

• i=1

b′(Ei) · b(A|Ei). Full conditionalisation is a special case of Jeffrey conditionalisation (JC). JC will also never make your extreme credences non-extreme. It is also

• rder-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 = {w1, w2, w3, w4}, A = {w1, w2} and B = {w2, w3} 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 = {w1, w2, w3, w4}, A = {w1, w2} and B = {w2, w3} 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 = {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′ 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)

2

as K. The belief update function which minimizes global expected inaccuracy in the sense of Leitgeb & Pettigrew (2010) is fully determined by the two constraints and the following condition: b′(A ∩ B) =      if K < 0 min {p, q} if min {p, q} < K K

• therwise.

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

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”; ¬RC RC R¬C ¬R¬C b1 .2 .4 .1 .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): ¬RC RC R¬C ¬R¬C b

′

1

.1 .5 .2 .2

Leszek Wroński (JU, Poland) 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, ¬RCB ¬RC¬B RC¬B R¬C¬B ¬R¬C¬B b2 .1 .1 .4 .1 .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: ¬RCB ¬RC¬B RC¬B R¬C¬B ¬R¬C¬B b

′

2

.043 .043 .514 .186 .214 Let us bring together the initial and updated credences for the two agents: ¬RC RC R¬C ¬R¬C b1 .2 .4 .1 .3 b

′

1

.1 .5 .2 .2 ¬RCB ¬RC¬B RC¬B R¬C¬B ¬R¬C¬B b2 .1 .1 .4 .1 .3 b

′

2

.043 .043 .514 .186 .214

Leszek Wroński (JU, Poland) Problems with minimizing GExpInacc 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

• r 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 b

′

2(φ) < b

′

2(ψ) while b

′

1(φ) > b

′

1(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: ¬RC RC R¬C ¬R¬C b1 .2 .4 .1 .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: ¬RC RC R¬C ¬R¬C b

′′

1

.3 .4 .0 .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?

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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 = ∅. Suppose that the evidence received by A between times t and t′ entails (only) her credences b′(A) and b′(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) = 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 = ∅. Suppose that the evidence received by A between times t and t′ entails (only) the following constraints on her belief function b′ at t′: b′(A) = p and b′(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) p − b′(A ∩ B) + b(¬A ∩ B) q − b′(A ∩ B) = b(A ∩ B) 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).

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