Phil 735: Homework 4

Due on Fri Dec 1.

All of these problems are adapted from Titelbaum.

Your credences are towards a language with only two atomic propositions B and C. These are logically independent, and suppose at t0 you assign equal credence to each of the four possible assignments of truth-value to each of them. Between t0 and t1 you perform a Jeffrey Conditionalization on the B:¬B partition, that has the end result of setting your posterior credence New1(B) = 2/3. Between t1 and t2, you perform a Jeffrey Conditionalization on the C:¬C partition, that has the end result of setting your posterior credence New2(C) = 3/4.

Calculate your full distributions New1(.) and New2(.).
Does your credence in B change between t1 and t2? Does your credence in C change between t0 and t1?
By talking about probababilistic (in)dependence at the different times mentioned, explain the changes or lack of changes you observed in (b).
Starting again with your distribution at t0, do first a Jeffrey Conditionalization the C:¬C partition, that has the same C:¬C Bayes Factor as when you moved from New1(.) to New2(.). What is the credence distribution you end up with in this case?
Is your final credence in C after the update described in (d) the same or different from your credence New2(C) at t2? Does this always happen if you do an update with the same Bayes Factor but starting with a different distribution? If so, why? If not, why did it happen in this case?

You’re guarding Stephen Curry in a basketball game and he is about to attempt a three-point shot. You have to decide whether to foul him to stop the shot.

If you don’t foul him, he will attempt the shot. In recent seasons, he’s made 44.3% of his three-point shot attempts. What is the expected number of points he will score if you decide not to foul him?
If you do decide to foul him, you are sure he won’t be able to attempt his three-point shot. However, he’ll then get to make three free throws, each worth one point if he makes it. In recent seasons, he’s made 91.4% of his free-throw attempt. Assuming that the result of each free-throw attempt is probabilistically independent of the others, what is the expected number of points Curry will score if you foul him?
Given your answers to (a) and (b), should you foul Curry when he attempts a three-point shot?

1. Suppose an agent is indifferent between two gambles with the following utility outcomes:
```
         | P | ¬P
---------|---|---
 Gamble1 | x | y
 Gamble2 | y | x
```
  where x and y are utilties where x ≠ y. Assuming this agent maximizes utilities in Savage’s way, what can you infer about the agent’s credence in P?
2. Suppose the agent is also indifferent between these two gambles:
```
         | Q | ¬Q
---------|---|---
 Gamble3 | d | -d
 Gamble4 | m | m
```
  where the agent’s credence in Q is 1/2. What can you infer about m?
3. Finally, suppose the agent is indifferent between these two gambles:
```
         | R   | ¬R
---------|-----|---
 Gamble5 | 100 | 20
 Gamble6 | 80  | 80
```
What can you infer about the agent’s credence in R?
Ash’s credences don’t satisfy the probability axioms. For some mutually exclusive P and Q, he assigns cr(P) = cr(Q) = 0.3, but cr(P ∨ Q) = 0.8. Construct a Dutch Book against Ash. Describe what bets compose your package, why Ash should find each one acceptable, and why your package of bets guarantee Ash a loss in every possible world.
Dot’s credence distribution includes these particular values for propositions H and G (don’t suppose that these propositions are mutually exclusive):
```
cred(H ∧ G) = 0.5
cred(H) = 0.1
cred(G) = 0.5
cred(H ∨ G) = 0.8
```
1. Show that Dot’s credences violate the probability axioms.
2. Construct a Dutch Book against Dot. Since Dot’s credences don’t satisfy the probability axioms, you cannot assume anything about her credences other than what’s given here. However, your package of bets need not take advantage of all four of the listed credences. Lay out the package of bets in the same way as in Problem 45.
3. Construct a Czech Book in Dot’s favor. Lay out the package of bets in the same way as before, explaining why the guarantee Dot a profit in every possible world.
Suppose you measure inaccuracy using the Brier Score.
1. An agent Alex assigns credences to two atomic propositions X and Y, and these credences are between 0 and 1 (inclusive). Draw a box diagram, like those in Titelbaum’s Figures 10.2, 10.3, and 10.4, illustrating possible distrubutions Alex may have over these two propositions. Then shade in the part of the box where cr(Y) ≤ cr(X).
2. As in (a), but now suppose that Y logically entails X. Use your diagram from (a) to show that if Alex’s credences violate the rule that cr(Y) ≤ cr(X), then there will exist a different distribution that is more accurate than Alex’s in every logically possible world. (Hint: when Y entails X, one of the three corners of the box diagram in (a) is no longer logically possible.)
3. Consider Dot’s credence in Problem 46, focusing just on the first two credences listed there. Construct an alternate credence function over H and G that is more accurate than Dot’s in every logically possible world. (Hint: let H ∧ G and H play the roles of Y and X in result (b) of this Problem.) To show that you’ve succeeded, calculate Dot’s inaccuracy and your alternative’s inaccuracy in each of the three available possible worlds.