Generating quasi-strictly proper scoring rules

A scoring rule s assigns to a credence c on a space Ω a score s(c) measuring the inaccuracy of c. The score is itself a function from Ω to [−∞,M] (for some fixed M), so that s(c)(ω) measures the inaccuracy of c if the truth of the matter is that we are at ω ∈ Ω. A scoring rule is proper provided that E_ps(p) ≤ E_ps(c) for any probability p and any other credence c: i.e., provided that the p-expected value of the score of p is at least as good as (no more inaccurate than) as the p-expected value of any other score. It is strictly proper if the inequality is strict. It is quasi-strictly proper if the inequality is strict whenever p is a probability and c is not.

Here’s a fun fact about scoring rules.

Theorem: Let s be any continuous bounded proper scoring rule that is defined only on the probabilities. Then s can be extended to a continuous bounded quasi-strictly proper scoring rule defined on all credences.

This result works in the finite-dimensional case with standard Euclidean topologies on the space of credences (considered as elements of [0,1]^PΩ) and on the scores (considered as values in [−∞,M]^Ω). But it also works in countably-infinite-dimensional contexts in the right topologies (ℓ^∞(PΩ) on the side of the credences and the product topology on the side of the scores), regardless of whether by “probabilities” we mean finitely or countably additive ones.

The proof uses three steps.

First, show that the probabilities are a closed subset of the space of credences.

Second, apply the Dugundji extension theorem to extend the score s from the probabilities to all credences while maintaining continuity and ensuring that the range of the extension is a subset of the convex hull of the range of the original score. Let s₀ be the extended score. The convex hull condition and the propriety of the original score on the probabilities implies that s₀ is proper, though not necessarily quasi-strictly so.

Third, let s₁(c) = d(c,P) + s₀(c), where Q is the set of credences that are probabilities and d(c,Q) is the distance from c to Q in the ℓ^∞(PΩ) norm. This equals s₀(c) and hence s(c) for a credence c.

I don’t know if there is much of a philosophical upshot of this. Maybe a kind of interesting upshot is that it illustrates that quasi-strict propriety is easy to generate?

Theism and emotional attitudes to adversity

Here are two three possible emotional attitudes towards great adversity:

1. Judaeo-Christian: hope

2. Stoic: calm

3. Russellian: anger/despair.

Now consider this argument:

1. The appropriate attitude towards great adversity is Judaeo-Christian or Stoic.

2. If naturalism is true, the appropriate attitude towards great adversity is Russellian.

3. So, naturalism is false.

The reason for (1) is the obvious attractiveness of the hopeful-to-calm part of the emotional spectrum as a way of dealing with diversity.

The reason for (2) is that emotions should fit with reality. But as Russell argues, a naturalist reality does not care about us: we came from the nebula and we will go back to the nebula, and the darkness of our life makes Greek tragedy the supreme form of human art. The most we can do shake our fist at the injustice of it all.

Rowe-style inductive arguments from evil

The examples, like Rowe’s, of evils in the inductive argument from evil are chosen to make them have a certain epistemic feature F. And the claim is that P(E has F | God) < P(E has F | no God), with the background information containing the occurrence of E (i.e., the evidence isn’t that the evil has occurred, but that the evil has F). Exactly what F is differs from paper to paper, but roughly the feature is that after investigation we don’t have a plausible candidate theodicy.

But not every evil has F. If every evil had F, then the examples in the literature wouldn’t run as heavily as they do to lethal harm to children and animals. The examples used by the atheological arguers are chosen to be particularly compelling and what makes them compelling is that they have F—nobody runs an inductive argument from evil based on robber barons getting stomachaches from too much caviar, because such evils do not have F.

So there are evils that don’t have F. And then P(E has ∼F | God) > P(E has ∼F | no God) by Bayesianism. So checking whether an evil has F sometimes yields an argument against the existence of God (namely when the evil does have F) and sometimes yields an argument for the existence of God (when the evil doesn’t have F).

And we do not know (as far as I know, Tooley is the only one to have made a serious attempt to figure it out, and his account fails for technical reasons) what the result is once the evidence is consolidated.

Arguing for divine simplicity

I want to defend this argument:

1. If God is not simple, then some of God’s parts are creatures.

2. If some of the parts of x are creatures, then x is partly a creature.

3. God is not even partly a creature.

4. So, God is simple.

I think (2) is very plausible. Premise (3) follows from the transcedence of God.

That leaves premise (1) to argue for. Here is one argument:

5. If God is not simple, then God has a part that is not God.

6. Anything that is not God is a creature.

7. So, if God is not simple, then God has a part that is a creature.

Premise (5) is true by definition of “simple”. Premise (6) follows from the doctrine of creation: God creates everything other than God.

But perhaps one doesn’t believe the full doctrine of creation, but only thinks that contingent things are created. I think we can still argue as follows:

8. If God is not simple, then God has contingent parts that are not God.

9. Anything contingent that is not God is a creature.

10. So, if God is not simple, then God has a part that is not God.

Why think (8) is true? Well, let’s think about the motivations for denying divine simplicity. The best reasons to deny divine simplicity are considerations about God’s contingent intentions or God’s contingent knowledge, and the idea that these have to constitute proper parts of God. But that yields contingent parts of God.

Now, what if one rejects even the weaker doctrine of creation in (9)? Then I can argue as follows:

11. If God is not simple, then God’s contingent thoughts are proper parts of God.

12. God is contingently the cause of each of his contingent thoughts.

13. Anything that God is contingently the cause of is a creature of God.

14. So, if God is not simple, then God has a part that is not God.

Again, the idea behind (11) is that it flows from the best motivations for denying divine simplicity.

What do we learn about God's existence from the clear failures to find justification for evils?

Suppose we know of some evil e₁, and careful further investigation clearly fails to turn up justification that God would have to allow that evil. Let F₁ be that clear failure.

Now, those who defend the inductive argument from evil claim that F₁ is evidence against the existence of God. But now presumably sometimes after investigation of some other evil, say e₂, we do not clearly fail to turn up justification for an evil (i.e., we either find a justification or it is unclear whether we have done so). Presumably, in that case the clear failure F₂ to turn up a justification would have been evidence against the existence of God. But now it is a basic Bayesian result that the absence of F₂ is evidence for the existence of God.

In practice, the results of investigation vary from evil to evil. Sometimes we clearly fail to find a justification and sometimes we don’t clearly fail. When we clearly fail, that is Bayesian evidence against theism. When we don’t clearly fail, that is Bayesian evidence for theism. What does the totality of the evidence say?

We don’t know. We just don’t have the numbers. We don’t have good numbers as to how many investigations resulted in a clear failure to turn up a justification and how many have not. Nor do we have any really good estimates of the crucially important conditional probabilities for any particular evil, namely how likely we are to clearly fail to find a justification assuming God exists and how likely we are to clearly fail to find a justification assuming God doesn’t exist.

The answer to the question in the title of the post, then, is: Not much.

One might object that I stacked the deck by talking of clear failures to find a justification. Perhaps I should instead have talked of failures to clearly find a justification. Failures will then be much more common, and there will be very few cases of clear finding of a justification. However, at the same time, our expectation that we be able to clearly find a justification given God’s existence will not be that strong. For these are controversial matters, where clarity is hard to have, and we would expect them to be such. Indeed, it is not clear that assuming the existence of God we would ever expect to clearly have found a justification, because there could always be further evil consequences down the road that we did not take into account.

Learning of secret wickedness

When people learn that some apparently really decent person was a hypocrite who secretly practiced execrable vices, that tends to shake people’s faith in God. We can explain this by noting that the case provides one with a vivid case of moral evil, which provides evidence against the existence of God.

But we already knew that there was a lot of moral evil out there. So the effect on belief in God should not be very significant. And it is worth noting that learning of cases like the above can actually help with the problem of evil. In our time, we are very hesitant to use the punishment theodicy for evils that happen to people. But learning that there are more people with terrible hidden vices than we thought increases the probability that any particular evil befalling an apparently decent adult might actually be a well-deserved punishment.

Of course, a punishment theodicy will only go so far. It doesn’t apply to animals or small children. And the Book of Job teaches that it doesn’t apply to all cases of adults either. But realizing the dark truth that people who appear to be exemplars of virtue can be quite wicked should open us to the possibility that the punishment theodicy applies to a lot more cases than we thought.

Of course, the more cases we have to which the punishment theodicy applies, the more moral evils we have that need a theodicy as well. But free will considerations can help a lot with moral evils.

So it may well be that learning of someone’s secret evils is a wash in terms of the evidential import of the evil for God’s existence.

Towards a great chain of being

Here is one way to generate a great chain of agency: y is a greater agent than x if for every major type of good that x pursues, y pursues it, too, but not vice versa.

Take for instance the cat and the human. The cat pursues major types of good such as nutrition, reproduction, play, comfort, health, life, truth, and (to a limited degree) social interaction. The human pursues all of these, but additionally pursues virtue, beauty, and union with God. Thus the human is a greater agent than the cat.

Is it the case that humans are at the top of the great chain of agency on earth?

This is a difficult question to answer for at least two reasons. The first reason is that it is difficult to identify the relevant level of generality in my weaselly phrase “major type of good”. The oak pursues photosynthetic nutrition, the dung beetle does its thing, while we pursue other forms of nutrition. Do the three count as pursuing different “major types” of good? I want to say that all these are one major type of good, but I don’t know how to characterize it. Maybe we can say something like this: Good itself is not a genus but there are highest genera of good, and by “major type” we mean these highest genera. (I am not completely sure that all the examples in my second paragraph are of highest genera.)

The second reason the question is difficult is this. The cat is unable to grasp virtue as a type of good. A cat who had a bit more scientific skill might be able to see an instrumental value in the human virtue—could see the ways that it helps members of communities gain cat-intelligible goods like nutrition, reproduction, health, life, etc. But the cat wouldn’t see the distinctive way virtue in itself is good. Indeed, it is not clear that the cat would be able to figure out that virtue is itself a major type of good, no matter how much scientific skill the cat had. Similarly, it is very plausible that there are major types of good that are beyond human knowledge. If we saw beings pursuing those types of good, we would likely notice various instrumental benefits of the pursuit—for the pursuit of various kinds of good seems interwoven in the kinds of evolved beings we find on earth (pursuing one good often helps with getting others)—but we just wouldn’t see the behavior as the pursuit of a major type of good. Like the cat scientist observing our pursuit of virtue, we would reduce the good being pursued to the goods intelligible to us.

Thus, if octopi pursue goods beyond our ken, we wouldn’t know it unless we could talk to octopi and they told us that what they were pursuing in some behavior was a major type of good other than the ones we grasp—though of course, we would still be unable to grasp what was good in it. And as it happens the only beings on earth we can talk to are humans.

All that said, it still seems a reasonable hypothesis that any major type of good that is pursued by non-human organisms on earth are pursued by us.

Some possible progress on continuous scoring rules and dominance in an infinite case

On finite sample spaces, we have the Pettigrew-Nielsen-Pruss domination theorem for strictly proper scoring rules that are continuous when restricted to the probabilities that shows that the score of any non-probability is dominated by the score of a probability. Last year, I showed that for a reasonable sense of “continuous”, this is not true on countably infinite sample spaces (when we take probabilities to be countably additive; for if we take probabilities to be finitely additive, there are no strictly proper scoring rules).

In the comments, Ian then suggested that we want our scoring rule to be continuous on all credences, not just the probabilities.

Here are two preliminary responses, though not all the details of the proof of the second have yet been checked, so I could just be wrong.

First, what happens seems to depend on the topology on the space of credences. Credences can be thought of as functions from PΩ to [0,1]. One possibility is to take the space of credences to get the product topology on [0,1]^PΩ. In that case, there is no continuous strictly proper (or even quasi-strictly proper) scoring rule. This follows from the uncountability of PΩ which shows that any countable intersection of neighborhoods of a probability function will contain infinitely many non-probability functions, so that any continuous score will have the property that for every probability there is a non-probability that gets the same score.

But, second, another reasonable topology on [0,1]^PΩ is the ℓ^∞(PΩ) topology. This topology is easily seen to be equivalent on the probabilities to the ℓ¹(Ω) topology (where a probability p on PΩ corresponds to a function p^* ∈ ℓ¹(Ω) defined by p^*(x) = p({x})). The example in my earlier post was a score s that was equal to the spherical score on all the probabilities and s(c)(n) = 1/2(n+1) for any non-probability credence, where we identify Ω with the natural numbers.

Let Q be the space of probability functions on PΩ. Let d(c) = inf_p ∈ Q∥c − p∥_∞ be the distance from c to Q. We can prove that d(c) = 0 iff c is a probability, and d is continuous in our topology. Let ϕ(c) = 0 if d(c) ≥ 1/4 and ϕ(c) = 4d(c) if d(c) < 1/4. This will be a continuous function. Now define s(c)(n) = ϕ(c)/2(n+1) + (1−ϕ(c))c^*(n)/∥c^*∥₂, where c^*(n) = c({n}), and where the second summand is deemed to be zero if ϕ(c) = 1 (regardless of the denominator). I haven’t checked all the details yet, but this s looks continuous to me in the relevant norm, but the domination result is false for any non-probability c. The important point is that the function c ↦ ∥c^*∥₂ is continuous and non-zero for c such that d(c) < 1/4, and that’s one of the points I might yet have an error in.

Transworld depravity is false

Plantinga’s transworld depravity thesis holds that in every world that God is contingently capable of actualizing (i.e., every “feasible” world), either there is no significant freedom or there is at least one free wrong choice. I will argue that transworld depravity is in fact false, assuming Molinism.

But consider a possible situation A where the first significantly free choice runs as follows. Eve has a choice whether to eat a delicious apple or not, while knowing that God has forbidden her from eating the apple. Eve comes into the choice with a pretty decent character. In particular, she is so constructed that she is unable to take God’s prohibitions to be anything but reasons against an action and God’s commands to be anything but reasons for an action. Nonetheless, she is free: she can choose to eat the apple on account of its deliciousness, despite God’s prohibiting it.

By Molinism, if enough detail is built into the situation, either:

1. in A, Eve would eat the apple, or

2. in A, Eve would not eat the apple.

If (2) is true, then transworld depravity is false, because God could simply take away freedom after Eve’s first choice, and so we have a feasible world where there is exactly one significantly free choice, and it’s right.

Suppose then (1) is true. Now imagine a situation A^* where just before Eve is deliberating whether to eat the apple, God announces that the prohibition on eating the apple is now changed into a command to eat the apple. If in A, Eve would eat the apple on account of its deliciousness despite its being forbidden, she would a fortiori eat the apple if God were to command her to do so. Thus:

3. in A^*, Eve would eat the apple.

But then transworld depravity is false, because again God could take freedom away after Eve’s first choice.

The argument as it stands does not show that transworld depravity is necessarily false. I try to do that here with a similar but perhaps less compelling argument.

Consequentialism and probability

Classic utilitarianism holds that the right thing to do is what actually maximizes utility. But:

1. If the best science says that drug A is better for the patient than drug B, then a doctor does the right thing by prescribing drug A, even if due to unknowable idiosyncracies of the patient, drug B is actually better for the patient.

2. Unless generalized Molinism is true, in indeterministic situations there is often no fact of the matter of what would really have happened had you acted otherwise than you did.

3. In typical cases what maximizes utility is saying what is true, but the right thing to do is to say what one actually thinks, even if that is not the truth.

These suggest that perhaps the right thing to do is the one that is more likely to maximize utility. But that’s mistaken, too. In the following case getting coffee from the machine is more likely to maximize utility.

4. You know that one of the three coffee machines in the breakroom has been wired to a bomb by a terrorist, but don’t know which one, and you get your morning coffee fix by using one of the three machines at random.

Clearly that is the wrong thing to do, even though there is a 2/3 probability that this coffee machine is just fine and utility is maximized (we suppose) by your drinking coffee.

This, in turn, suggests that the right thing to do is what has the highest expected utility.

But this, too, has a counterexample:

5. The inquisitor tortures heretics while confident that this maximizes their and others’ chance of getting into heaven.

Whatever we may wish to say about the inquisitor’s culpability, it is clear that he is not doing the right thing.

Perhaps, though, we can say that the inquisitor’s credences are irrational given his evidence, and the expected utilities in determining what is right and wrong need to be calculated according to the credences of the ideal agent who has the same evidence.

This also doesn’t work. First, it could be that a particular inquisitor’s evidence does yield the credences that they actually have—perhaps they have formed their relevant beliefs on the basis of the most reliable testimony they could find, and they were just really epistemically unlucky. Second, suppose that you know that all the coffee machines with serial numbers whose last digit is the same as the quadrilionth digit of π have been rigged to explode. You’ve looked at the coffee machine’s serial number’s last digit, but of course you have no idea what the quadrilionth digit of π is. In fact, the two digits are different. You did the wrong thing by using the coffee machine, even though the ideal agent’s expected utilities given your evidence would say that you did the right thing—for the ideal agent would know a priori what the quadrilionth digit of π is.

So it seems that there really isn’t a good thing for the consequentialist to say about this stuff.

The classic consequentialist might try to dig in their heels and distinguish the right from the praiseworthy, and the wrong from the blameworthy. Perhaps maximizing expected utility is praiseworthy, but is right if and only if it actually maximizes utility. This this still has problems with (2), and it still gets the inquisitor wrong, because it implies that the inquisitor is praiseworthy, which is also absurd.

The more I think about it, the more I think that if I were a consequentialist I might want to bite the bullet on the inquisitor cases and say that either the inquisitor is acting rightly or is praiseworthy. But as the non-consequentialist that I am, I think this is a horrible conclusion.

Defining horrendous evils

Marilyn Adams famously defines horrendous evils as follows:

1. Evils the participation in which (that is, the doing or suffering of which) constitutes prima facie reason to doubt whether the participant’s life could (given their inclusion in it) be a great good to him/her on the whole.

This epistemically-based definition doesn’t really capture the relevant category because of the background-dependence of reasons.

Note that the quotation above is ambiguous as to who is to have the prima facie reason: the participant or the observer. If the observer, then whether an evil is horrendous is observer-dependent, which seems quite mistaken.

If the participant, then we still have a problem (and in fact the following problems apply, with different wording, in the observer-dependent case as well). For suppose that I have fully internalized the absurd view that suffering is always good. Then no matter what the suffering is, it does not give me any prima facie reason to doubt that my life is a great good to me on the whole, and so I can never suffer a horrendous evil—yet that seems mistaken. Or, on the contrary, suppose I have internalized the nearly as absurd view that the only good life is a life without any suffering. Then any suffering gives me prima facie reason to doubt whether the my life could be a great good to me on the whole, and hence a mosquito bite is a horrendous evil—which again seems mistaken.

What if we de-epistemicize the definition, by saying something like this?

2. Evils the participation in which make the participant’s life not be a great good to him/her on the whole.

But now suppose that apart from one mosquito bite, Alice’s life is just about the “great good” line, and the mosquito bite brings the life below that line. Then by (2), the mosquito bite is a horrendous evil—and that seems mistaken.

We could try for something like this:

3. Evils such that it is metaphysically impossible for the participant’s life to be a great good to him/her on the whole.

But if we did that, then Adams’ other commitments would force her to deny that there are any horrendous evils (since given her picture of God’s love and power, God ensures everyone’s life is a great good to them). That, I guess, would be good news. But that’s not Adams’ view.

I don’t have an alternative, besides the unrigorous:

4. Really bad evils.

Respecting vs. not violating free will

God could pretty much guarantee that you freely choose to love him. Here’s how. God puts you in a situation where you have a non-infinitesimal probability of freely choosing to love him (this may require grace, etc.). Such situations clearly exist, since a significant number of people have freely chosen to love God. If you don’t freely choose to love him, then God resets your memory and character to how they were before your choice, and puts you in the same situation again. No matter how small the probability of freely choosing to love God, as long as it’s not infinitesimal, the probability that eventually you would freely choose to love God is one, or at least within an infinitesimal of one.

I think this scenario illustrates something interesting: There is a difference between (a) not violating someone’s freedom and (b) respecting someone’s freedom. If God engaged in the above course of action, he wouldn’t be violating our freedom, but he also wouldn’t be respecting it.