Buchak's review of my Epistemic Risk and the Demands of Rationality

May we all be blessed with interlocutors as brilliant and interesting and constructive as Lara Buchak! Her review essay on my Epistemic Risk and the Demands of Rationality has just appeared in Mind. In it, she gives a better survey of the book’s thesis than I think I could manage, and then goes on to raise a number of issues with it. I’ll try to say something about one of her points in this post, and I’ll pick up on another in a later post.

In this post, I spell out one of Buchak’s criticisms of my argument as it appeared in the book, agree with it, and then sketch a different argument that has occurred to me over the past couple of years, too late for inclusion in the book—esprit de l’escalier is alive and well in philosophy.

As Buchak notes, the book is fundamentally an attempt to spell out with formal precision one of the central claims of William James’ The Will to Believe. It is worth quoting the famous passage in full:

There are two ways of looking at our duty in the matter of opinion,—ways entirely different, and yet ways about whose difference the theory of knowledge seems hitherto to have shown very little concern. We must know the truth; and we must avoid error,—these are our first and great commandments as would-be knowers; but they are not two ways of stating an identical commandment, they are two separable laws. […]

Believe truth! Shun error!—these, we see, are two materially different laws; and by choosing between them we may end by coloring differently our whole intellectual life. We may regard the chase for truth as paramount, and the avoidance of error as secondary; or we may, on the other hand, treat the avoidance of error as more imperative, and let truth take its chance. […] We must remember that these feelings of our duty about either truth or error are in any case only expressions of our passional life.

The first thing to note about this passage is that James proposes a teleological approach to epistemology: beliefs have goals and we evaluate them by how reasonable they are, or how successful they are, as means to achieving those goals.

The second thing to note is that the goals are not pragmatic. They are: (i) believe the true; (ii) do not believe the false. And these are purely epistemic goals. Of course, it’s reasonable to think that our beliefs better guide our actions the better they achieve these goals, and so pursuing the epistemic goals is a good means to achieving the pragmatic goals. But the goals specified are fundamentally epistemic ones. So James has a reasonable claim to be the founder of epistemic utility theory of the sort that has flourished in the past twenty-five years (as I describe here, here, and give an overview here).

The third thing to note is that these goals are in tension. I am guaranteed to fully achieve the first—Believe truth!—by believing everything; I am guaranteed to fully achieve the second—Shun error!—by believing nothing. Every time I try to achieve the first by forming a belief, I risk failing to achieve the second; every time I ensure I’ll achieve the second, I preclude fully achieving the first. James’ point is that our beliefs have both goals; and, since they are in tension, we must weigh them against each other. However, he thinks, there’s no single right way to set these weights. How we do it is only an expression of our “passional life”. And, we presume, rationality does not demand that our passional life be one way rather than another in this matter.

In a context in which our evidence determines unique evidential probabilities for each proposition we entertain, and in which we want to use these evidential probabilities to set our beliefs, Thomas Kelly points out that we can formalise the Jamesian thesis as follows: a belief has some positive epistemic value—let’s say R—if it is true and a negative epistemic value—let’s say -W— if it is false, while suspending belief has a neutral value of zero either way. We can interpret R as the weight you give to James’ first goal of belief—Believe truth!—and W as the weight you give to the second—Shun error!. So, for instance, perhaps James’ interlocutor, W. K. Clifford, will set W high and R low; while, on the other hand, someone who, in James’ words, regards “the chase for truth as paramount, and the avoidance of error as secondary” will set R high and W low.

Kelly then suggests that, having set the epistemic values for believing and suspending in this way, we choose whether to believe or suspend by maximizing expected epistemic value from the point of view of the evidential probabilities. For instance, if I set R = 1 and W = 2, and the evidential probability of a proposition is 3/4, then the expected epistemic value of suspending is ((3/4) x 0) + ((1/4) x 0) = 0, while the expected epistemic value of believing is ((3/4) x 1) + ((1/4) x (-2) = 1/4, which is greater than 0, and so we’re required to believe the proposition. And in general, the more weight you give to avoiding believing falsehoods and the less weight you give to believing truths, the higher the evidential probability of a proposition must be before believing it has greater expected epistemic value than suspending on it.¹

As Kelly notes, this gives a version of epistemic permissivism. If you set R = 1 and W = 3, while I set R = 1 and W = 1, and our shared evidence dictates an evidential probability of 2/3, then rationality requires you to suspend and me to believe. So we have different rational responses to the same body of evidence.

As I said, Kelly is working in a framework in which: (i) evidential probabilities are available; (ii) we choose by maximizing expected utility by the lights of the evidential probabilities; and (iii) we are choosing whether to believe or suspend. What happens if we are choosing not whether to believe a proposition or suspend judgment on it, but which credence to assign to it? Or even which credences to assign to a whole set of propositions?

To answer that, we need to say how we are to measure the epistemic value of a single credence or an assignment of credences. Fortunately, epistemic utility theory has a standard line on this: you use an immodest epistemic utility function or strictly proper scoring rule. An epistemic utility function is immodest or strictly proper if, for any probabilistic assignment of credences, that assignment uniquely maximizes expected epistemic utility from its own point of view; that is, any alternative assignment of credences has lower expected epistemic utility from the original assignment’s point of view than the original assignment has from its own point of view. But, as Sophie Horowitz points out, if we measure epistemic utility in such a way, then whatever the evidential probabilities are, they will demand that you have credences that exactly match them, since those are the ones that will maximize expected epistemic utility from the point of view of the evidential probabilities. So, in this context, we don’t get epistemic permissivism.

And yet the permissivist should not despair. For we might think about the weight you give to James’ two goals not as specifying your epistemic values at all, but rather as specifying your epistemic risk attitudes, which feed into your decision-making in a different way.

Since the 1950s, decision theory has taught us that there are two ways in which we might incorporate our attitudes to risk into our preferences: first, by including them in our utilities, and then combining those with our credences in a particular way to give a preference ordering over options; second, by including them in the weight given to utilities when we combine them with our credences to give our preference ordering. So suppose I prefer £4 for sure to a gamble that gives £10 with 50% probability and £0 with 50% probability. Then this might be because I set my preferences in line with expected utility, and my utility function is concave in money, so that, for instance, my utility for £0 is 0, for £4 it’s 4, but for £10 it’s 7. Or it might be because my utility function is linear in utility, so that, for instance, my utility for £0 is 0, for £4 it’s 4, and for £10 it’s 10, but I combine my credences with my utilities so that the weight given to the utility of the best-case outcome is less than expected utility theory requires, and the weight given to the utility of the worst-case outcome—theories that allow us to do the latter include John Quiggin’s rank-dependent utility theory, Lara Buchak’s risk-weighted expected utility theory, Chris Bottomley and Timothy Luke Williamson’s linear-weighted utility theory. Or, and this will become important later, it might be a combination of the two: my utility function is concave in money, but not quite concave enough to make the sure thing better in expectation; and my weighting of the best- and worst-case utilities is not quite different enough to make the sure thing better if I have a linear utility function; but in combination my slightly concave utility function and my slightly risk-averse way of weighting best- and worst-case utilities ensures that the sure thing is better.

How does all this play out in the epistemic case? One proposal, which I didn’t explore in the book, sticks with evidential probabilities and strictly proper epistemic utility functions, and uses something like risk-weighted expected utility theory or linear-weighted utility theory to set our credences. And if you think there are such things as evidential probabilities, this would be an interesting approach. I don’t, partly for reasons I spell out here and partly because I think there’s no room for them in a Jamesian approach, and so in the book I explore an alternative approach.

The idea is that our attitudes to epistemic risk only determine the way we set our prior credences; thereafter, having set them in that way, we do everything else by maximizing expected utility, which I take to be the risk-neutral theory of decision-making under uncertainty. We act in order to maximize expected pragmatic utility, and we set our new credences in the light of evidence in order to maximize expected epistemic utility.

In the absence of evidential probabilities, when we pick our priors, we need a decision rule that does not appeal to probabilities. Fortunately, there are plenty around. Perhaps the most well-known is Maximin, which says that we should pick an option whose worst-case utility is highest. As I argued in this paper, which was the starting point for the project that later became Epistemic Risk and the Demands of Rationality, if we use this decision rule, we get the Principle of Indifference, which says that you should divide your credences equally over all the possibilities you conceive.

But the Maximin rule encodes the most extreme possible risk-averse attitudes. And, while it might be permissible to use it, with James, we want to allow that other attitudes to risk are permissible. It’s permissible to be risk-seeking; it’s permissible to be risk-neutral; it’s permissible to be risk-averse but not quite so risk-averse as Maximin requires you to be. In the book, I argue for a particular decision rule that allows this. I call it the Generalized Hurwicz Criterion, not least because it’s a generalization of the Hurwicz Criterion, whose epistemic consequences I explored in this paper. The idea is this: Suppose you entertain just three ways the world might be. Then your attitudes to risk are encoded by three numbers. The first is the weight you give to the best-case utility of an option; the second is the weight you give to its second-best-case (or second-worst-case) utility; the third is the weight you give to the worst-case utility. The weighted sum of the utilities of with these weights applied gives the generalized Hurwicz score of an option, and then options are ordered by their generalized Hurwicz score, and we pick one at the top of the ordering.

I won’t go into the consequences of this decision rule in the epistemic case, other than to say that everything from extreme risk aversion to risk neutrality leads to the Principle of Indifference, but beyond that, as we become more and more risk-inclined, it is no longer demanded that we give equal credence to all possibilities: we are permitted to pick a single possibility and put greater weight on that than on the others, or do that for a set of possibilities, and so on. So, on this view, epistemic permissivism is recovered. Different attitudes to epistemic risk give rise to different priors, which are our credences before we collect any evidence.

So we use the Generalized Hurwicz Criterion, fed with our attitudes to risk, to set our priors. How about our posteriors? How should we respond to evidence? Roughly speaking, I say that, once we’ve picked our priors, the role of our attitudes to epistemic risk is over. Thereafter, we should choose by maximizing expected utility. And then I appeal to an argument by Dmitri Gallow that we should therefore respond to learning a new proposition by conditioning on it just as the Bayesian says we should.

In the book, I argue for this stark split between picking priors by appealing to our attitudes to epistemic risk and picking posteriors in a risk-neutral way by saying that, if we were to use our attitudes to epistemic risk also in the choice of posteriors, then we would be double-counting them:

If you were to use a risk-sensitive decision rule to pick your priors, and then a risk-sensitive decision rule also to pick your posteriors using your priors, you would double count your attitudes to risk…[W]e encode our attitudes to risk entirely in the decision rule we use to pick our priors.

Buchak objects:

It’s true that if our attitude towards risk is encoded entirely somewhere, then we don’t need to encode it elsewhere. But in the practical case, we allow for variation in both the decision rule and the utility function (for example, we allow for a convex risk function and a concave utility function), precisely because neither of these functions entirely exhausts the basis for our risk-preferences. And this is because the risk function and the utility function concern different attitude-types.

I think Buchak is right. I haven’t given a good enough reason to think that we don’t or shouldn’t divide our attitudes to risk between the rule we use to set our priors and the one we use to set our posteriors. But, since writing the book, I think I’ve come upon an argument that might give a better reason.

My view invokes two decision theories: Generalized Hurwicz Criterion to pick priors; Maximize Subjective Expected Utility to do everything else. Here’s a noteworthy feature they share: they are both self-recommending in the sense that, if you were to ask each which decision theory you should use, it will recommend itself. I discuss this property in this paper, so I won’t go into all the gory technical details, but I’ll try to sketch out what I mean.

Take a decision theory, and add whatever ingredients it needs to select an option from any decision problem it faces: if it’s expected utility theory, it’ll need credences and utilities (as well as a tie-breaking function, which selects a single option if there’s more than one that maximizes expected utility); if it’s Maximin, it’ll just need utilities (and again a tie-breaking function); if it’s risk-weighted expected utility theory, it’ll need credences, utilities, and a risk function (and again a tie-breaking function). Now set up what I’ll call a meta-decision problem. In this, the (meta-)options are decision theories themselves, each equipped with whatever ingredients it needs to pick an option in each decision problem. And the (meta-)states of the world specify not only the state of the world, but also a decision problem you’ll face. Then we can easily define the utility of a meta-option at a meta-state of the world: the utility of using a decision theory (plus necessary ingredients) to face a decision problem at a particular state of the world is the utility, at that state of the world, of whichever option the decision theory (equipped with ingredients) would pick were it to face that decision problem. And we can say that a decision theory (with ingredients) is self-recommending if it takes itself (with the same ingredients) to be among the rationally permitted options in this meta-decision problem; and it is self-undermining otherwise.

Being self-recommending doesn’t tell in favour of a decision theory. After all, think of the decision theory that says every option is rationally permissible. Then it is surely self-recommending, but that says nothing in its favour. However, being self-undermining does seem to tell against a theory. As I note in the linked paper, expected utility theory and the Generalized Hurwicz Criterion are both self-recommending, while risk-weighted expected utility theory and linear-weighted utility theory are both self-undermining.

Now, this is not a knockdown argument against dividing up our attitudes to risk so that some are incorporated into the decision rule we use to pick our priors, while others are incorporated into the rule we use to pick our posteriors using our priors. After all, perhaps there is a self-recommending decision theory out there that is both sensitive to risk (in the way expected utility theory is not) and sensitive to probabilities (in the way that the Generalized Hurwicz Criterion is not); and if there is, perhaps this is the decision theory we should use, in combination with our prior credences, to determine how we should update. But it does suggest that it’s going to be difficult to find such a theory.

1. See Kenny Easwaran and Kevin Dorst on this approach to epistemic utility for beliefs.